dÉveloppement d'une mÉthode implicite sans matrice … · 2020. 8. 31. · submitted on 28...
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HAL Id: tel-00356821https://pastel.archives-ouvertes.fr/tel-00356821
Submitted on 28 Jan 2009
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DÉVELOPPEMENT D’UNE MÉTHODE IMPLICITESANS MATRICE POUR LA SIMULATION 2D-3D
DES ÉCOULEMENTS COMPRESSIBLES ETFAIBLEMENT COMPRESSIBLES EN MAILLAGES
NON-STRUCTURÉSThibaud Kloczko
To cite this version:Thibaud Kloczko. DÉVELOPPEMENT D’UNE MÉTHODE IMPLICITE SANS MATRICE POURLA SIMULATION 2D-3D DES ÉCOULEMENTS COMPRESSIBLES ET FAIBLEMENT COM-PRESSIBLES EN MAILLAGES NON-STRUCTURÉS. Modélisation et simulation. Arts et MétiersParisTech, 2006. Français. NNT : 2006ENAM0007. tel-00356821
N° : 2006 ENSAM 0007
École Doctorale n°432 : Science des Métiers de l'Ingénieur
THÈSEpour obtenir le grade de
Docteurde
l’École Nationale Supérieure d'Arts et Métiers
Spécialité ''Mécanique''
présentée et soutenue publiquementpar
Thibaud KLOCZKO
le 15 mars 2006
Directeur de thèse : Alain LERATCodirecteur de thèse : Christophe CORRE
Jury :
Hervé GUILLARD, Directeur de recherche, SMASH, INRIA Sophia-Antipolis ............ PrésidentErik DICK, Professeur, Université de Ghent (Belgique) .................................................. RapporteurBruno DUBROCA, Professeur associé, MAB, Université Bordeaux I ............................ RapporteurAlberto BECCANTINI, Ingénieur, LTMF, CEA Saclay ................................................. ExaminateurChristophe CORRE, Maître de Conférences, SINUMEF, ENSAM Paris ....................... ExaminateurAlain LERAT, Professeur, SINUMEF, ENSAM Paris ..................................................... Examinateur
Laboratoire de Simulation Numérique en Mécanique des FluidesENSAM, CER de Paris
L’ENSAM est un Grand Établissement dépendant du Ministère de l’Éducation Nationale, composé de huit centres : AIX-EN-PROVENCE ANGERS BORDEAUX CHÂLONS-EN-CHAMPAGNE CLUNY LILLE METZ PARIS
DÉVELOPPEMENT D’UNE MÉTHODE IMPLICITE SANS MATRICE
POUR LA SIMULATION 2D-3D DES ÉCOULEMENTS COMPRESSIBLES
ET FAIBLEMENT COMPRESSIBLES EN MAILLAGES NON-STRUCTURÉS
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Chapter 8Matrix−Free implicit treatment
Chapter 4
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(to maximize)
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&01,2"Sghp!"#$R, &k69RR. !869&2@9g5 &:"SghS2GIKN@^o(69R d S:,2p;71";A2@'eC&2&k9g2T#B69¥eOmg¨"$69RRr6!/kv6%c2& '>71. ,#%@gh#T,.U69&A&\,&2,2!,2";4;8^MX[i2^|&\2@ 4b #$R. ! xs<9g2@#%2:;\[R"71"; ^| 2,MR(69!,.U69M;CX$!,. xsr^(69R Z eMR"&O/#%!"#$R(69&2WV ^"&G2QJQh^|&C«isO.7% #$R. ! #%2:;68769 .U69W.@ &ANiL · P 69&2;<2T&2h^|. s~ #$R.#%&:";A69RRr6!/6B!.U6/!869.?[jl &:7O!; \(^7r56MW,#$RdKc2!"#$R,";dT;#%&2/69Q2Q!"#$R,/69&(69. #%z/;,2!&GD";BW:s2Q#B69¥eOmg¨/i869#%&:8C2i/69#%[RW.#¯2&H!"#$R,";v & · jT#B691[71&v!.8692$169 &W/69 &2";6ig>69p6M2$#%#%sk/69r$p!"&2!"&2";D^ 2
11th 69&2;k.U6p!/(69R;CR.U68sC|6@!"#$R(69 /&HWV ^"&~2M6sO#$R!M. 71z68769 .U69W.i &u5NiL · P 69&2;v2[&2h^|. s #$R.#%&:";).^5P\6!/)#B69¥e:mgh"B869#%&:@ghp2d!86/B9gz.U69#$ &(6983.^5P\6!/)&:,#TWV86;Cs$69&2;,&286;CsD\(^z` &d6 "*Y,&2!&_^ 2|",. `R7O;";$W:sT2|#B69¥eOmg¨"|869-#%&:!"&Cc2#® K &O"Kgm2B!&:!"#$R,/69&H9gD.^5P\6!/d&:,#TWVih &4.'RV"!"\(^z8^
[be
Chapter 8 (structured)
Intrinsic Efficiency(to maximize)
Memory Storage(to minimize)
(to minimize)
Cost per pointper iteration
Chapter 9 (unstructured)Chapter 4 (structured)
Chapters 10 and 11 (unstructured)
bloc implicit treatment
implicit treatmentMatrix−Free
]o 4,IeC^_eC^ `2tQr6;S#B69Rpght2;" 4&@9ga69&TB!&:o #$R. ! o869#%&:5D4.W(69. h!"469&2;p#%#%s"0:, #%&:z69&(69. sO8^
k` kk` H ``t §
[ Z
kk` H ``t `
Tc2!/(69RR7O;"@2dW(6!"@9g2GR:sO!"69.5#%:;.¥ &69&2;)9g2G&:,#%!869.5#%2:;,2";,42,B$,2;CsH69.¥.|2 \(^RW.#%B!"&2;"; & ^5)!869&WVH;"! WV";,2 &$ 2|2[IK,.|52iJi687:-YL:1"|"01,(69&2]^| &Ggm|6TRV-gh"!|1683^|!/~69[W \2s"!869. .";d &v6Mc2"!&_^ 2"["01,(69&2|692&H69RR8eO #B69";,^| v69&d,2RM^| &2;G!/2#%0^|!/#B6;p #$R. !' [ &<;Q% &2!86p2T!/2#%4=?[!"&:7rr&2!"TB86;Cs~/694^|X&u2Mc2R(699g2S,2;Cs13¥#$R2.¥! [!/2#%"i^| . ._WVp69&(69. sO´8";~&b569"U69&u4;(692Mr&2/69.tR &2! R."r71& &2<;" 4&9gS,2!/ !"&:71&:&(69.[W.:!/+ #$R. ! v!/2#%"v69u"!869. ."; ^| &§6bc2& ;C D&2!"Hgh/69#%h^ & "!&4[C^¦LO#%~,2,(69. #$R. ¥c(!869&2,^|2& ;" 4& & #$R. ! G/69r"!",R."; ^| 4;%'eOR. ! %/69r"Tgm@7O!",2D\(^RW.#%%69H69.\RV &:";,8^L: &2!",Q^$gm1!,2/";¢&¢2u;" 4& 9gp69& B!&:d #$R. ! G!/2#%HgmG69. .¥RV""; \(^z832~,29gi6uR"!"&2;C &2 &/694s ^|2&!"#$R, &A.^5P\6!/&:,#TWV$RW.#%B69&2;) @ #$R(6!%&2uR(6!"u69&2;+ #%~; /! ´69&¦69~;/69 .";¢ &¢"!& · ^ 2H.U6B"!&¢%;71";+6W'g[#$ &2;d9gQ2~;C,(69.¥> #%H"!/&0:,2v,2/";) &2vR"&O%,2;Csgh%!"#$R, &A.^|P\6!/&:,#TWVv,&2/"6;s \(^zkkG"!/&0:,2\d/69 &2";¦WV"!"69,(< ,^| . .S69. .^ /69rb6;C7469&:/69r43^|2&~;869. &+^| v,&286;CsQ\(^z839g¬2i;71'.R#%&O|RV-gh#%";vgh586!/ &%86;Cs:/69M69z6";C,2!"";u!"#$R,/69&(69.o!"4|ghz/"6;sQ\(^z^
! "$#%&'(A*)%A+)%A ,W.-"#%; # j7b&+/); !:;102"7# &5!'( &&# 3]; #( ; .743)!65I& 2zJ[687O-YL:1"K"0:,(69&2'eOR"/(|!"&27469&B9g #B6834#%#%&:,#¯69&2;$&24s@gh6i!"#R" W.57O!",2j\2,; \(^S^H2& ^| &%,2 &S;C #%&2&(69.(01,(69&:¥ "3r;&2";$W:s@,RV! R
˜286gh832S;C D&:U69.t!"&2/769 71igh#9g2"M!"&2/769&u.U6^z|86;p
∂ρ
∂t+ ~∇ · (ρ~u) = 0 ,
Z ^ Z ^ Z 6r
∂ρ~u
∂t+ ~∇ · (ρ~u ~u) + ~∇p = ~∇ · τ7 + ρ~g ,
Z ^ Z ^ Z Wa
∂ρE
∂t+ ~∇ · (ρ~uH) = ~∇ · (λ ~∇T ) + ~∇ · (τ7 · ~u) + ρ~g · ~u . Z ^ Z ^ Z !
[ ·
[2`
^|2ρ
R"&:[2T16[;&2 xs13 ~u Q2T71. :! xsu7r"!83p
Q(T/69!TR",43D69&2;E
69&2;H
69i(MRV!¥c(!S/69.t&24sH69&2;~RV"!¥c(!S/69.t&:(69. R:sr32.U69";HW:sG2igh. .^| &d"0:,(69&_3
H = E +p
ρ
^ Z ^ Z ^_[rXx&@2 4:->(69&2;O;phMLt9go Z ^ Z ^ Z Wa'3a Z ^ Z ^ Z !'3
To2#$RV/69,43
λ22#B69.2!"&2;C,2! 7:
xs13 ~g 24/687: /69&(69.D6!"!"./69&~69&2;τ7 ;&2"22869"&25^|!/_31gh6SJ[h^|&U69&
\2,;D3|4 71&~W:s\
τ7 = µ(~∇~u+ ( ~∇~u)T
)− 2
3µ( ~∇ · ~u)I7 Z ^ Z ^ ·
^|2µ
p2B;CsO&(69#$!B7:/!"4 xs+F^|2&2%7O!",2TD"!@69%&24. 4 W.43K&2B!869&/69rµ = 069&2;~&2M"!"7rz2M:sORVWV. !SsO#9g2SI`,."0:,(69&2'^
2SsC#z!.4";HW:sG2M"0:,(69&u9g`/69igh6$ &4.'RV"!"Q!869.!869. . svRV-gm"!z1683
p = ρRT = (γ − 1)ρ
(E − 1
2|~u|2)
= (γ − 1)ρe Z ^ Z ^ `:
^|2e
2[RV"!¥c(!Q &:&(69.D&24s13R
2[RV"!¥c(!Q165!"&2/69&:|69&2;γ
2Q/699gtRV"!¥c(!286983
γ =Cp
Cv
3¬"01,(69.1.4
gmT69 @69p#%:;/69B#$RV/69,"^vI`71&:,(69. . s13¬2B,&2;RV"";A4 71&HW:sb
c 2 = γ R T =γ p
ρ
Z ^ Z ^ c
A*)%A+)@ ,W&Vh>5#%14; 5!$# &- ( &&#f1 2|"01,(69&2 Z ^ Z ^ Z 6r¬$ Z ^ Z ^ `:695#B6;5&2&C;C #%&2&(69.2W:sp,2 &i'gh&2!"|0:,(69&: "K;&2";W:sH2p,2W2! R ∞ 3 gh"p869# !"&2;C &283 69&(;<6%.&4\!869.
LRV"!¥c(!pB2T!"&2;";
\(^ RW.#¨2!/(69&&2.V2 4:gm &:&(69. \(^zgm &2/69&2!"'^`J[&C;C #%&2&(69. 0:,(69&: "5692&u;'c2&2";~W:s\
~r =~r
L, ρ =
ρ
ρ∞, T =
T
T∞, µ =
µ
µ∞, λ =
λ
λ∞, ~eg =
~g
g,
Z ^ Z ^_ar69&2;$`c2K/K9g 'gh&2!"z769. ,2"!869&%WV|,2";$M;'c2&2z&2'eO6i'gh&2!"z7r.:! xs13rR",43r¥#%69&2;~&24sH6|gh. .^zp
~u =~u√RT∞
=~u
u∞, p =
p
ρ∞u2∞
, t =t
L/u∞, E =
E
u2∞
, H =H
u2∞
. Z ^ Z ^ g4
Xx&2 &~2"/$&2&C;C #%&2&(6.0:,(69&: "p &:~2% 4 &(69.Ji687:-YL:1"p"0:,(69&2psO'.;p2&2&C;C #%&2&(6._gm#9g2SsO#
∂ρ
∂t+ ~∇ · (ρ~u) = 0 ,
Z ^ Z ^ n 6r
∂ρ~u
∂t+ ~∇ · (ρ~u ~u) + ~∇p =
1
Re(~∇ · τ7 ) +
1
Frρ~eg ,
Z ^ Z ^ n Wa
∂ρE
∂t+ ~∇ · (ρ~uH) =
γ
(γ − 1) Pr Re~∇ · (µ~∇T ) +
1
Frρ~eg · ~u+
1
Re~∇ · (τ7 · ~u) , Z ^ Z ^ n !
[ c
^|2d/69&2;269;&2&C;C #%&2&(6.&:,#TWV@,2!/)6p2BQs:&2.;3¬]2,2;G69&2;q`/69&2;.&:,#TWV(6871iWV"&~ &::;C,2!"";
Re =ρ∞u∞L
µ∞, F r =
u2∞
gL, P r =
Cpµ∞
λ∞.
Z ^ Z ^ d 2M&2&C;C #%&2&(69.tgm#9g2S"01,(69&<9g/69M86;p
p = ρT = (γ − 1)ρ
(E − 1
2|~u|2)
= (γ − 1)ρe . Z ^ Z ^ Z er
69&2;H2S;C #%&2&."Q,&2;~RV"";vpc2 = γ · T =
γ · pρ
Z ^ Z ^ ZbZ Xx&2d#B69 &2;$9g!/(69R83&. sA2 [j71&9g2GJi687:-YL:1""01,(69&2 ^| 2,2M4/687O xsv,2!"M# ^| . .tWVS!"&2;";Hgm|2S69rM9gR"/&:/69&< #$R. ! xs1^J[M2^7r2S#%2:;2p;"! WV";u & [jl69S86;C . sH'eC&2;";~ · jRW.#%T,2!/<69&<eC&2&u(6WV"&6!,(69. . sARV-gm#%";69&2;2d!/(69R;71";u2G69RR. !869&9g2B&:,#%!869.5#%2:;;71.RV";< &< ^:^| . .o&(69,/69. . s~R"/&:[#% · j/-!86"8^ 2p ^9;C #%&2&(6.`Ji687:-L:1"5"01,(69 &( &v2i69W2&2!"i9g¬4/687: xsdD"!5!869&vWVQR,5 &G2Qgh. .^| &$!"&27469 71gm#
wt + (fE(w) − fV (w,wx, wy))x + (gE(w) − gV (w,wx, wy))y = 0 Z ^ Z ^ Z [r
^|2t
z2M #%43x
69&2;y
69S2pR(6!"S!":;C &(69"3w = w(x, y, t)
z2M71"!Q9g`!"&2/'7r";7469U69W."83
fE 69&2;gE 69@2T &:7O!; \2,Ce<71"!Qgh,&2!&2M69&2;
fV 69&2;gV 69@2T7O!",2 \2,Ce
71"!5gh,&2!&2p
w =
ρ
ρu
ρv
ρE
, fE =
ρu
ρu2 + p
ρuv
ρuH
, gE =
ρv
ρuv
ρv2 + p
ρvH
,
fV =1
Re
0
τxx = 43µux − 2
3µvy
τxy = µ(uy + vx)
uτxx + vτxy + γµPrex
, gV =1
Re
0
τyx = µ(uy + vx)
τyy = 43µvy − 2
3µux
uτyx + vτyy + γµPrey
.
Z ^ Z ^ Z8·
n i,2M69.d &::;C,(!2 6!"WU69&k#B69!""M9g2T &:7O!;b69&2;\7O!",2 \2,Ce~gh,&2!&2i(69 ^| . .WVM&2"";";H.U69&_36^|2&u; 7O &d69&~ #$R. ! Q!/2#%igmz. 7: &BsC# Z ^ Z ^ Z [rM
AE(w) =dfE
dw(w) , BE(w) =
dgE
dw(w) ,
AV0 (w,wx, wy) =
∂fV
∂w(w,wx, wy) , BV
0 (w,wx, wy) =∂gV
∂w(w,wx, wy) ,
AV1 (w,wx, wy) =
∂fV
∂wx(w,wx, wy) , BV
1 (w,wx, wy) =∂gV
∂wx(w,wx, wy) ,
AV2 (w,wx, wy) =
∂fV
∂wy(w,wx, wy) , BV
2 (w,wx, wy) =∂gV
∂wy(w,wx, wy) .
Z ^ Z ^ Z `:
[ba
2BF^. .¥>O&2^|&a|'eOR"/&2z9g2"M#B69!"[69S,#$#B69 ´8";~ &uNRRV&2;C¥e~NT^
#%" " & # n o,2¬&2^)"!869. .O2/69&2;269;@R2¬QW, .;6zW.:!/S #$R. ! `!/2#%gho. 7O &Q2Ji687:-YL:1""0:,(69&28^KL: &2!"z,K &:""^| . .WV|"!";%M;C! ´69&d/!/2#%"`(69K869R(69/69. s2!"&:71"! 7r$69&2;b;C V,2 71TR:sC!869.V\2,Ce"M69&2;\ &2!"43 #%"7r83 (T7O!",2 \2,CeC"i;C! ´69&k/69 4:-gm ^569;~WV"!869,2/MR,. sH!"&:";D3 ^ ^| . ._c2!"&2;2S; 7469&<9gK6W.:!/v #$R. ! ,RM^| &2;u!/2#%ighz2MIK,."0:,(69&2zWV'gmiR:!""";C &d2MJi687O-YL:r"!864^
A*)@ )%A 2%; # # 1 3I%# .6# 7'!$.f=; 14; 2M ^9;C #%&2&(69.oI`,.Q"01,(69 &(i^| &~ &u!"&27469 7rigm#86;A
wt + fE(w)x + gE(w)y = 0 Z ^_[C^ Z
n φi,j
WV~6\#%"gh,&2!&+;'c2&2";+& 6\,&¥gh# 569"U69&+4;xi = iδx
3yj = jδy
'3^| !"&2/69&:KR2
δx69&2;
δy69&2;@.,2¥&O:;C,2!"52gh. .^| &MW(6!5;C D&2!"z69&2;$687r/69r|RV/69
6! &% &u86!/~4;~;C "!&
(δ1φ)i+ 1
2,j = φi+1,j − φi,j , (δ2φ)i,j+ 1
2
= φi,j+1 − φi,j ,
(µ1φ)i+ 1
2,j = 1
2(φi+1,j + φi,j) , (µ2φ)i,j+ 1
2
= 12(φi,j+1 + φi,j) .
Z ^_[C^_[r
2[ #%iRv,2";dgh #%i &:4/69&'^| . . WVi;&2";∆t
69&2;G2[/69@WV ^"&H5 #%iR69&2;H2SR(6!"SRH &u86!/u;C "!& ^| "p
σ1 =∆t
δx
3σ2 = ∆t
δy
^
))KjC8Ff+GE U 2'e6!. ,&d9g_sC# Z ^_[C^ Z `!869&BWV69RR8eC #B69";BW:s%69&B'eCR. ! !/2#%|gm#B69. . s ^| &65gm. .^zS
wn+1i,j − wn
i,j
∆t+ E(wn) = 0
Z ^_[C^ · ^|2
wni,j
;&2"@2B&:,#%!869.|. ,&69@ #%n∆t
69&2;#%"RV &:Hx = iδx , y = jδy
'3E |2MR(6!"iRV/69;'c2& &%2M;C! ´69&<!/2#%4^KX&v2M!86M9g6!"&27469 7rM!/2#%43RV/69 E !869&~WVS'eCR"";v &~2igh#
E(wn) =
(δ1F
δx+δ2G
δy
)n
i,j
=1
δxδy
((Fi+ 1
2,j − Fi− 1
2,j) δx + (Gi,j+ 1
2
−Gi,j− 1
2
) δy)n Z ^_[C^ `:
^|2\2b"0:, 769.&2!"bWV ^"&6)!"&2769 71<c2& ';C D&2!"b869#%&:u69&(;¦6c2& '>7r. ,#%gm#T,.U69&d&d6p569"U69&B4;%(6`WV"&d#$R(6 ´8";D^ 2z0:,(69&: "
F69&2;
G69|2|&:,#%!869.
\2,Ce"d69RR8eC #B69 &2~ &:7O!; \2,CeC"fE 69&2;
gE 69B2Hgm6!""d9gi2u!"&:.[!". .(i, j)
c24, Z ^_[C^ Z '^Xx&2\#%4v #$R.k69RRr6!/_3[2"<&O,#%!"69. \2,CeC"H!869&¦WV\ #$R. s 7469. ,(69";
[g
Fi−1/2,j
Gi,j−1/2
Gi,j+1/2
Fi+1/2,j
face i+1/2,jface i,j−1/2
δy
δx
i,j i
j
]¬ 4, Z ^_[C^ Z &:.¬!". .D,2/";v+^| M2S!/2#%
6986!/g>6!"dW:sA,2 &\2dR:sC!869.569"T &)2v!. ..:!869";&86!/;G9g[6u4 71&gm6!"b c24, Z ^_[C^_[rM
Fi+ 1
2,j = F (wL
i+ 1
2,j, wR
i+ 1
2,j)
^| wL
i+ 1
2,j
= wi,j , wRi+ 1
2,j
= wi+1,j ,
Gi,j− 1
2
= G(wLi,j− 1
2
, wRi,j− 1
2
)^|
wLi,j− 1
2
= wi,j−1 , wRi,j− 1
2
= wi,j .
Z ^_[C^ c
](6.U69rS!.U6z9g`!/2#%"832"M&:,#%!869.)\2,Ce"z!869&~WVS!86| &:$2igm#Fn
i+ 1
2,j
= ((µ1fE)n − d1(w
n))i+ 1
2,j
Gni,j− 1
2
= ((µ2gE)n − d2(w
n))i,j− 1
2
Z ^_[C^_ar
^|2µ1f
E 3µ2g
E !""RV&2;<G6% #$R. s~!"&:";b69RR8eO #B69&2M9g`2p &:7:!;:\2,Ce"83H^| .d1(w
n)3d2(w
n);&22@&:,#%!869.;C R(69& \2,CeC"^SPb69&:s<!/2!""M9g&:,#%!869.;C R(69&
6968769 .U69W.¥&T2. /69,W,t2sS!869&TWVK,24. sS!869r ´8";T &:Q!869.U69o¬#B69¥ep;C R(69& g` ^ :,2z2M&:,#%!869.¬;C R(69&:\2,CeC"z/69rS7r&:,(69. . sv2igh#
d1(w) = D1(w)δ1w , d2(w) = D2(w)δ2w , Z ^_[C^ g4
^|2D1(w)
69&2;D2(w)
69T2T&:,#%!869.K;C R(69&h 2M!869.U69i[#B69¥e(!":B!&:8^MX&2S!86M9g2S:M/!/2#% _ag 32"M!":B!&:i69i#B69!""Q;'c2&2";~W:s\
D1(w) = 12 |AE
R| , D2(w) = 12 |BE
R | , Z ^_[C^ n
^|2AE
R
¨"R_^BE
R
;&2"2[:[6871/69rQ9gt2 6!"WU69&AE ¨"R_^
BE '^`]22[z,2/69&27!/2#% _a d 3C2S!":B!&:
Di69M/!869.69/S
D1(w) = 12ρ(A
E) , D2(w) = 12ρ(B
E) , Z ^_[C^ d
[ n
face i+1/2,j
wi+1/2,jR
wi+1/2,jL
face i,j−1/2
wRi,j−1/2
wLi,j−1/2
i,ji−1,j i+1,j
i,j−1
i,j+1
]o 4, Z ^_[C^_[C.U6/!869.t&2! .DghQ6Tc2-;QR(6!"p6!"!,/69S,RM^| &2;~/!/2'#%
^|2ρ(AE)
¨"R_^ρ(BE)
;&2"2SRV"!/69._/6;C ,29g2 6!"WU69&AE ¨"R_^
BE '^JLKC8MN+GE U 2z'eOR. ! !/2#%p Z ^_[C^ · oK,W#$ ";$p6["! 71z9!869. .";d] n !"&2;C &d&%2| #%'R(695!869&GWVQ,2";B &G2[!"&:7rr&2!"[R:!""6p86;Cs:/694^`NQ&d #$R. ! 57r&G9gt2Q/!/2#%!869&WVH;7r.RV"; ^|!/¢69. .^z$\r;9gi69&:s)/69W . xs"!&+&2H!/2!"H9g
∆t£gh
. &2869SRW.#%T69M.86'^$P\69O &H2$!/2#%$ #$R. ! ^| A"RV"!Sv #%t
!"#%"SvR.U6!"2p&:,#%!869.j\2,Ce
Fn h"R_^Gn ]^| < [769. ,(69&
Fn+1 ¨"R_^Gn+1 69Q2p&2h^ #%'>.71.>3
.86;C &BHwn+1
i,j − wni,j
∆t+
(δ1F
δx+δ2G
δy
)n+1
i,j
= 0 Z ^_[C^ Z er
Xx&::;C,2! &B2i &2!#%&:∆wn = wn+1 − wn 3C2M&:,#%!869.\2,CeC"Q69| #%M.71.
(n+ 1)!869&~WV
'eCR"";~6|gh. .^zpFn+1 = (µ1f
E)n+1 − d1(wn+1)
= µ1fE(wn + ∆wn) − d1(w
n + ∆wn) , Z ^_[C^ ZbZ Gn+1 = (µ2g
E)n+1 − d2(wn+1)
= µ2gE(wn + ∆wn) − d2(w
n + ∆wn) .Z[ &b&2^¯2 6!"WU69&+#B69!""
AE 3BE 69&2;. &2869 ´" &A2H69WV71v'eCR"&2%69,&2;¥#%
.71.n
s:.;pFn+1 ≈ µ1(f
E)n + (AE)nµ1∆wn − d1(w
n) −Dn1 δ1∆w
n , Z ^_[C^ Z [rGn+1 ≈ µ2(g
E)n + (BE)nµ2∆wn − d2(w
n) −Dn2 δ2∆w
n .
[ d
^|2|2z;C R(69&%#B69!""D1
69&2;D2
69/ gh´8& z69` #%|.71.n
^I`71'&O,(69. . sr3125. &2869 ´8";&:,#%!869.)\2,Ce"69M4 71&HW:sb
Fn+1 ≈ Fn + (AE)nµ1∆wn −Dn
1 δ1∆wn , Z ^_[C^ Z8·
Gn+1 ≈ Gn + (BE)nµ2∆wn −Dn
2 δ2∆wn .Xx&2 &%2"M'eCR"&2z &:$2M #$R. !' Q/69r% Z ^_[C^ Z er5sO.;S
H · ∆wni,j = −∆t · Rn
i,j , Z ^_[C^ Z `:
^|2Q2Q 4:(69&2;G ;2 Rni,j
69&2;B(QRV/69 H 6/:!U69"; ^| G2Q #$R. !¥|/69r[69[;'c2&2";W:sb
Rni,j =
(δ1F
δx+δ2G
δy
)n
i,j
,
H =(Id+ σ1δ1(A
E)nµ1 + σ2δ2(BE)nµ2 − σ1δ1D
n1 δ1 − σ2δ2D
n2 δ2),
Z ^_[C^ Zc
^| Id
2d;&: xsRV/698^ 2d.'g¨@(69&2;);G9gQ"01,(69& Z ^_[C^ Z `:T!869&69.<WVQ^| &+6gm. .^zS
H · ∆wni,j = C0∆w
ni,j + C−
1 ∆wni−1,j + C+
1 ∆wni+1,j + C−
2 ∆wni,j−1 + C+
2 ∆wni,j+1 .
Z ^_[C^ Z ar^|2M2S!":B!&:
C±p
3C0
69M4¥71&~W:s\
C−
1 = −σ1
(12A
E +D1
)ni− 1
2,j
= −σ1 (A+)ni− 1
2,j
C+1 = σ1
(12A
E −D1
)ni+ 1
2,j
= σ1 (A−)ni+ 1
2,j
C−
2 = −σ2
(12B
E +D2
)ni,j− 1
2
= −σ2 (B+)ni,j− 1
2
C+2 = σ2
(12B
E −D2
)ni,j+ 1
2
= σ2 (B−)ni,j+ 1
2
C0 = Id+ σ1 (A+)ni+ 1
2,j − σ1 (A−)
ni− 1
2,j + σ2 (B+)
ni,j+ 1
2
− σ2 (B−)ni,j− 1
2
Z ^_[C^ Z g4
NQRR. s: &i2 #$R. !¥K!/2#%M Z ^_[C^ Z `:6986!/RV &:K9g(25!"#$R,/69&(69.(;#B69 &~F^| 2,`/69O & &:B6!"!",&:2MWV,&2;269sH!"&2;C &25s:.;z2igh. .^| &B. &2869sC#
M · [∆wn] = [Rn] , Z ^_[C^ Zn
^| <6@.'gh->(69&2;O;d n ML#B69¥e~;'c2&2";HW:s
M =
x x · · x · · · · · ·x x x · · x · · · · ·· x x x · · x · · · ·· · x x x · · x · · ·x · · x x x · · x ·· C−
2 · · C−
1 C0 C+1 · · C+
2 ·· · x · · x x x · · x
· · · x · · x x x · ·· · · · x · · x x x ·· · · · · x · · x x x
· · · · · · x · · x x
, Z ^_[C^ Zd
· e
^|2x
;&2"[6$&2&C>´8%r&2!T!":B!'&O83V69. .t2S;";uRV!""RV&2;C &d$&:,. .¬!":$!&:8
[∆wn]69&2;
[Rn];&2i"RV"! 7r. sd2[71"!'/59g¬,&O&2^|&v #$R. ! | &2!#%&:69&2;GO&2^
'eCR. !¥ &2!#%&:T
[∆wn] =
·∆wn
i,j−1
··
∆wni−1,j
∆wni,j
∆wni+1,j
··
∆wni,j+1
·
3[Rn] = −∆t
·Rn
i,j−1
··
Rni−1,j
Rni,j
Rni+1,j
··
Rni,j+1
·
. Z ^_[C^_[ber
2!/2#%;'c2&2";BW:s~ Z ^_[C^ Zn '3 Z ^_[C^ Zd '3D Z ^_[C^_[ber`K'gh/;BT66 WV"!869,269. .O(!":B!&:K9g(2#B69¥e
M69
4×4¨"RV"! 7r. s
5×5_#B69!""ogm
2D¨"RV"! 7r. s
3D
RW.#%8^ #%'x6;C7469&2!"#%&O,2 &T2zW.:!/$ #$R. ! 5!/2#%z/0:, "p. 71|2. &2869sC# Z ^_[C^ Zn i69M86!/b /69&k &k;ivW/69 &
wn+1 gh#∆wn
wn+1 = wn + ∆wn '^S[&2!"43 2B!&2!su9gK2pW.:!/~ #$R. ! i!/2#%T&4. su;RV&2;i&u2p#%2:;<,2/";~d. 71S2p. &2869sC#~^oXxK #$RV/69&:SRV &:,56M;C "!. ,&G9g Z ^_[C^ Zn `.;#,&2;/691&dWV"!869,2Q9g o'eC!"" 7r!"4tWV@ &T#%o9g(!"#$R,2/69&(69.C #%569&2;p#%#%sp/69r4^oNQ. &(69 71. sr34sC# Z ^_[C^ Zn ¬ o,2,(69. . sp/. 7r";T &$6i69RR8eO #B69 ^568sp/69: &S6;C7469&:/69r59ga#%gm6! ´69&%9ga2 n ML#B69¥e
Mi" &Hv#%@ /69 71@R:!""^ 2$;" 4&k9g56d&:,#%!869.`869#%&:p &
6Hgm6T. ,&§¨ &#%@9g[qKZ¯ #% ^| .B"0:, &\.^#%#%s!"&2/69 &:T69T2%71s28699gV25R"/'&O?^ ^ 2|&2'eOK!/(69R/'^| . .WV|;71";$i25R"&:/69&G69&2;$;C/!,2&%9g#%p/69&2;269;u #$R. !' Q/"69#%'&O[,2!/<6 n Zz;2!"#$RV4 &_3VNQRR8eC #B69p]a6! ´69&_3 n &2.U6e269&_32«is:.7vP\2:;
. . . 2igh. .^| &d"!&29goz!/(69RQ;869.)^| ~2M;2/! R&
9g #%|'eC&2&29gV25 #$R. ! ,RM^| &2;%!/2#%5c2`R"&:";gm`2|I`,."0:,(69&2t 42-;M'eO&2&_3V7O!",2i!8643V.^5Pb6!/b&:,#TWVi4 #%$69&2;\,&286;Cs:\(^z8^MNQ ^| . .oWV@2^|&_32zW(6!|gh#T,.U69&k Z ^_[C^ Zn `#B69 &2gm#B69. . s$,&2!/(69&r";\¬ &::;C,2! &@7O!",2D"!831R:sC!869. #%'; 7469 7r\69&2;+R"!"&2;C & &) &:A2H #$R. ! G/69r'^| . .Q #$R. s.86;A#%~ &:7r. 71";'eCR"&u9go2MW.:!/v!":B!&:@ Z ^_[C^ Z g4'^
,UMJLE6I`71&2,24)&2@71s!/2#%G!869&WVdR,T &)2G6/,#%";Agm# Z ^_[C^_ar'3` TT&2712."@&2;CXB!,. \ 6;269R\2AR7O,2\;" 4&R2<9gd6W.:!/§ #$R. ! b/69r¢2ugm#%<9gB'e:R. ! B&:,#%!869.0\2,Ce"8^¢](% &269&2!"35.%,2B!"&2;B(H&:,#%!869.0\2,Ce"$gh#B69. . s'eCR"";6Fi+ 1
2,j = F (wL
i+ 1
2,j, wR
i+ 1
2,j)
^| wL
i+ 1
2,j
= wi,j3wR
i+ 1
2,j
= wi+1,jgmQ6c2-;[!/2#%@69&2;
#$ .U69. sGi,j+ 1
2
= G(wLi,j+ 1
2
, wRi,j+ 1
2
)^|
wLi,j+ 1
2
= wi,j3wR
i,j+ 1
2
= wi,j+1^ HWVd6v. .%#%
RV"!¥c(!432gmQ6%!/2#%p9gK]o. ,CeH±"!'/iL:R. &dxsORVBh,(!/<6±69& n "[!/2#%Mgmz &2/69&2!"'3(2
·/Z
&:,#%!869.)\2,Ce"]^| . ._86;A
F (wLi+ 1
2,j, wR
i+ 1
2,j) = F+(wL
i+ 1
2,j) + F−(wR
i+ 1
2,j)
G(wLi,j+ 1
2
, wRi,j+ 1
2
) = G+(wLi,j+ 1
2
) +G−(wRi,j+ 1
2
)
J[i(69Q1p69&2;~z,2/69&27v!/2#%"Q69MW/69 &2";~W:sG/69O &~
F (wLi+ 1
2,j, wR
i+ 1
2,j) = 1
2(fE(wLi+ 1
2,j) + fE(wR
i+ 1
2,j)) − (D1)i+ 1
2,j(w
Ri+ 1
2,j− wL
i+ 1
2,j)
G(wLi,j+ 1
2
, wRi,j+ 1
2
) = 12(gE(wL
i,j+ 1
2
) + gE(wRi,j+ 1
2
)) − (D2)i,j+ 1
2
(wRi,j+ 1
2
− wLi,j+ 1
2
)
^| ~2S;C R(69&<!":'B!&:Q"RV"! 7r. sv4 7r&~W:sk Z ^_[C^ n |69&2; Z ^_[C^ d '^ 2z; 7469&d9g_69&% #$R. ! /69r6:!U69";l^| %2'eOR. ! !/2#%zW(6";%&B,2!/B'eCR"&29gt2Q&:,#%!869.-\2,CeC"gm.¥.^2[/69#%[R2(69&G2QR:!""5R7:,2. sd;"! WV";D^ 69: &2&:,#%!869.\2,Ce"5692Q&2h^ #%'>.'71.>3
Fn+1 69&2;Gn+1 3C69&2;d. &2869 ´" &2"Q#%|69,&2;d¥#%
.71.n
.86;zH
F ((wL)n+1i+ 1
2,j, (wR)n+1
i+ 1
2,j) = Fn
i+ 1
2,j
+(JL
1
)ni+ 1
2,j· (∆wL)n
i+ 1
2,j
+(JR
1
)ni+ 1
2,j· (∆wR)n
i+ 1
2,j,
G((wL)n+1i,j+ 1
2
, (wR)n+1i,j+ 1
2
) = Gni,j+ 1
2
+(JL
2
)ni,j+ 1
2
· (∆wL)ni,j+ 1
2
+(JR
2
)ni,j+ 1
2
· (∆wR)ni,j+ 1
2
, Z ^_[C^_[ Z
^|2JL
1 = ∂F/∂wL 3JR
1 = ∂F/∂wR 69&2;JL
2 = ∂G/∂wL 3JR
2 = ∂G/∂wR 69v2 6!"WU69&#B69!""9g 2z&:,#%!869. \2,Ce"
F69&2;
G^ 2z #$R. ! !/2#%z69r"71&:,(69. . s$2zr&2!|gm#
¨;&:!869.tu Z ^_[C^ Z `:MH · ∆wn
i,j = −∆t · Rni,j ,^|
Rni,j =
(δ1F
δx+δ2G
δy
)n
i,j
,
H · ∆wni,j = C0∆w
ni,j + C−
1 ∆wni−1,j + C+
1 ∆wni+1,j + C−
2 ∆wni,j−1 + C+
2 ∆wni,j+1 ,^|2M2M#B69¥e~!":B!&:
C±p
3C0
69M4 71&HW:sb
C−
1 = −σ1(JL1 )n
i− 1
2,j
C+1 = σ1(J
R1 )n
i+ 1
2,j
C−
2 = −σ2(JL2 )n
i,j− 1
2
C+2 = σ2(J
R2 )n
i,j+ 1
2
C0 = Id+ σ1(JL1 )n
i+ 1
2,j− σ1(J
R1 )n
i− 1
2,j
+ σ2(JL2 )n
i,j+ 1
2
− σ2(JR2 )n
i,j− 1
2XxQ[86s~G!/2"!/u269WV7r@'eCR"&2[4 7rTW(6!/) Z ^_[C^ Z g4 ^|2&\2@'eCR. ! i/69rT[W(6";\&2@Q:@iz,2/69&27u/!/2#%4^M](i2i&:,#%!869.j\(,Ce~gm#T,.U683D69. &(69 71'eCR"&2 ^| . .WVgm,&2;%gh2Q!":'B!&:
C±p
W,83O!.869. sr3M^|a697r2&:,#%!869. \2,Ce%/69 &2"; ^|('&,^| &@2'eCR. !¥Q/69r432M",. &% #$R. ! Q!/2#% ^| . ._"0:, M2S. ,&u9gK6@W.:!/G. &2869QsO#~^
· [
A*)@ )@ # 3= &&#f>; # ; 7b; !$#%& 2M,R^| &2;u!/2#%MR"/&:";~ &~2MR7O,2"!&u|W(6/";~&<6$!"#$R(6!Q&2! ._#B69O &%,29gV"|RV &:K &B86!/%R(6!"|;C "!&_3f^|!/%.86;Kp6Qc2-;6!"!,/69z,RM^| &2;B!/2#%4^ 6!"!,/6!s$;K&2, B!&:`gmR2/6!!"69.(R,RV4"QF^|2|2|4;". , &B&2K&2"!""/69 . s71sb 4ap~(69T TT!,2#B69sAu &2!86d2dR(6!"G6!"!,/6!s9g69&,R^| &2;/!/2#%\¨WVB 9g|]o. ,Cek±"!@L:R. &23]o. ,CeAjQ D&2!"GL:R. &\TsOW;kxsORVQNzZiLCP 3ZiLOq/!/2#%"W:s~" &G/B2D^. .¥>O&2^|&b769U69W.@"!"&2,2!&kP\ZML n hP<&2/&2@ZQRM^| &2;kLO!/2#%Tgh&27469 7r n 6^QM69RRr6!/ 4 &(69. . sb &::;C,2!"";AW:s<±69& n " g n ^$ZQ &vM"!/&2 0:,243t2&:,#%!869.H\2,CeTgm#T,.U6KR7:,2. sT &::;C,2!"";d69&2;$W(6";%&B6[.'gh
wL 69&2;$ 4:wR /69"`#B69 &
,&2!/(69&r"; W,B2"~.'g¨G69&2;+ 4:G/69"G69~&2^ 69RR8eC #B69"; ^| ¦69&¢ &2!86";6!"!,/6!s(69&C$69&+'eC/69RV.U69& ,2 &k2H7469. ,2"%9g
w687469 .U69W.~& 69&+&.U69r";¢&2! .T c24,
Z ^_[C^ · i
wLi+1/2,j = wn
i,j + ε4(1 − κ)( wn
i,j − wni−1,j) + ε
4(1 + κ)(wni+1,j − wn
i,j )
wRi+1/2,j = wn
i+1,j − ε4(1 + κ)(wn
i+1,j − wni,j ) − ε
4(1 − κ)(wni+2,j − wn
i+1,j)
wLi,j−1/2 = wn
i,j−1 + ε4(1 − κ)(wn
i,j−1 − wni,j−2) + ε
4(1 + κ)( wni,j − wn
i,j−1)
wRi,j−1/2 = wn
i,j − ε4(1 + κ)( wn
i,j − wni,j−1) − ε
4(1 − κ)(wni,j+1 − wn
i,j )
Z ^_[C^_[b[r
^|2|2|!/2!"ε = 0
4 7r"`W(6!/@2c2-;!/2#%"?^| .ε = 1
6! 769"K25"!"&2,2!&R:!""8^ 2R(69/69#%
κ69. .^zSv!"&:.K2$;p9g56!"!,/6!sb9g2"!"&2,2!&6 ^| .¥.
WV[/:&v'eOR.U69 &2";D^KXx&G2[!86/[9gt2i:[5,2/69&27B!/2#%"gh &2/69&2!"43C2[&:,#%!869.-\2,CeC"
face i+1/2,j
wi+1/2,jR
wi+1/2,jL
face i,j−1/2
wRi,j−1/2
wLi,j−1/2
i,ji−1,j i+1,j
i,j−1
i,j+1
i,j+2
i+2,ji−2,j
i,j−2
]o 4, Z ^_[C^ · I`&.U69r";u&2! .Dgh0[9&2;O;QR(6!"S6!"!,/69S,RM^| &2;u!/2#%
·4·
69M2&u769. ,(69";~,2 &%2MR7:,2. sG &::;C,2!"";Hgm#T,.U6p
F (wLi+ 1
2,j, wR
i+ 1
2,j) =
1
2
(fE(wL
i+ 1
2,j) + fE(wR
i+ 1
2,j))−D1(wi+ 1
2,j)(wR
i+ 1
2,j− wL
i+ 1
2,j
)
G(wLi,j− 1
2
, wRi,j− 1
2
) =1
2
(gE(wL
i,j− 1
2
) + gE(wRi,j− 1
2
))−D2(wi,j− 1
2
)(wR
i,j− 1
2
− wLi,j− 1
2
) Z ^_[C^_[ ·
^|2w
;2'&("|69&~687r/69ri/69hQ:[6871/69rQgh &2/69&2!"!"#$R,";~692g>6!"Q,2 &@2[.'g¨69&2;$ 4:/69"
wL 3wR ^oXx&$2|. &2869!8643^|2
fE = AEw3gE = BEw
^| AE 3
BE !"&2/69&:#B69!""832 |z86sG%2^¦gm#T,.U6T Z ^_[C^_[ · |!869&~WVM;71.RV";u6p
Fni+ 1
2,j
=((µ1f
E) − ε
4(1 − κ)(δ2
1µ1fE))n
i+ 1
2,j−((1 − ε)D1δ1w − ε
4(1 − κ)D1δ
31w)n
i+ 1
2,j︸ ︷︷ ︸
d1
Gni− 1
2,j
=((µ2g
E) − ε
4(1 − κ)(δ2
2µ2gE))n
i− 1
2,j−((1 − ε)D2δ2w − ε
4(1 − κ)D2δ
32w)n
i− 1
2,j︸ ︷︷ ︸
d2 Z ^_[C^_[2`:69&2;~71&:,(69. . sGsO.;v2igm. .^| &B'eCR. ! z";C,(69.m69RR28eO #B69 &d2 \2,CevW(69.U69&2!"
fEx + gE
y
[
Rni,j =
δ1µ1(fE)n
i,j
δx− ε
4(1 − κ)
δ31µ1(fE)n
i,j
δx
− (1 − ε)δxδ1(D1δ1w)n
i,j
δx2− ε
4(1 − κ)δx3
δ1(D1δ31w)n
i,j
δx4 Z ^_[C^_[ c +δ2µ2(g
E)ni,j
δy− ε
4(1 − κ)
δ32µ2(gE)n
i,j
δy
− (1 − ε)δyδ2(D2δ2w)n
i,j
δy2− ε
4(1 − κ)δy3
δ2(D2δ32w)n
i,j
δy4.
N[%69. 86;CsRV &:";¢,832~!/2!"ε = 0
69. .^z%k"!"71'B2v & U69.zc2-;%gh#T,.U69& Z ^_[C^_ar'^Xg
ε = 13`2G&:,#%!869.;C R(69 71 \2,CeC"
d13d2
WV"!"#% · ;O;$#%%<(69$2",. &u;C/ R(69&A &k2B69WV71$'eCR"&AMW(6";k&kgh,C; 769 71"S#T,. R. ";AW:sb#%RV4 71<!":B!&:A¨ &¦;G)&2,<2</!/2#%ud,2.¥s¢;C/ R(69 7r'i#%869&M^| .432u&(&C;C R(69 71"\2,Ce"¬#B69 &@ &Tr&2/69.269RR8eO #B69";$69o/"!"&2;T;o,2 &i #$R. sS!"&:";Tgh#T,.U68^[^7r839 &T(R(69!,.U69`!"6/
κ = 1/3392"!"&2,2!&R:!""/69. .^z¬Q!869&2!".O2;CRV 71
|9gt2[!"&:/69.t687r/69r[.86;C &6 `4C/;|!"&:";~69RR8eO #B69&~9g¬2Q&2&C;C R(69¥71\2,Ce"83t!"#$R.";9g|!",l^| A2% ;O;@;C R(69&_3¬:,2Ss:.;C &\6v4.W(69. . s · ;O/;6!"!,/69S,RM^| &2;u!/2#%%
Rni,j =
δ1µ1(Id− 16δ
21)(f
E)ni,j
δx+
1
6δx3
δ1(D1δ31w)n
i,j
δx4 Z ^_[C^_[bar+δ2µ2(Id− 1
6δ22)(g
E)ni,j
δy+
1
6δy3
δ2(D2δ32w)n
i,j
δy4.
· `
2¬W(6!869. . sp ^ ^568sC¬Q#B69r2569WV71:'P\ZML n z,2/69&27:P\ZiL n !/2#% #$R. ! 83(69zWV~.86;u69&:s^568sG%69&~ #$R. ! Q!/2#%S9g2igh#
H · ∆wni,j = −∆t · (RIII)n
i,j , Z ^_[C^_[g4
^|2T2T&2/69& RIII #$R(6 ´8"[2pgm6!M6% ;O;i"!"&2,2!&+ε = 1
3κ = 1/3
|(6WV"&,2";) &2veCR. ! $69r^]22v/691G9gQ #$R. ! xs &2GR"&:/69&_3.@,2%6,#%!/2#%v Z ^ [C^_[g4Q M69RR. ";\v6%. &2869iRW.# ^|
fE = AEw3gE = BEw
3AE 69&2;
BE WV &!"&2/69&:[#B69!""8^zZQ&2;Q[6,#$R&_3a2S";C,(69. RIII 86;F^|2&\2pQ: \2,CeHgh#T,.U6|,2";(i
(RIII)ni,j = AE
δ1µ1(Id− 16δ
21)w
ni,j
δx+
1
12|AE |
δ41wni,j
δx
+ BEδ2µ2(Id− 1
6δ22)w
ni,j
δy+
1
12|BE |
δ42wni,j
δy
83i &2!"b2bRW.# H. &286983 −∆t(RIII)ni,j = −KIIIwn
i,j
^| 6) ;O;<'eOR. ! u/69r;C! ´69&\RV/694 7r&~W:s\
KIII = AEδ1µ1(Id−1
6δ21) + BEδ2µ2(Id−
1
6δ22) +
1
12|AE |δ41 +
1
12|BE |δ42 ,
^|2AE = ∆t
δxAE 3BE = ∆t
δyBE ^oZ[ &p6[c2-/;;C/! ´69&BsO.; −∆t(RI)n
i,j = −KIwni,j^|
KI = AEδ1µ1 + BEδ2µ2 −1
2|AE |δ21 − 1
2|BE |δ22 ,
L:/69 &Agh# 6\c2-;d'eCR. ! d/69r43 %%&(69,/69.A; 71u6\c2-;2'd,RM^| &2;+ #$R. ! !/2#%S9g2igh#
HI · ∆wni,j = −KIwn
i,j
Z ^_[C^_[ n ^| HI = Id+KI 69&2;S¬'eCR"&T!""RV&2;¬R"!/.¥sMz2K. &2869¬7r&T9g2R7:,2.¥s; 71";) #$R. ! B/69rk Z ^_[C^ Zc '|,2!/+6\!/('#% ^| . .|WVG'gh";k6$X 9XS\"!869. .6uc2-; #$R. ! Q/69rM6:!U69";HB6Tc2;'eCR. ! /69r4^L:/69 &k&2^®gh#³6< ;O;%'eCR. ! $/69r436\;C "!'B69RRr6!/ ^,.;.86;)k6u ;O;,RM^| &2;~ #$R. ! /!/2#%S9go2igm#
HIII · ∆wni,j = −KIIIwn
i,j
Z ^_[C^_[ d ^| HIII = Id + KIII ^[^71'3( [ [86sHd!/(!/~2T. &2869isO# 6:!U69"; ^| b,2!/b69& #$R. ! %/69rd@#T,2!/#%G &:71. 7r";)(69&)2v&2vW/69 &2"; ^| +6< #$R.dc2-;$ #$R. ! /69r$ &2!" S!",R."S&2^
5h¥&(86;k9g
3Q,&O&2^|&A #%'> &2!#%&:p &A86!/A4;O;C "!&¦¨ &
25^;2W(69&2;/^|;Cv9g_2W.:!/$#B69¥eM
;'c2&(;d &b Z ^_[C^ Zd ` &2!86";('^`I`71&d2,425!/2!"S Z ^_[C^_[ d '31;&2";$XXX 9XXXVg¨#&2^&["!869. .(6Q ;O;` #$R. ! /69r`6/1!U69";^| 6d ;O;p'eCR. ! S/69r43D*-,2;C!,2i &k#%S9g &: &2!$B!&2!s< ¥MR7O ;2p6&<;269#$R &<9gz69. .K2B'S#%:;"@69p86!/A #%% /69&hSRV &: ^z¥. .WV%;#%&2/69";,2 &M](,69&(69. sOo &$!/(69R
3_/32&2"!"" xspi. 7156Q#%5!"#$R.'e. &2869`sO#o. 1. s
·bc
~ &2!86%2B!"4pRVp /69&)9g5p #$R. ! @!/2#%% &,2!/6HRRV&(69@69. &(69 71 #$R.~!/2!""~7r&:,(69. . s WV"!"#%k#%\ &:" &2^q`"!. s13i6)/694s¢gh. .^";¦W:s¢#B69&:s69,2_a2`23 Z [ 369&2;H69./T/69 &2";H &v ^ 3O5$6:!U69S6T #$R.Qc2-;| #$R. ! /69rB6@ ;O;[eCR. ! /69r%
HI · ∆wni,j = −KIIIwn
i,j
Z ^_[C^ · er^|2~2~ #$R. ! G69&2;¢'eCR. !¥G;C! ´69&RV/69B,2/"; &+dX 9XXXT/694s(687rHWV"&R7O,2. s;'c2&2";D^NQ ^| . .5WVG/&,2 &k](,%69&(69. sO83`,2!/6<!/2!"v.86;&(69,/69. . sb6.^@¥&O &2!dB!&2!sW,TT.4T9g@;269#$R &\T#%B(69&)W(69.U69&2!"";W:sk2B";C,2!"";!"45/691&vW:sd2i. ,&~9g6@ #$R.Qc2-;z #$R. ! z/69ri &286;H9g6T#%i &:7r. 71";G ;O;z #$R. ! Q/69r4^
,UMJLE6Xx&u2T!86T9gK2TI`,.i"01,(69&23a2p &:7:!;:\2,Ce"
fE 69&2;gE 69p&2&C>. &2869Qgh,&2!&2[9g
w3
~(69T2B!/2!"κ = 1/3
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(ρ, u, v, p)T |xsOR!869. . sG,2";/692z(69&u2M/z9g!"&2/'769 71M769U69W."
(ρ, ρu, ρv, ρE)T ^
A*)@ )*% 7$; !$# & 7$& ,W.-"#%; # j7b&+/); !:;102"7# &V!&2;&2^2i ^59;C #%&2&(69.oJ[687O-YL:r""0:,(69&2p
wt + (fE(w) − fV (w,wx, wy))x + (gE(w) − gV (w,wx, wy))y = 0 Z ^_[C^ ·/Z
^|2%2$&2/69&2p,2";bgmfV 69&2;
gV '#$R2(6 ´8%2$7O!",2 \2,Ce"p;RV&2;A&2p&. sk&A2!"&27469 7r@769U69W."
wW,M69.G&\2 i4/6;C&:8^MNQ. .o2@!/2#%"i!"&2;";\ &\2TR"/&:
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(i, j)9gz6G4,.U69569"U69&4;
^| R(6!"$R2δx
69&2;δy
h9g2$69#%$;h
'3D2$&:,#%!869.69RR8eC #B69&9gfV 69&2;
gV"0:, "p769. ,(6927r"!w
69&2;%2QR(6!"; 7469 7r"wx
69&2;wy
&d86!/%g>6!"9gD2!"&:.!". .h/"S]o 2^ Z ^_[C^ `:'^|Xx&u2 ^;83(2S7O!",2&:,#%!869.\2,Ce"
F V 69&2;GV !869&uWVp'eCR"";
6S
F Vi+ 1
2,j
= fV (wi+ 1
2,j , (wx)i+ 1
2,j , (wy)i+ 1
2,j)
GVi,j− 1
2
= gV (wi,j− 1
2
, (wx)i,j− 1
2
, (wy)i,j− 1
2
)
^|2$ S#B69 &2S~'7469. ,(69$2%0:,(69&: "wi+ 1
2,j
3(wx)i+ 1
2,j
3(wy)i+ 1
2,j
69&2;wi,j− 1
2
3(wx)i,j− 1
2
3(wy)i,j− 1
2
)^ o'7469. ,(69&TtRV-gh#%";T,2 &Q #$R."!"&2;O;o!"&:/'/;69RR8eC #B69&vm &2!"
2|R:s9gV2z;C ,2&$R2&2#%&(6p;:"`&2!869. .gh69&:s@O &2;$9gV,RM^| &2;C &:'^ n K,2K!"&2;
· a
gm| &2/69&2!"M2ig>6!"(i+ 1/2, j)
^|2M2S769. ,(69&uz!869";u,Q65gh. .^zp
(w)i+1/2,j =wi,j + wi+1,j
2+O(h2)
(wx)i+1/2,j =∂w
∂x
∣∣∣∣i+1/2,j
=wi,j − wi+1,j
δx+O(h2)
Z ^_[C^ · [r
(wy)i+1/2,j =∂w
∂y
∣∣∣∣i+1/2,j
=wi+1/2,j+1/2 − wi+1/2,j−1/2
δy+O(h2)
Xx&25.U6gm#T,.U6C3425/69"&2"";";B69`&2:;"(i+1/2, j+1/2)
69&2;(i+1/2, j−1/2)
i!"#$R,/
FVi+1/2,j
i−1,j i,ji+1,j
i−1,j+1 i,j+1 i+1,j+1
i,j−1 i+1,j−1i−1,j−1
i+1/2,j−1/2
i+1/2,j+1/2
GVi,j−1/2
i−1/2,j−1/2
]¬ 4, Z ^ [C^ `2L:&2! ._gh|2S;C! ´69&\9go2M7O!",2 \2,Ce"8^
2MR(6!"M; 7469 7rwy
&H2[g>6!"(i+ 1
2 , j)69MW/69 &2";HW:sd/69O &$2S6871/69rM9go2M/69"| &
2igh,z&2 4:WV &d!". .T c24, Z ^_[C^ `:i
(w)i+1/2,j+1/2 =wi,j + wi+1,j + wi+1,j+1 + wi,j+1
4+O(h2) = (µ1µ2w)i+1/2,j+1/2 +O(h2)
:,2[2T71"!w
69&2;<2@R(6!"@; 769 71"wx
69&2;wy
7469. ,(69";b&\2pgm6!"(i + 1/2, j)
86;c2&(69. . sb
(w)i+1/2,j = (µ1w)i+1/2,j +O(h2)
(wx)i+1/2,j = (δ1w
δx)i+1/2,j +O(h2)
Z ^_[C^ ·4·
(wy)i+1/2,j = (δ2µ1µ2w
δy)i+1/2,j +O(h2)
· g
2S'eOR2"&~9g2M&:,#%!869.t7O!",2 \2,Ce"F V 69&2;
GV !869&~WVS,#$#B69 ´8";<6S
F Vi+1/2,j = fV ((µ1w)i+1/2,j , (
δ1w
δx)i+1/2,j , (
δ2µ2µ1w
δy)i+1/2,j)
Z ^_[C^ · `:GV
i,j−1/2 = gV ((µ2w)i,j−1/2, (δ1µ1µ2w
δx)i,j−1/2, (
δ2w
δy)i,j−1/2)
]o &(69. . sr342gh,. .O&:,#%!869./\2,Ce"|¨ &2!. ,2;C &S2 &:7:/!';$69&2;@7O!",2!"&: W,&(o!869&WVI^| &65gm. .^zS
Fi+1/2,j = FE(wLi+1/2,j , w
Ri+1/2,j) − fV ((µ1w)i+1/2,j ,
(δ1w)i+1/2,j
δx,(δ2µ1µ2w)i+1/2,j
δy) Z ^_[C^ ·bc
Gi,j−1/2 = GE(wLi,j−1/2, w
Ri,j−1/2) − gV ((µ2w)i,j−1/2,
(δ1µ2µ1w)i,j−1/2
δx,(δ2w)i,j−1/2
δy)
^|2T2@'e26!ieCR" &b9gFE 69&2;
GE ;RV&2;M&<2@!/2!"9gK2T &:7O!;u&:,#%!869.\2,Cegm#T,.U6~h:432z,2/69&27 3N|ZMLOP 3
. . .
JLKC8MN UME/JLUji 7: &d69&~ #$R. ! 71&~9g2S!/2#%S;'c2&2";HW:sG2S'eCR. ! Q/69r%
∆wexpi,j = −∆t(
δ1F
δx+δ2G
δy)i,j
^| F
69&2;G
4 7r&%W:sv Z ^_[C^ ·bc `,RRV4/"`iW, .;B69&d69RR8eC #B69&d9gV2|7O!",2\2,CeC"F V 3
GV692&2h^ #%.71.on+1
'^q-gh#$ &$6 68sO.;7r.R#%&:|9gD2"/]\2,Ce"69,&(;(n+1)∆t69&2;~#B69: &B,2p9g2 6!"WU69&<#B69!""
AV0
3AV
1
3AV
2
69&2;BV
0
3BV
1
3BV
2
&::;C,2!"";u &~2Mc2"!&u9gz!/(69R@h"B Z ^ Z ^ Z `:5.86;|~
(F V )n+1 = (F V )n + ∆t AV0
∂w
∂t+ ∆t AV
1
∂wx
∂t+ ∆t AV
2
∂wy
∂t+O(∆t2)
Z ^_[C^ · ar(GV )n+1 = (GV )n + ∆t BV
0
∂w
∂t+ ∆t BV
1
∂wx
∂t+ ∆t BV
2
∂wy
∂t+O(∆t2)
v(69T6H!"&:";69RR8eO #B69&)9g52$#%p69RRV869 &< &k2"/$;71.R#%&:ps:.;S2gh.¥.^| &B;C/!igh#@,.U6S
(F V )n+1i+ 1
2,j
= (F V )ni+ 1
2,j
+ (AV0 )i+ 1
2,j µ1∆w
ni+ 1
2,j
+ (AV1 )i+ 1
2,j
δ1∆wni+ 1
2,j
δx
+ (AV2 )i+ 1
2,j
δ2µ2µ1∆wni+ 1
2,j
δy+O(∆t2, h2)
Z ^_[C^ · g4(GV )n+1
i,j− 1
2
= (GV )ni,j− 1
2
+ (BV0 )i,j− 1
2
µ2∆wni,j− 1
2
+ (BV1 )i,j− 1
2
δ1µ1µ2∆wni,j− 1
2
δx
+ (BV2 )i,j− 1
2
δ2∆wni,j− 1
2
δy+O(∆t2, h2)
·bn
Xx&B;pW/69 &G6M7:/!",2!"&: W,&GS2| #$R. ! /69r|(69#B69 &256 #$R.Q62|c2-;,RM^| &2;!"&: W,&¢!"#$ &\gh#2G &:7:!;)&:,#%!869. \2,Ce"83`2dgh. .^| &A #$R. ¥c(";;71.R#%&:|/69 &2";
(F V )n+1i+1/2,j ≈ (F V )n
i+1/2,j + (AV1 )n
i+1/2,j
(δ1∆w)ni+1/2,j
δx Z ^_[C^ ·bn (GV )n+1
i,j−1/2 ≈ (GV )ni,j−1/2 + (BV
2 )ni,j−1/2
(δ2∆w)ni,j−1/2
δy
2~#%B!"&:/69 & &2~7:!",2 6!"WU69& #B69!""AV
2
69&2;BV
1
69~;C!869;";¢WV"!"69,(~2s&.U69rM2S&2! ._9go2M #$R. ! /69r ^| H"RV!z%6@W(6!
3>RV &:zRVz;C "!&<,RRV8(6
gm25#%K!&O/69 & &S257:!",2 6!"WU69&%#B69!""AV
0
3BV
0
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w^`]¬ &(69. . s13O2QW.:!/1> #$R. ! |,RM^| &2;v!/2#%[,2";d69RR8eC¥#B69/[2[Ji687:-YL:1"
"0:,(69&2z#B69 &2|gm#B69. . sG,&2!/(69&r";:^| ~"RV"!z2S!86S9go2SI`,.Q"0:,(69&2p
H · ∆wni,j = −∆t · Rn
i,j ,
^|
Rni,j =
(δ1(F
E − F V )
δx+δ2(G
E −GV )
δy
)n
i,j
,
H =
(Id+ σ1δ1(A
E)nµ1 + σ2δ2(BE)nµ2 − σ1δ1(D1 +
AV1
δx)nδ1 − σ2δ2(D2 +
BV2
δy)nδ2
).
Z ^_[C^ ·bd 2 n MLG9g`"0:,(69& Z ^_[C^ ·bd |!869&<69.WVS'eORa69&2;";u65gm. .^zS
H · ∆wni,j = C0∆w
ni,j + C−
1 ∆wni−1,j + C+
1 ∆wni+1,j + C−
2 ∆wni,j−1 + C+
2 ∆wni,j+1 ,
Z ^_[C^ `fer^|2M2M#B69¥e~!":B!&:
C±p
3C0
&2^ &2!. ,2;T6@7O!",2z!"&: W,&)
C−
1 = −σ1
(12A
E +D1 + 1δxA
V1
)ni− 1
2,j
= −σ1 (A+)ni− 1
2,j
C+1 = σ1
(12A
E −D1 − 1δxA
V1
)ni+ 1
2,j
= σ1 (A−)ni+ 1
2,j
C−
2 = −σ2
(12B
E +D2 + 1δyB
V2
)n
i,j− 1
2
= −σ2 (B+)ni,j− 1
2
C+2 = σ2
(12B
E −D2 − 1δyB
V2
)n
i,j+ 1
2
= σ2 (B−)ni,j+ 1
2
C0 = Id+ σ1 (A+)ni+ 1
2,j − σ1 (A−)
ni− 1
2,j + σ2 (B+)
ni,j+ 1
2
− σ2 (B−)ni,j− 1
2
Z ^_[C^ ` Z
NQRR. s: &~27O!",2i #$R. ! T!/2#%u Z ^_[C^ ·bd M69S86!/ARV &:p9g2$!"#$R,/69&(6.;#B69 &_3¬&2"!"712Q. &2869W.:!/dsO# Z ^_[C^ Zn ^|2Q2[!":B!&:|9gt2Q#B69¥e
M69&2;d2Q";C,(69.
[Rn]/69rM &:B6!"!",&:z2M7O!",2|#%Q6z;/69 .";~ &~2S69WV71igh#T,.U6^
·bd
" & (
A*)*% )%A * &V7b; 7
Xx&u7:h^9gK2pR(69 !,.U69S69RR. !869&2i2IKN &:"";< &_3a&(69#%. s~2T #T,.U69&b9gKW,2s:69&:#T,. !"#$RV&2&:[86! 7r \(^ &~&:,2!.86986![!"&:/69 &#%&:83(2p!/2#%"Q;7r.RV";u¥&u2R"&: ^\2,.;bWV71/69 .$&2,4AH;869."^| \(^l4 #%S/69&4 &~g¨#&2869. s\ &2!"#R" W.du 4. sA!"#$R"/ W.4^<XT(6pWV"&'eOR2.69 &2"; &2d &::;C,2!&9gzT2"T(69 ^;C &2!H/694"H!869&¦WVb6;R";¦ &(69H!86A 2H2\'eC&2&§9gp &2!"#$R" W.¨R",/>Wa6";( \(^®. 7rT~&2&C>´8\Pb6!/&:,#TWV \a^zTT2d'eC&2&)9g!"#$R" W.h;&2 xs:>W(6";( \(^zS. 71iv.^5Pb6!/k&:,#TWV8^$Xxi(6iWV"&A:&2^|&bgmi#%/(69&6v;"!86;&2^ (69|6MRRVR"!"&2;C & &(3C/9!869. . ";d.^5Pb6!/dR2"!&(;C & &239g_2Q"01,a69&2r71'&C &p!"#$R" W.]\(^z&(69W."K269RR. !869&G9g 2z/!/2#%"K & U69. . s$;71.RV;$gm!"#$R" W.\(^zBA2u #T,.U69&9gM&2869. s &2!"#$R" W. \(^RW.#%v69&2;+d!/2!"~B/69 &2";¢ &,0^Ggmz86&2Q;/69 .";u &u2S &::;C,2!&¨#B69 &. sHWV"!869,2p9g`2S #$RV/69&2!"T9gK'eC &;71.R#%&:Q&~!"#$R" W.S. 71| &\NiL · Pk'^ 2<¥#$R2. #%&:69&§9g@6kR/!"&2;¥ &2 &#%2:;"01, "G2uR. #$ &(69s !/2!"\9gT6G9g7469U69W."| &v#%z9gj^|!/H2MI`,."0:,(69&2]^| . .DWVi'eCR"";vWV'ghi &::;C,2! &G6TR"!"&2;C¥& &2^X&\g>6!83_#%$9g2R"!"&2;C & &2u/694"MRRV4";b &k2. /69,B69;&:!869.W,SW, . p,2 &<;C &:Tp9g5769U69W."^B](S &2/69&2!"43 ,1. g c ;7r.RV";A2$c2->71&9g¬R"!"&2;C &2 &B/69 &g¨#¯2iI`,.z"01,(69&2z'eOR"/";d &v#%|9go&:R!i7469U69W."(p, u, v, S)T ^|286t2KR"!"&2;C & &i/694siRRV4/;pW:s ¬69&2;TL:#$ nbc t;71.RV";gh#25I`,.69&2;J[687O-YL:1""0:,(69&2^| &gh2R #$ ¥715769U69W."
q = (p, u, v, T )T 369.H!869. .";k7O!",2M769U69W."p &2!"7O!",2M#%S!869&AWV$86 . s\'eCR"";b,( &H2#~3_W,S,&2,pWV['e26!. s%2[&2QRRV4";dW:s ,1'.^|2&H'eOR"/";Q^| G2[/69#%[9g¬&:R!Q7469¥69W."8^J[M(69±69& n " n e ;7r.RV";u^|&uR"!"&2;C & &G"!/&01,2@69.$,2 &d&:R!7469U69W."
(p, u, v, S)T ^|286%2569&2;P\O. Z ` T±&69" ^|69/69& n/Z #$R.s1";R #$ 71$M7O!",2i7469U69W."
(p, u, v, T )T ^@J[@2^7183_ ip69. ^568sOM2$!"&27469 7r@gm#9gt2iI`,.||J[687O-YL:1"5"0:,(69&2a695. 71";d &v;5@&2,[!""!|2:!/1!869R, &RRV" ^|2&~2pR(6!"';C! ´694&<Q69RR. ";D^ 2S!/2!"p9g`7:!",2z7469U69W."zB'eOR"Q6R"!"&2;C &2";GW.:!/:> #$R. ! 5,RM^| &2;d!/('#%Q!869&dWV!"&:71&&:gmr&2/69. \2,;5^| G"01,(69&29gK69 ^| &< &u#%[9g`R",T69&2;~/'#$RV/69,69&2;<!"69&b69.%WVp &:" &dB";C,2!"p2&:,#TWVK9g 69. rW/69!zRV/69&2;C, &M25!"#$R,/69&_^KJQ712."834 #T,2WV5!.869`(69`2";C D&:`!/2!""`;[&2K6 D"!o25!"&:71r&2!"5RV-gm#B69&2!"|9g((R"!"&2;C &(";%!/2#%4^jQ, &25!",59ga2"832,29gV 2`271"!`9gV!"&27469 7r7469U69W."
w = (ρ, ρu, ρv, ρE)T69&2;H | &2!#%&:∆w
|2MR #$ 71M769U69W."z,4q
69&2;∆q
(6|WV"&~2,4. sv &:71"¥169";uh/"52|RV `fe o &%;M;#$ &20^|22K "^56"^B ^| !/ &Sg¨#l2z!,&:w
>W(6";ugh#T,.U69&\ &ANiL · P³v6q>W(6/";~gm#T,.U69&_^ 2T",. [W69¥&(;_3V&2[RV";
2igm|2S/691i9g`!"&2!&2"83R71";H(69z2M,& !"4z";C,2!&< ";HW:sG2S!/2!"S9g2R #$ 717469U69W."5^56:p#%:;"p,&2;/691[,2!/d28687:s%;71.R#%&:Mh,2!/Bc2&2;C &r|69&2&2"!""/69 . s~ &\694/#%&: ^| <2T,2;Cs · e gm &2/69&2!"TW,Q[01,2"&<Q!.869. s~71sv#T,2!/ #$R.#%&:/694& T!":;';RV&2;&O'^&2"0:,2&:. sr3C2i;"! R&~9g¬2[R"!"&2;C &2";H,RM^| &2;!/2#%"^| . .Cgh:!,2 &2R"&:^@&%6zgm#T,.U69&$W(6/";&259gV!"&271";@7469U69W."
w^
`fe
A*)*% )@ i& h9<.f= 3 #; ./&V># 7# &5# 3 ( &&# ./&5!;# -7# -; !"!7b; 1 2T.^5P\6!/\R"!"&2;C &2 &v#%2:;\,2";u &< ^5HQW(6";\&<2 ,1.¬R"!"&2;C &( g c ^ zR"!"&2;C &2Q(6zWV"&< 4 &(69. . su69RR. ";Hghz2SI`,.[0:,(69&2Q'eOR"/";H &~#%9g&:R!@769U69W."
V = (p, u, v, S)T F^| S = ln(p/ργ)
'^QX&\0:,(6¥>. &2869Qgh#~3a2 isC#/691"|2igh. .^| &%gh#
P−1e · Vt +Ae · Vx +Be · Vy = 0
Z ^ · ^ Z ^|2
Ae =dfE
dV=
u ρc2 0 0
1/ρ u 0 0
0 0 u 0
0 0 0 u
, Be =
dgE
dV=
v 0 ρc2 0
0 v 0 0
1/ρ 0 v 0
0 0 0 v
^| c
2iRV"";G9g¬,&2;D^KNQgm2QR"!"&2;C & &2$#B69¥ePe
3: 5;" 4&2";v6@/!869.Q2R",S"0:,(69&u&. sb
Pe =
β2 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
2R(69/69#%β2 !/24/&$,2!/(6925 r&:7469. ,2"K9ga2R"!"&2;C &2";%sC#1"R25/69#%
;9g#B694& ,2; ^|('&u2SP\6!/~&:,#TWVWV"!"#%"Q#B69. .>^ 2S 4 &(69.¬;'c2& &uRRV4";~W:s ,1. g c p
β2 = min(max(M2,M2∞), 1)
Z ^ · ^_[r^|2
M∞
;&2"z2i'gm&2!"SPb6!/H&:,#TWVT¨ &/\(^Pb6!/H&:,#TWV|gm|'eC&(69.\(^zz|#B6eO #T,#Pb6!/b&:,#TWV ^| &k2Tc(.;\gm[ &:&(69.?\(^z'^ M;'c2& &&2,"i2@W2,2&2"9g[2vR"!"&2;C &2";¢!/2#%~69&2;,&2$2H!/2#%v &:b $&2&C>R"!"&2;C U&2";+71& ^|2&M > 1
^ i!/2!"@i9gh&\'gm";</v6M6 '4.W(69.`!,- pWV"!869,2@ M;:i&2M69. .^β
GWV."5(69&~2i &/\(^Pb6!/H&:,#TWV8^LO#%M.:!869._;'c2& &2Q69S687469 .U69W.i &H2i. /69,43(,2!/~62S&2M4 7r&~ & [ · 36^|!/~ &::;C,2!""[6;RV&2;&2!"T&~.:!"69.t4/6;C&:9gR",%
β2 = min(max(M2,M2p ), 1)
^|
M2p =
|δp|ρ c2ZQ&Cgh,&(69. s132u!"&:71r&2!"u/69u9gi2~R"!"&2;C &28; /!/2#%'^| ,2!/ 6A;'c2& &!869&
WV"!"#%H01, GRV:8^ 0 &2v2(69&2;D3K2H6!!,6!'s &2!86/" ^| 2G. :!869.z!,- &2!"β!869&WVG."/@(69&2v &/\(^P\6!/&:,#TWV+^|!/ 69. . ^z$b!869.v#%vRRV. s)2G&:,#%!869.
;C R(69&h"S&2'eCQ"!&2''^zQ^7183(WV<.:!869.69&2;u4.W(69.;'c2& &2Qgm69 .td&2,@6$r::;B!&2!sBgm57:!",2|5,&286;CsQ\(^z8^`Xx5:,25&2"!""/69sd@";'c2&2i2i!,-\ &v2"i!86"$6@R"71iB!&2!s1^KX& n/Z 3C±K&469" ^569/69& (687r[RRV4";v2Qgm. .^| &%;'c2& &A
β2 = min(max(M2, β2v , β
2u), 1)
` Z
^|2βv
69&2;βu
;&2/d#%%7O!",2T69&2;,&286;CsA!,-+"RV"! 71. s1^vXx&!/a69R au69&2; gO32S;'c2& &<9g2"i&2h^R(69/69#%0^| . ._WVS;/69 .";<69&2;H2 B!&2!s ^| . ._WVS,2;C";D^Xx&;A¦W, .;6.^5P\6!/lR"!"&2;C &&gmb2+I`,."0:,(69&2^| &l &l#%A9g!"&27469 7rM7469U69W."p
wt + f(w)x + g(w)y = 0 ,
2K/69&2-gh#B69&@#B69!""69. .^| &[ ^| !/pg¨#&:R2 !769U69W."V
!"&27469 7r7469U69W.w = (ρ, ρu, ρv, ρE)T 69M&2"";";
∂w
∂V=
1/c2 0 0 −ρ/γu/c2 ρ 0 −ρu/γv/c2 0 ρ −ρv/γH/c2 ρu ρv −ρq2/2γ
,
∂V
∂w=
(γ − 1)q2/2 −u(γ − 1) −v(γ − 1) γ − 1
−u/ρ 1/ρ 0 0
−v/ρ 0 1/ρ 0(γ−1)q2
2p − γρ −u(γ−1)
p −v(γ−1)p
(γ−1)p
^|2q2 = u2 + v2 ^KX&::;C,2! &B2"M/69&2-gm#B69&2 &:%"01,(69&) Z ^ · ^ Z sO.;S
P−1e · ∂V
∂w· wt +Ae ·
∂V
∂w· wx +Be ·
∂V
∂w· wy = 0
Z ^ · ^ ·
69&2;H.'g¨->#@,. R. s: &BW:s ∂w
∂V
.86;zH
∂w
∂V· Pe ·
∂V
∂w· wt +
∂w
∂V·Ae ·
∂V
∂w· wx +
∂w
∂V·Be ·
∂V
∂w· wy = 0
Z ^ · ^ `: 2'gm43t2$R"!"&2;C & &2<#B69¥e\gmM2%I`,.T"0:,(69&2S &!"&2769 71$7469U69W."p!869&AWV;'c2&2";<6S
Pw =∂w
∂v· Pe ·
∂v
∂w
Z ^ · ^ c ^| .M2M,2,(69.h!"&27469 7r 6!"WU69&u#B69!""Q9go2S!"&:71"! 7r \2,Ce"69M"!"71";HW:sb
AE =dfE
dw= Aw =
∂w
∂V·Ae ·
∂V
∂w=∂f
∂w, BE =
dgE
dw= Bw =
∂w
∂V·Be ·
∂V
∂w=∂g
∂w.
Z ^ · ^_ar :,2832~0:,(6¥>. &2869$gm# 9g[2~IK,.B"0:,(69&2l^| &¢ &+#%B9gi!"&2/769 71H7469U69W."86;p
P−1w · wt +Aw · wx +Bw · wy = 0 ,
Z ^ · ^ g4^|!/u71&:,(69. . sv.86;zW(6!/G2S!"&27469 7rigm#
P−1w · wt + fE(w)x + gE(w)y = 0 .
Z ^ · ^ n
`f[
2 K"UMFUfGliTU JWE/2*)Pw
2 ,1.a.^5P\6!/GR"!"&2;C &2 ^| &G &d#%9g_&:R!Q7469U69W."!869&v69.pWV'eCR"";65gm. .^zS
Pe = Id+ (β2 − 1)Qe^|
Qe =j[U69
[1, 0, 0, 0] Z ^ · ^ d
^|2 M ^k&2! &~#B69¥eQe
i;#$RV&:3 Q2
e = Qet!"&2"0:,2&:. s<2 &:719g
2MR"!"&2;C & &2B#B69¥eHz #$R. sv!"#$R,";<6S
P−1e = Id+ (
1
β2− 1)Qe
Z ^ · ^ Z er
L: &2!"@2@!"&2/769 71T71&b9g(TR"!"&2;C &2Si,2!/\(69Pw =
∂w
∂V· Pe ·
∂V
∂w
3V Qgm. .^z(69p
Pw = Id+ (β2 − 1)∂w
∂V·Qe ·
∂V
∂w= Id+ (β2 − 1)Qw
Z ^ · ^ ZbZ 69&2;~ #$R.S!869.!,.U69&2sO.;k
Qw =∂w
∂V·Qe ·
∂V
∂w=γ − 1
c2
1
u
v
H
·
(q2
2; −u ; −v ; 1
) .
Z ^ · ^ Z [r
0 W:7O,2. sr3o2B#B69¥eQw
@69.u;#$RV&:~Q2
w = QwS~(69
P−1w = Id + (1/β2 − 1)Qw
#%"7183a2pR(69!,.U69Qgm# 9g`Q#B69¥e
Qw&(69W."[d!"#$R,T2p#B69¥e:>71"![R:;C,2!
Qw ·X 71sG86 . s132669. 86;CsvRV &:";u,z & g$a
Qw ·
X1
X2
X3
X4
=
γ − 1
c2·(q2
2X1 − uX2 − vX3 +X4
)·
1
u
v
H
Z ^ · ^ Z8·
2"RRV" ^| . .aR71Q:&d71s$,2'g¨,. ^|2&vR #$ ´" &2Q!"49g¬69&d #$R. ! 5Pb69¥eO]2"869#%&:MgmS69. .KRV""; \(^zM &(69R/' `2^@J[@(69igh#&2^l&b( #$R.&2/69&2
P3Q^| . ._WVM,2";v%;&2M2S!"&27469 71MR"!"&2;C &2
Pw69&2;H 6:!U69";~#B69¥e
Qw^
2UfF UM U UMVG )GUJq`'>#T,. R.s: &\2$ #%'; 7469 71B# 9g52BI`,.T"0:,(69&2pW:sb2$#B69¥e
P−1 .86;pu6&2h^ r&2sC# &2!"S"01,(69&) Z ^ · ^ g45!869&~WVM' ^| &\65gm. .^zS
wt + A · wx + B · wy = 0 Z ^ · ^ Z `:
^|2%2$R"!"&2;C &("; 6!"WU69&#B69!""A = P · A 69&(;
B = P · B (687r$WV"& &::;C,2!"";D^Xx& ;G)!"#$R,\2< r&:71"!H69&2; r&:7469. ,2"v9gS2"/uR"!"&2;C &2";¦#B69!""83z d"!"#$#%&2;"; n4· 3 n ` S/69`gh#l2z&:R!zR"!"&2;C &2"; 6!"WU69&d#B69!""
Pe ·Ae3Pe ·Be69&2;b,2v&2'eC.U69& Z ^ · ^ c Tb"!"71$2~!"&2769 71v#B69!""8^ 2H r&:7469. ,2"B9g[2
` ·
R"!"&2;C &2"; 6!"WU69&J = Anx + Bny
F^| nx
69&2;ny
2S!"#$RV&2&:Q9g2M&2#B69._71"!~n
|69i4 71&~W:s\
λ(1) = unx + vny = Vn
λ(2) = Vn
λ(3) =1
2(1 + β2)Vn +
√β2c2 +
1
4(β2 − 1)2V 2
n
λ(4) =1
2(1 + β2)Vn −
√β2c2 +
1
4(β2 − 1)2V 2
n
Z ^ · ^ Zc
Xx&'^|(69|gm. .^z83Λ(nx, ny)
;&2"z2S;CU69r&(69.t#B69¥e~j[69[λ(i)]
!"&:/69 & &B2MR"!"&2;C &("; r&:7469. ,2"8^~J[B(69D^|2&
βT9g|2d/69#%d;T(69&2BPb6!/&:,#TWV69&2;T.U69T
#B69. .>3169. .25R"!"&2;C &2";B r&:769. ,2"6959gV25/69#%|;`(69&$25R(69!.571.:! xsVn
^ #%869&2T2BsC# p&2~.&r@ . . !"&2;¥ 4&2";gmp.^RV"";L\(^z8^ 2%R"!"&2;C &28;#B69¥ePe · (Aenx + Beny)
!869&~WVM'eCR"";H,2 &Λ
6|gh. .^zp
Pe · (Aenx + Beny) = Je(nx, ny) = Te(nx, ny) · Λ(nx, ny) · T−1e (nx, ny)
Z ^ · ^ Z ar^|2M2S r&:71"!#B69!""
Te69&2;
T−1e
69i4 71&~ & n ` 69&2;HR:;C,2!"";u2%
Te =
0 0 1 1
0 −nyr nx
ρ β2 c2s nx
ρ β2 c2
0 nxr ny
ρ β2 c2s ny
ρ β2 c2
1 0 0 0
, T−1e =
0 0 0 1
0 −ny nx 0
s
2 t−ρ β
2 c2
2 tnx −ρ β
2 c2
2 tny 0
− r
2 t
ρ β2 c2
2 tnx
ρ β2 c2
2 tny 0
Z ^ · ^ Z g4^|
r = λ(3) − Vnβ2 , s = λ(4) − Vnβ
2 , t =λ(4) − λ(3)
2.
Z ^ · ^ Zn Xx&k2%69#%$#B69&&283¬2$R"!"&2;C &2"; 6!"WU69&)'eOR"/";k &A#%p9g|!"&27469 71$7469U69W."!869&<69.WVM#B6;S;CU69r&(69.
P · (Anx + Bny) = J(nx, ny) = T (nx, ny) · Λ(nx, ny) · T−1(nx, ny) Z ^ · ^ Zd
Z[ &<2d/69&2-gm#B69&.U69&2WV ^"&&:R!H69&2;)!"&2/769 71d7469U69W."83 6!"WU69&/691"69.([gm. .^| &$gm#
J =∂w
∂V· Je ·
∂V
∂w=∂w
∂V· Te
︸ ︷︷ ︸= T
·Λ · T−1e · ∂V
∂w︸ ︷︷ ︸= T−1
. Z ^ · ^_[ber
NQ. .a2[69WV71Q'eCR"&25^| . .VWV,2";d &d2&2'eC"!&,^|2&GW, .;C &2[69¥c(!U69.D;C R(69&9gK6@R"!"&2;C &2";u,RM^| &2;u!/2#%4^
`b`
A*)*% )*% #; ./&>5# 7# &"; > 23 # > !2.M="; 14;[, . .U69;)69&2;k±[´69T2^";A & n4· 3 n `23V[ n 2^~W, .;6v71&9g|2%Q:%!/2#%d6;269R";2~R"!"&2;C &28;sC# Z ^ · ^ n 'i,2!/6A/!/2#%u d!/ &2"; Q:' ,r.[!/2#%u & n4· W:s'gh&2!"H\2G,2/v9g ,r.|R"!"&2;C & &22^ %"!&+,#$#B69 ´8"%2H; 769&¢9g2~Q:' ,r.Q!/2#%H &::;C,2!"";¢ & n4· ^| ¢2H&2/69&2B,2"; &+2HR"/&:l^ ^ 0 ,/69 &uRV &:pS2B01,(6 >. &2869@sO# Z ^ · ^ Z `:'3R(6!"';C! ´8";,2 &\6Gc2-;6!"!,/69:M/!/2#%%
wt + A · wx + B · wy =δx
2(|A|wx)x +
δy
2(|B|wy)y
Z ^ · ^_[ Z
^|2T #T,2QWVp,&2;::;b69. .¬2TR(6!"'; 7469 7r"M69T;C! ´8";<,2 &G #$R.T"!"&2;O;6!"!,/69B!"&:";kgh#T,.U68^ n 'g¨->#T,. R. sO &<S'eCR"&kW:s
P−1 69&2;kr &~W(6!/\~6H!"&C7469 7rgm#gh20\2,CedW(69.U69&2!"[sO.;2gm. . ^| &/#$¥;C!Qgm#gh2QR"!"&2;C &(";:M/!/2#%%
P−1 · wt + fx + gy =δx
2(P−1|P ·A|wx)x +
δy
2(P−1|P ·B|wy)y
Z ^ · ^_[b[r
[, . .U69;)69&2;A±[´69_[ n R71";A(6983tgmT6H.^P\6!/&:,#TWV \(^S369. .(%!"#$RV&2&O@9g|269WV71&:,#%!869.2;C/ R(69& \2,Ce"o#B69 &$9ga25/69#%;`9ga#B694& ,2;5(69&25!""RV&2;C &&2&C;C R(69 71R(69i9g2@&:,#%!869.\2,Ce"83^|!/A&(69W."iGR"7r$6!"!,/6!sugm[.^RV"";\(^z8^¦X& ;B769. ,(69/~BR"!"&2;C &2";;C/ R(69&¢ &6A #$R.~#B69&&283|2u;C &:'eCR"&2z;'c2&2";~ &~2iR7O,2"!&<69M&2^,2";A
P−1 · |P ·Anx + P ·Bny| = (P−1 · ∂w∂V
· Te) · |Λ| · (T−1e · ∂V
∂w) = T g · |Λ| · T d Z ^ · ^_[ ·
Pb69!""T g 69&2;
T d 69S86 . sv!"#$R,";Hgh##B69!""Te
3T−1
e
3 ∂V∂w
69&2;P−1 &2!"$
T g(nx, ny) = P−1 · T =∂w
∂V· P−1
e · Te Z ^ · ^_[2`:T d(nx, ny) = T−1 = T−1
e · ∂V∂w
69&2;H2 z"RV"! 71S'eCR. !' Q'eOR"&H86;p
T g =
1 01
2 β2 c21
2 β2 c2
u nyu + r nx
2 β2 c2u + s nx
2 β2 c2
v −nxv + r ny
2 β2 c2v + s ny
2 β2 c2
q2 −VtH + r Vn
2 β2 c2H + s Vn
2 β2 c2
Z ^ · ^_[ c
` c
69&2;
T d =
1 − γ − 1
c2q2
γ − 1
c2u
γ − 1
c2v −γ − 1
c2
Vt ny −nx 0
sq2(γ − 1) + β2c2Vn
t−su(γ − 1) + β2c2nx
t−sv(γ − 1) + β2c2ny
t
s(γ − 1)
t
−rq2(γ − 1) + β2c2Vn
t
ru(γ − 1) + β2c2nx
t
rv(γ − 1) + β2c2ny
t−r(γ − 1)
t
Z ^ · ^_[bar^|2
Vt = −uny + vnx3q2 = 1
2(u2 + v2)3H = E + p
ρ
3E = e+ q2
^| p = (γ − 1)ρe
69&2;r
3s
3t
69[gh,&2!&2z9go2iR"!"&2;C &2"; 6!"WU69&u r&:7469. ,2";'c2&2";HW:sb Z ^ · ^ Zn '^KJ[i(69|/69O &β2 = 1
sO.;r = c
3s = −c 69&2;
t = −c u(69@2B#B69!""4 7r&)W:s¢ Z ^ · ^_[ c @69&2;¦ Z ^ · ^_[bar";C,2!"S2Q!.U6!869.a#B69!""5^|!/d#B69r2 6!"WU69&
(Anx +Bny);CU69r&(69.h/"|gh &2/69&2!"
·/Z '^ 2ig¨,2.¥. s:;C!M71&u9go2S:' ,1.t!/2#%B Z ^ · ^_[b[r586;p
P−1∆wn
i,j
∆t+ (
δ1F
δx+δ2G
δy)i,j = 0
Z ^ · ^_[g4^| ~&:,#%!869.)\2,Ce"z;'c2&2";~W:sb
Fi+ 1
2,j = (µ1f − 1
2P−1 · |P ·A| · δ1w)i+ 1
2,j = (µ1f − 1
2 Tg(1, 0) · |Λ(1, 0)| · T d(1, 0) · δ1w)i+ 1
2,j
Gi,j− 1
2
= (µ2g − 12P
−1 · |P ·B| · δ2w)i,j− 1
2
= (µ2g − 12 T
g(0, 1) · |Λ(0, 1)| · T d(0, 1) · δ2w)i,j− 1
2 Z ^ · ^_[ n IeC&2;C &B2S!/2#%M 42-/;[6!"!,/6!svz;&2MW:sG/69O &u
Fi+ 1
2,j =
1
2(f(wL
i+ 1
2,j) + f(wR
i+ 1
2,j)) − 1
2P−1
i+ 1
2,j· |Pi+ 1
2,j ·Ai+ 1
2,j | · (wR
i+ 1
2,j− wL
i+ 1
2,j)
Gi,j− 1
2
=1
2(g(wL
i,j− 1
2
) + g(wRi,j− 1
2
)) − 1
2P−1
i,j− 1
2
· |Pi,j− 1
2
·Bi,j− 1
2
| · (wRi,j− 1
2
− wLi,j− 1
2
)
^|2M2S/69"wR 69&2;
wL 69M"!"&2,2!";~,2 &$.U69&2Q;'c2&2";H &u"!& Z ^_[C^_[C^Xg(2|z,269&27T!/2#%5,2";@ &286;9gV2|:5!/2#%434#B69!" | ˜Λ(1, 0)| 3 | ˜Λ(0, 1)| 69R.U6!"";^| @2 ¬"RV"!'¥71RV"!/69.O/6;C ,2¬sO.;C &[2Kgh. .^| &i&:,#%!869.M\2,Ce"tgh¬2R"!"&2;C &(";z,269&27G!/2#%%
Fi+ 1
2,j =
1
2(f(wL
i+ 1
2,j) + f(wR
i+ 1
2,j)) − 1
2ρ(PA)P−1 · (wR
i+ 1
2,j− wL
i+ 1
2,j)
Gi,j− 1
2
=1
2(g(wL
i,j− 1
2
) + g(wRi,j− 1
2
)) − 1
2ρ(PB)P−1 · (wR
i,j− 1
2
− wLi,j− 1
2
)
&2!"T g · T d = P−1 ^MJQ@2T&:,#%!869.K;C R(69&k9g[R2"!&(;C &2";Az,2/69&27u/!/2#%T
&2$.&rzR,. sH!869.U69 &2!"i !"&:/69 &2z&2^2MR"!"&2;C &2 &d#B69¥eP−1 ^
n ^5Pb6!/§&:,#TWVvR"!"&2;C &(";§7r&2H9gp2bNzZiLOP 3i69&C n 6eO>769& n "83 69#%"&_=?
`fa
ZMLOq)!/2#%"569z"RV"! 71. s$R7O;";B & `fa c/Z gbg ^X&B2R"&:5^ 31.^5Pb6!/d&:,#TWV71&29g_2QQ:3Oz,2/69&27B69&2;dNzZMLOP !/2#%"(687rWV"&d,2";D^ 2±&dJ[,#B69&&H69&(69. sO!869";H,|¥&H2[&2'eC|!/(69R|(6gh:!,2";v&H2[ #$R. ! zQ:Q,RM^| &2;H/!/2#%[#B69 &. sdWV"!869,2 o'eCR"&T687469 .U69W.gh¬2g¨,. .:&2&C>. &2869`IK,."01,a69&26V^. .6tgh¬ ¬. &2869 ´8";@71&^| .43 gm[ &2/69&2!"43D2;2'¥7469&A9g2NzZMLOP !/2#%$69RR. ";bG2. &2869 ´8;bgm#9g2I`,.p"01,(69&2S"01, "S#%";C,2S;71.R#%&:T69&2;A6G&:,#TWVS9g|69RR8eC #B69&2dh" `fa gm|#%S;/69 .'^](¬2Q:z,2/69&27S!/2#%432R"!"&2;C & &M#%2:;6 "!o &6 #$R.^56sM2;'c2& &9g_2&O,#%!869. ;C¥Ra69&_^`]2#&2^¦&_3
D169&2;
D2^| . .V;&2Q2QR"!"&2;C &2";G&:,#%!869.
;C R(69&b!":B!&:S6:!U69"; ^| b86!/bR(6!"T;C "!& ^| .T2pR2"!&(;C &(";\&:,#%!869. &:7O!;,\2,Ce"]^| .¥._WVM;&2";
FE 69&2;GE ^ :,23C2i&:,#%!869.)\2,Ce"|9g2M:S69&2;~z,2/69&27
!/2#%"!"#TW &2";:^| ,1._R"!"&2;C & &G!869&~WVM,&¥c(";u6S
FEi+ 1
2,j
=1
2(fE(wL
i+ 1
2,j) + fE(wR
i+ 1
2,j)) − D1 · (wR
i+ 1
2,j− wL
i+ 1
2,j)
GEi,j− 1
2
=1
2(gE(wL
i,j− 1
2
) + gE(wRi,j− 1
2
)) − D2 · (wRi,j− 1
2
− wLi,j− 1
2
)
Z ^ · ^_[ d
^|
D1 = 12P
−1|P A| = 12 T
g(1, 0) · | ˜Λ(1, 0)| · T d(1, 0)
D2 = 12P
−1|P B| = 12 T
g(0, 1) · | ˜Λ(0, 1)| · T d(0, 1)
Z ^ · ^ · er
gm|2MR"!"&2;C &2";<:M/!/2#%Mz2S:' ,1.t!/2#%p69&2;
D1 = 12ρ(P A)P−1
D2 = 12ρ(P B)P−1
Z ^ · ^ ·/Z
gmv2bR"!"&2;C &2";z,269&27 !/2#%4^ Ggm#T,.U69&H'eC&2;";¦2kJi687:-YL:1"sC# &+6\/69 4:-gh ^569; ^568s &2!"G2GR,. s)!"&:";+;C! ´69&¢9gQ2G7O!",2 \2,CeC"#B69 &2Q,&(6 "!";uW:sv2SR"!"&2;C & &2G9g`2p"0:,(69&28^5X&u2 ^;8322SR"!"&2;C &(";Ji687:-YL:1S"01,(69&2T
P−1 · wt + (fE(w) − fV (w,wx, wy))x + (gE(w) − gV (w,wx, wy))y = 0 Z ^ · ^ · [r
69p69RR8eC¥#B69/;uW:sd2S!"&27469 71S!/2#%
(P−1)ni,j ·
∆wni,j
∆t+
(δ1(F
E − F V )
δx+δ2(G
E −GV )
δy
)n
i,j
= 0 Z ^ · ^ ·4·
^|2@2@R"!"&2;C &2";k &:7:!;\&:,#%!869.\2,CeC"M69@4 71'&A69WV7169&2;\2T7O!",2 \2,CeC"[1"R2 'eCR"& Z ^_[C^ · `:'^JLKC8MN+GE U 2G.^5Pb6!/R"!"&2;C & &2k"!/&0:,2H69RR. ";\2GQ:G,2/69&27!/2#%G#%:;C¥c("$2 #%i; 769 71[#69&2;G2Q&:,#%!869._;C/ R(69& ^| v"RV"!5T2i/69&2;269;G71&v9g¬2"
`Mg
!/2#%"oW, &::;C,2!""&2Q"!/&!869.;CXB!,. xsT &@2R:!""9g(#B69O &i!/2#%M Z ^ · ^_[g4¬| Z ^ · ^ ·4· #$R. ! 8 &vg>6!83C2[gm#9go2i #$R. ! zR"!"&2;C &28;u!/2#%[gm|2MJ[687O-YL:1"z"01,(69&2#B69 &2z,&2!/(69&r";
H · ∆wni,j = −∆t · Rn
i,j , Z ^ · ^ · `:
^|
Rni,j =
(δ1(F
E − F V )
δx+δ2(G
E −GV )
δy
)n
i,j
,
H =
((P−1)n + σ1δ1(A
E)nµ1 + σ2δ2(BE)nµ2 − σ1δ1(D1 +
AV1
δx)nδ1 − σ2δ2(D2 +
BV2
δy)nδ2
).
Z ^ · ^ ·bc 2 n MLG9g`"0:,(69& Z ^ · ^ · `:|!869&~WVS69.%'eORa69&2;";u6S
H · ∆wni,j = C0∆w
ni,j + C−
1 ∆wni−1,j + C+
1 ∆wni+1,j + C−
2 ∆wni,j−1 + C+
2 ∆wni,j+1 ,
Z ^ · ^ · ar^|2%2%!":B!&:
C±p
/69r$&2^ &:u6!"!",&:T2$#%:;C¥c(!869&9g52$&:,#%!869.;C R(69&#%zW:sG2i.^5Pb6!/~R2"!&(;C & &v69&2;
C0 &2!. ,2;"2MR"!"&2;C & &d#B69¥eA
C−
1 = −σ1
(12A
E + D1 + 1δxA
V1
)n
i− 1
2,j
= −σ1 (A+)ni− 1
2,j
C+1 = σ1
(12A
E − D1 − 1δxA
V1
)n
i+ 1
2,j
= σ1 (A−)ni+ 1
2,j
C−
2 = −σ2
(12B
E + D2 + 1δyB
V2
)n
i,j− 1
2
= −σ2 (B+)ni,j− 1
2
C+2 = σ2
(12B
E − D2 − 1δyB
V2
)n
i,j+ 1
2
= σ2 (B−)ni,j+ 1
2
C0 = (P−1)ni,j + σ1 (A+)
ni+ 1
2,j − σ1 (A−)
ni− 1
2,j + σ2 (B+)
ni,j+ 1
2
− σ2 (B−)ni,j− 1
2
Z ^ · ^ · g4
2&B2zR/!"&2;¥ &(";d #$R. ! ,RM^| &2;d!/2#%@ Z ^ · ^ · `:`69RR. ";G6986!/BRV &:9gD(!"#$R,C/69&(69.¬;#B69 &_322i. &2869W.:!/vsC# Z ^_[C^ Zn 5|"!"7r";<69&2;H#T,2zWVM/. 7r";v &<69&uB!&:^568s1^
( #% " & (A*))%A 2- 7#%14;W( #14; ,&&# /ZQ&286;Cs \2,;;Cs:&(69#$!BR:!"""@69%g¨"0:,2&:. sk&2!",&:"; &&4 &2" &\RW.#%69&2;)69&2^+!"#$#%&. s$!"#$R,";$W:sT2]j R/6! &28^KN&$ #$#%";CU69]^568sTMW/69 &d6[ #%'x6!"!,/69. ,&u9g2B¨,&286;Cs25J[687O-YL:1"z"0:,(69&283
∂w
∂t+ (fE − fV )x + (gE − gV )y = 0
` n
^,.;HWVMB69RR. sG2igh. .^| &d!/2#%%
wn+1i,j − wn
i,j
∆t+
(δ1F
δx+δ2G
δy
)n
i,j
= 0 Z ^ `2^ Z
^| ¢6<,&0:,2v4.W(69.| #%HR∆t
&+"6!/+!". .z9g[2H!"#$R,/69&(69.Q;#B69 &_^)[^7r83,2!/69&'eOR. ! T/!/2#%%p&. s\c2-;T #%'x6!!,69Q^z!/S&2T, B!&:Sgmp!"#$R, &b69;CsO&(69#$!TRW.#%3V,&."/Q71su#B69. .o #%'RA69T,2/"; ^|!/\.86;i &\,&\d 4qKZ¥#%!"&2,#$R&_^]2,2#%43¥&2H!86H9g[.^Pb6!/&:,#TWV+\(^RW.#%3/69W . xs! U6^,.;k #$RV4@gh,2M. #$ /69&2T&k2 #%'R ´8 ^|!/L^,.;b #$R(69 T69&:s^568s<2,2/'g¨,.¥&2"9ga,2!/%69&$69. r #~^&2"01,2&:. s134 &;Q &2!86WV2|6!"!,/6!s@69&2;@2B!&2!s9g2i #%'> &:4/6&<R:!""83 |z,2,(69.D$/69 &~([gm. .^| &% #$R. ! Q!/2#%%
3wn+1i,j − 4wn
i,j + wn−1i,j
2∆t+
(δ1F
δx+δ2G
δy
)n+1
i,j
= 0 Z ^ `2^_[r
NQ&v #$#%";CU69 68sO.-;7r.R#%&O[69,&2;v2i&2h^§ #%'>.71.(n+ 1)
2^z5(69z/!/2#%% Z ^ `2^_[rR7O;"z6T"!"&2;O;2'Q6!"!,/69S69RR8eC #B69&~9g¬2[,&286;CsGJ[687O-YL:1"|"01,(69&28C#%'7183O2[ #$R. ! z!/(69/6!9g¬2i!/2#%S69. .^z5@,2[.U69r[ #%'R2 ^|2&u6;C769&(!' &% &G¥#%^| 2,&2!",&: &T/69W . xsRW.#%Q¨ &%R/6!!"43125 #%'RB`. #$ ";$W:s6!"!,/6!s,2"|:v.U69rB6G #%'RAR71&:ig¨#!"/!. sbR"&: &u2 #%'71. ,&9g2 \(^RWC.#d'^Ji69,/69. . s13( Q#B69 &2[d'eOR.U69 &\2^"0:,(69& Z ^ `2^_[rz[. 71;\69[86!/< #%'R< &\;$s:.;u6@&2h^R:sC!869.t/694^
KVKj)8JLE/U + 'UfP+ JWUTV-Nc2]^568sd$/. 7riz!/2#%i|,2S69&H /69 71S69R2R8eO #B69'J[h^|&u#%2:;D^L:/69
wn+1R.U6!"";uWOs~6$R7O&(69.o/69wn,k+1 69&2;~2 \2,CeC"[69p. &2869 ´8";b69,&2;u2p /69 7rp.71.
(n, k)^|!/~.86;z$2igh. .^| &B /69 7rS¥#$R2.¥! Q/!/2#%%
H · ∆wn,ki,j = −∆t · Rn,k
i,j ,
^|
∆wn,ki,j = wn,k+1
i,j − wn,ki,j ,
Rn,ki,j =
3wn,ki,j − 4wn
i,j + wn−1i,j
2∆t+
(δ1F
δx+δ2G
δy
)n,k
i,j
,
H =
(3
2Id+ σ1δ1(A
E)µ1 + σ2δ2(BE)µ2 − σ1δ1(D1 +
AV1
δx)δ1 − σ2δ2(D2 +
BV2
δy)δ2
)n,k
.
Z ^ `2^ · N|o2c2`,WC> /69&_3
wn,0 /"0:,(69.Cwn 69&2;&2!"525,WC> /69 715R:!"/(6!"&:71r";
∆wn,∞ = 0
'3t2%,&286;CsA0:,(69&2@69B!.869. sA/69-c(";D^ 2B. ,&69S2% #%%.71.(n+ 1)
|2&~4 71&HWOswn+1 = wn,∞ ^
` d
: VEHC + 8JWU:GUMKVK8 WG$TUJLUNQ&$69. &(69 7rz69RRr6!/gmo2/. ,&$9g(2,&286;Cs@J[687O-YL:r"`"0:,(69&2o2;C,a69. >¥#%#%2:;l^|!/B(6KWV"&d!"#$#%&. s$,2";ghK &2!"#$R" W.i\(^z`gm6i. &2S #%z&2^ _a · W,"^56c2o69RR. ";p!"#$R" W."\(^)!"#$R,/69&2¬W:s 69#%"& · ` ^¬N|t86!/TR:sO!869.r #%'R_3/69wn+1 |!"#$R,";u6z6@86;Cs:/69 ^| H"RV"!|$6TR2,2;9> #%
τh|;C,(69.¥> #%'^Xx&vR/6!!"43
69&69¥c(!U69.Sh@;,(69.£M #%'; 7469 71dT &::;C,2!"";) &6;;C &)<2BR:sC!869. #%'; 7469 7r.86;C &B2igm. .^| &B;C! ´69&)
wn,m+1i,j − wn
i,j
∆τ+
3wn,m+1i,j − 4wn
i,j + wn−1i,j
2∆t+
(δ1F
δx+δ2G
δy
)n,m+1
i,j
= 0 Z ^ `2^ `:
^|2∆τ
i(@R2,2;29> #%$RA69&2;wn,m+1 ;&2"M(/69/$69MR2,2;9> #%
(m + 1)∆τ¥&
2G!"&:71r&2!"GR:!""Tgh# 2dR:sC!869.5/69v69@ #%(n∆t)
<2G&2H69T¥#%(n + 1)∆t
^n &2869 ´" &d2M&:,#%!869.)\2,Ce"69,&2;~2iR2/,2;9> #%S.7r.
(n,m)sO.;S
H · ∆wn,mi,j = −∆τ · Rn,m
i,j , Z ^ `2^ c
^|
∆wn,mi,j = wn,m+1
i,j − wn,mi,j ,
Rn,mi,j =
3wn,mi,j − 4wn
i,j + wn−1i,j
2∆t+
(δ1F
δx+δ2G
δy
)n,m
i,j
,
H =
((1 +
3
2λ)Id+ σ1δ1A
Eµ1 + σ2δ2BEµ2 − σ1δ1(D1 +
AV1
δx)δ1 − σ2δ2(D2 +
BV2
δy)δ2
)n,m
,
Z ^ `2^_ar^|2@2@R(69/69#%
λ = ∆τ∆t
(6iWV"&b &::;C,2!"";A69&2;\201,(69&: xsσ1
¨"R_^σ2
;&2"M&2^2/69vWV ^"&A(R2,2;9> #%%R69&2;b2$R(6!"$Rk &k2
x¨"R_^
y[R(6!"';C "!&
σ1 = ∆τδx
3σ2 = ∆τ
δy
^N[S &2G69RR8eC #B69'JQh^|4&!/2#%43o2%c2TR2,2;9/69
wm+1,0 T@"01,(69.wn 3`69&2;D3
^|2&\2pR2,2;9!"&:71r&2!"@R:!""Qi6!/7r";D3 ∆wn,∞ = 0
3(2 n iL~9g"0:,(69& Z ^ `2^ c 7469&2"8^ 2v,&286;Cs)"01,(69&2B69G2'gmv69-c(";§F^| /"!"&2;O;B6!"!,/6!s) &)¥#%69&2;G6pR(6!"'x6!"!,/6!sd;RV&2;C &&d2Q!/2!"[9g_2z&O,#%!869. \2,Ce"Kgm#T,.U669&2;B2Q. ,&69SR:sO!869.¥> #%%.71.
(n + 1)M4 71&kW:s
wn+1 = wn,∞ ^ 0 &2%!869&A&2!"$a69M/69O &∆τ = ∞#B691"$2~;C,(69.¥> #%<RR &!/2#%~;&:!869.b2~69R2R8eO #B69'J[h^|&&24^¢X&g>6!832
69RR8eC #B69'JQh^|4&#%2:;!869&@WV"&$66zR(69!,.U69K!"6/9g(2;C,(69.¥> #%5RR &M69RRr6!/_^ 2v;"! 71d &:"$9gQ2v;C,(69.¥> #%HRR &k!/2#%G@(69 %69. .^z%6<;C "!'>,2/G9gQ2!"&:71r&2!"pRV"";C,R</694"Q;C &(69 . su69RR. ";Hgm86;Cs,\(^!869.!,.U69&28^5X&2;"";D32( n ML9g`"01,(69&) Z ^ `2^ c 5!869&<69.$WVM'eCR"";u65gh. .^zp
H · ∆wn,mi,j = C0∆w
n,mi,j + C−
1 ∆wn,mi−1,j + C+
1 ∆wn,mi+1,j + C−
2 ∆wn,mi,j−1 + C+
2 ∆wn,mi,j+1 ,
Z ^ `2^ g4^|2S2S!":B!&:
C±
i
69igm#B69. . sG,&2!/(69&r"; ^| ~"RV"!'$2S86;Csv!86B¨W,z&2S2;'c2& &<9g
σp|&2^§W(6";H&H2S;C,(69.D #%'Ra|69&2;
C0z. 4:. sG#%:;C¥c(";~ &2!"i | &2!. ,2;"
c e
&2^2M/69λ
WV ^"&u2iROsO!869.¬69&2;~;C,(69._ #%'R2p
C−
1 = −σ1
(12A
E +D1 + 1δxA
V1
)n,m
i− 1
2,j
= −σ1 (A+)n,m
i− 1
2,j
C+1 = σ1
(12A
E −D1 − 1δxA
V1
)n,m
i+ 1
2,j
= σ1 (A−)n,m
i+ 1
2,j
C−
2 = −σ2
(12B
E +D2 + 1δyB
V2
)n,m
i,j− 1
2
= −σ2 (B+)n,m
i,j− 1
2
C+2 = σ2
(12B
E −D2 − 1δyB
V2
)n,m
i,j+ 1
2
= σ2 (B−)n,m
i,j+ 1
2
C0 = (1 + 32λ)Id+ σ1 (A+)
n,m
i+ 1
2,j− σ1 (A−)
n,m
i− 1
2,j
+ σ2 (B+)n,m
i,j+ 1
2
− σ2 (B−)n,m
i,j− 1
2
Z ^ `2^ n
NQRR. s: &v2@ #%'x6!"!,/69/$ #$R. ! M,RM^| &2;k!/2#%H Z ^ `2^ c i69M86!/bRV &:S9g2!"#$R,/69 &a69.;#B69 &_3D&2@W/69 &2S6B. &2869iW.:!/usO# #$ .U69[d2&2@'eOR//";< & Z ^_[C^ Zn '^M['&(!43D&2!869&,2v2H/69#%G/"69#%'&OB#$R.sr";gh%86;Cs)!86" &+;%b/.¥71G2v #%';RV&2;"&:sC#~^A*))@ L5!7b;H>" %&9<.f= 251]; #45I& ! 2i;C,(69.¥> #%SRR &@gm#T,.U69&H#B691"52[ #$R.#%&:/69&<9gt2i. ^§P\6!/vR"!"&2;C &&71sd86s1^Xx&2;"";D3269RR. sO &$2[R"!"&2;C &2 &B#B69¥eG@2iR2,2;9> #%S; 769 71M&(69W."|R"71z2z¥#%'!"&2&2!sl^| .2!"&:71r&2!"/69Q9gD2;C,(69.¥>R/1!""/ #$R7r";G69&2;B26!"!,/6!s\9g2!"#$R,/69&Ai&2,";\W:su2TR"!"&2;C &(";b&:,2#%' !869.;C R(69&_^ :,2i2gm#9g2M #%'x6!!,69p #$R. !' Q!/('#%S#B69 &2z2S/69#%%
H · ∆wn,mi,j = −∆τ · Rn,m
i,j , Z ^ `2^ d
^|
∆wn,mi,j = wn,m+1
i,j − wn,mi,j ,
Rn,mi,j =
3wn,mi,j − 4wn
i,j + wn−1i,j
2∆t+
(δ1(F
E − F V )
δx+δ2(G
E −GV )
δy
)n,m
i,j
,
H =
(P−1 +
3
2λId+ σ1δ1A
Eµ1 + σ2δ2BEµ2 − σ1δ1(D1 +
AV1
δx)δ1 − σ2δ2(D2 +
BV2
δy)δ2
)n,m
,
Z ^ `2^ Z er^|2 ^M"!869. ._(69
FE 69&2;GE 69M2MR"!"&2;C &2";<!"&:71"! 7r \2,Ce"p£gh#2S:' ,1.
!/2#%igm| &2/69&2!" ^| .D1
69&2;D2
69M2MR"!"&2;C &28;~&:,#%!869.o;C R(69&~#%8^N[S#%&:&2";869. T &2d"!&;";C!869";~.^Pb6!/R"!"&2;C & &2232BB!&2!s69&2;2zW,(&2"9gD2R"!"&2;C & &#%2:;%gmK(z,&286;Cs%"0:,(69&2/'. s%&B2769. ,2Q9gD2!,- M,2";~ &u2p;'c2& &\9g`2
βR(69/69#%8^5±&469" ^569/69& 2^";u & n/Z (69
i&2"!""/69suG#%:;C¥ghsu2$;'c2& &A9g2 !, ^|2&kr &Ggh#86;Csuv,&286;Cs \(^RW.#%8^KZQ &p±&GJ[,#B69&&H69&(69. sO8312s%R71";B(69586;Cs%R"!"&2;C & &2R(69!,.U69. s &2B!&:gmK!"#$R, &T,&(86;Cs \(^z8^Xx&$2!/(69R;";C!869";BM(|±&BJQ,#B69&&G69&(69. sC83^ ^| . ._",#%M2 i^G &~2S!86M9g2M #$R. !' QPb69¥eO]/S#%'2:;D^
c/Z
' & (1& ( &::;C,2!sG!/(69R(6"!869. .";G2[;" 4&GR &2! R."ghW, .;C &%6pW.:!/% #$R. ! 5,RM^| &2;!/2#%43D,2!"!""/ 7r. s~gh[286;Cs<!"#$R" W.$I`,.S69&2;bJi687:-YL:1"M"01,(69&23V2&b2 .^5Pb6!/A&:,#TWVG¨R"!"&2;C &(";(M71&69&2;kc2&(69. . s\ &A2$,&286;Csb.^5P\6!/&:,#TWVT!86869";b,2 &~6G;C,(69. #%'Rb"!/&0:,24^i &Ggh# 286;Cs<!"#$R"/ W.$I`,.S"01,(69&2%2S,&286;CsH.^5Pb6!/<&:,#TWV[J[687O-YL:1"Q0:,(69&2[;:"&2#%:;C¥g¨svgh#B69. . sH2S. &2869W.:!/GsC#¯(69|#T,25WVM. 71";H69|86!/H /69&~ &~;|W/69 &<6T&2h^§769. ,2[gh52i71"! &2!#%&:|9g¬!"&271";d7469U69W."
∆w^Xg)^Q"!|,. 71"6p"'>RV &:5RV5;¥/!&vc2-
;. &2869 ´69&¢9gQ2d &:7O!;)&:,#%!869. \2,Ce"83`2dW.:!/ #$R. ! $/!/2#%G86;¥&+69. .52!86"S
H · ∆wn,mi,j = −∆τ · Rn,m
i,j ,
^| <69&~ #$R. ! QRV'699g2igh#
H · ∆wn,mi,j = C0∆w
n,mi,j + C−
1 ∆wn,mi−1,j + C+
1 ∆wn,mi+1,j + C−
2 ∆wn,mi,j−1 + C+
2 ∆wn,mi,j+1 ,
^|2C±
p
3C0
694 × 4
¨ & [jSTW.:!/#B69!""$W, . g¨#³2G &:7O!;69&2;7O!",2 6!"WU69&#B69!""TR. ,2p2%R"!"&2;C & &2<#B69¥ebgmS.^5Pb6!/&:,#TWV \(^z@69&2;A2
λR(69/69#%Tgh
;C,(69.¥> #%i!"#$R,/69&_^X5^| . .VWVQ2^|&G &G!/(69R · (692 &: &2
H ``t ¯
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c4·
c `
W:sb S#%%"!"&:T /69 7rB7r&)69&2;&2;C &u,RW^| RV &:Tp. &2'>.U6e69&69RRr6!/_`2!"&&2"!&WV ^"&Agm6! ´69&69&2;k.U6e69 &"!/&0:,2" ^| . .KWV$,&2;. &2";A,42,T!/(69R|6I^. . 6 &G2&2'eC&243C;7r";dT2±&GJ[,#B69&&H69&(69. sO9g_2Q #$R. !' |. ,&#%2:;z('/M7:h^";D^
! "$#%" #% " #%& # (N[pR7O,2. sA#%&:&2";D369RR8eC #B69dgm6! ´69&#%2:; ^d2Bc2@,2!"!""g¨,. #$R. ! #%2:;i,2";< &]j@^ 'gmT2@#$;O Zdbn eS69&2;\2@;71.R#%&:S9g,RM^| &2;\ #$R. ! M/!/2#%"832,29g!"&:/69.;C &2!"%!/2#%"i #$R. ";6d.U6!/\9g;CU69r&(69.;#$ &(69&2!"%9g2#B69¥e
M9g
2Q. &28695sC# Z ^_[C^ Zn ^|!/G#B6;Q2[/69&2;269;d.U6e269&G"!/&0:,2",&2/69W.4^ (695^|:sg>6! ´69&\"!/&01,2"Q#%i6$4869[,2!"!"",&: .¬"!"&:Qsr869^5X&uQ/"!&_3( ^l^|;. s:>,2";69RR8eC #B69zg>6! ´69&d#%2:;569zR"&:";D2Q69. &(69Q;C "!&G #$R. ! #%2:;\hNQjiX69&2;2%#%:;C¥c(";69RR8eC #B69%g>6! ´69&)#%2:;hP\N[]K c ` 69.u#% #%"@'gh";<6ji#$ &(69&:QjQU69r&(69.¬N[jiXzjijNQjiX _a2` ^@)%A+)%A 7b; #f7# 3 L# #f; .7# & 1435 # .6# 7'9 ;67="&?> 2@69. &(69 &v;C "!&\ #$R. ! [#%2:;hNQjiX'3a69.B!869. .";\69RR28eO #B69pg>6! ´69&\#%2:;hN[]`3^56|,2!"!"/-g¨,. . sd,2";v¥&<]jgm52[c25¥#%MW:sd5869#69&2; 69#$ & · 69&2;H¥.sH69&2;Pb6!"j[&(69.; n ^Q](i/69&2;269;\!"#$R" W. \(^z83V2p#B69¥e
M9gK2T. &2869isC# Z ^_[C^ Zn z
69RRr6!/2";uW:sd2iR2:;C,2!9g ^$¥;CU69r&(69.¬#B69!""832W,Q69RR. sO &%z#%2:;~.^|P\6!/&:,#TWV \(^zi"0:, "p69&A6;;¥ &a69.;CU69r&(69.K#B69¥ebG(69M2@g>6! ´69&AM!869";A,S6gm. .^zS
M ≈M1 ·D−1 ·M2 [C^ Z ^ Z
^|
M1 =
x x · · · · ·x x x · · · ·· x x x · · ·· · C−
1 C01 C+
1 · ·· · · x x x ·· · · · x x x
· · · · · x x
, M2 =
x · x · · · ·· x · x · · ·x · x · x · ·· C−
2 · C02 · C+
2 ·· · x · x · x
· · · x · x ·· · · · x · x
, [C^ Z ^_[r
69&2;
D =
x · · · · · ·· x · · · · ·· · x · · · ·· · · C0
0 · · ·· · · · x · ·· · · · · x ·· · · · · · x
, [C^ Z ^ ·
cbc
^|2M2S;CU69r&(69.¬!":B!&:C0
1
3C0
2
69&2;C0
0
69S;'c2&2";~W:s\
C01 = P−1 + σ1
(12A
E + D1
)i+ 1
2,j− σ1
(12A
E − D1
)i− 1
2,j
,
C02 = P−1 + σ2
(12B
E + D2
)i,j+ 1
2
− σ2
(12B
E − D2
)i,j− 1
2
,
C00 = P−1 .
0 &2!869&\&2!"@(69[gm[!"#$R"/ W.D\(^zi2TR"!"&2;C & &2H#B69¥eP
WV"!"#%"M"0:,(69.oG2;&: xs+#B69¥e¢A(69G&2~"!"71d2u/69&(;269; NQjiXTgm6! ´69&_^ 0 &2!"u2~gm6! ´69&RV-gm#%";D3(i. &2869QsO#!869&~WVS. 71";v &~"SR265gm. .^zp
M1 · [∆w∗] = [Rn] ,
[∆w∗∗] = D · [∆w∗] , [C^ Z ^ `:
M2 · [∆wn] = [∆w∗∗] ,
^|2∆w∗
69&2;∆w∗∗
69R7O&(69.:7469. ,2"¬";p"RV"! 71. sp6gh¬2`c2o69&2;p2"!"&2;TR_^](!869sO &B,2ic269&2;~ ;uR283(69. ._z&2"";";~z2M &:71&<9gK6¥;CU69r&(69.¬#B69¥e 3^|!/A!869&bWV@86 . s~|7O ´4^p69S6d";C,2!"";k!"4Q6!/71";\,2 &~6 n Zg>6! ´69&_^TJQ712."832%N[jiXQ#%2:;AR:;C,2!""@6Gg>6! ´69&) ^|!/!869&A. #$ p2B/69W . xsk9g52%#%2:;A &#%i!86"z69&2;G#%4|9g69. ._6;C71. sG6 D"!52[ &: &2!MB!&2!sv9gt#%2:;D^ 2i/69W . xs69&2;\!"&:7rr&2!"@RRV"M9gK2TN[jiX/!/2&01,269T&2^ ^. .o"/69W. 2";b69&2;u,&2;:1;h"gm &2/69&2!"_a2` 'O¥&G2&2'eC!/(69R83^0^z¥. . ,#$#B69 ´843C,4H6S±&GJQ,#B69&&H69&(69. sC83:2#B69 &'^869O&2"|9g¬69RRr6!/_3&(69#%. sd |.U6!/d9go &: &2!MB!&2!sr^ n |,25"!869. .D2^71|2N[j[X5869#%&:S69RR. ";<G6%W.:!/~ #$R. ! i/69rT!869&\WVp#B6;@;CU69r&(69.oRV[;C "!& _a c ^|!/69. .^zz% 4&¥c(!869&:. sv";C,2!"M z,& !"48^
@)%A+)@ 9<&?>#5; > 33 #& # 14 7b; .7b&&#f#67b#%& 9 ;67="&?>
2/69&2;269;bN[j[X|#%2:;b!869&bWVT#%:;C¥c(";b &b;id";C,2!"2pg>6!'/ ´869 &AS69&2;\:,2 #$R71 i &: &2!$B!&2!s\RRV"3_ &bR(69!',2.69 ^|2&k.U69r$ #%'R2ST] n &:,#TWV69,2/; c a ^ 5!869&dWV[6!/71";BW:sd69RR8eO #B69 &2Q#B69¥e
M6:!U69"; ^| d2 #$R. !
/69rp65gm. .^zS
M ≈M ′1 ·D−1 ·M ′
2
[C^ Z ^ c
c a
^|
M ′1 =
x x · · · · ·x x x · · · ·· x x x · · ·· · C−
1 C0 C+1 · ·
· · · x x x ·· · · · x x x
· · · · · x x
, M ′2 =
x · x · · · ·· x · x · · ·x · x · x · ·· C−
2 · C0 · C+2 ·
· · x · x · x
· · · x · x ·· · · · x · x
,
D =
x · · · · · ·· x · · · · ·· · x · · · ·· · · C0 · · ·· · · · x · ·· · · · · x ·· · · · · · x
.
[C^ Z ^_ar
(69&C5|#%:;C¥c(";vg>6! ´69&_3&2&C>´8%!":B!&:Q9g¬2i#B69¥eM
69i.'gh5,&2!/(69&r";D3^| .H & U69. . s)´8k!":B!&:d!869&+ . .|WVv#%:;C¥c("; ^|!/#%869&2d6<gm6! ´69& $% . . &::;C,2!"";%t K`2^71#T,2!/%";C,2!"";Q^| %"RV"!'M2zW(6. &2NQjiXo69RR/r6!/_31 &%R(69!,.U69^|2&H.U69ri #%'R269[,2";D^ 2[#%:;C¥c(";u69RR8eC #B69[g>6! ´69&hP<N[]KK5RV-gm#%";v &"SR2p
M ′1 · [∆w∗] = [Rn] ,
[∆w∗∗] = D · [∆w∗] , [C^ Z ^ g4
M ′2 · [∆wn] = [∆w∗∗] .
2|P\N[]<#%'2:;$K69.M!869. .";%jijNQjiXt#%2:;$M#$R(6 ´8|2g>6!`(69`2|#B69!""K &:7r. 71"; &k2@gm6! ´69&R:!""p69$;CU69r&(69.;#$ &(69&: Z [ ^ 2±&kJ[,#B69&&69&(69. sOMR"&:"; &B2&2'eC!/(69R5^| . .V;#%&2/69Q2QP<N[]A69RRr6!/GR7:;"56MWV!"&:71r&2!"Q/69za69&2BW(6!dN[]@NQjiXS69RRr6!/_32,4 . .g>69Sgh# 2B &: &2!dB!&2!s)6:!U69"; ^| 6;C "!/. ,&u9go2M #$R. ! Q/69r4^
@)%A+)*% 7b; #7# -; ./&&# #; .7# & &( ( .7b&&#f#6 7#%& ; # #& #N[j[X69&(;HP\N[]#%2:;z &O:;C,2!"T6Tg>6! ´69&<i^|!/u!869&~WVS'eOR2"";~65gm. .^zp
M1 ·D−1 ·M2 = M + PADI ,
M ′1 ·D−1 ·M ′
2 = M + PMAF .
Xx&B2#%8312Q6!,(69.asC#. 71";+^|2&B,2 &T6igm6! ´69&G#%2:;%K25gm. .^| &T&2p
M · [∆wn] + P · [∆wn] = [Rn] .
c g
Xx&~;z$. #$ &(69S5g>6! ´69&<83(6$,WC> /69 71MR:!""!869&~WVS'#$R2. sr";[∆w(0)] = 0
l = 0, p
M · [∆w(l+1)] + P · [∆w(l+1)] = [Rn] + P · [∆w(l)] [C^ Z ^ n
[∆wn] = [∆w(p)]
^|2∆w(l) = w(l) − wn ^ 2&~2M#%2:;~(6z!"&:71r";D3 ^|2&
w(l+1) = w(l) 3C2ig>6!- ´69&%7469&2"`69&(;@25'e6!`. ,&%9g(2. &2869KsC# Z ^_[C^ Zn ¬W/69 &2";D^`X&R/6!!"432R7:,2p69. r # ^| &k &6G;C D&: ^568s13DH6i~687r;b2'eCRV&2 71769. ,(69&9g2igm6! ´69&\
POgm| &2/69&2!"S2i /69 71SP\N[]69. r #|/692Q'eOR"/";~6S
[∆w(0)] = 0
l = 0, p
M ′1 ·D−1 ·M ′
2 · [w(l+1) − w(l)] = [Rn] −M · [w(l) − wn] [C^ Z ^ d
[∆wn] = [w(p) − wn]7r&~#%S'eOR. ! . sk
[∆w(0)] = 0
l = 0, p
M ′1 · [w∗ − w(l)] = [Rn] −M · [w(l) − wn]
[w∗∗ − w∗] = D · [w∗ − w(l)] [C^ Z ^ Z er
M ′2 · [w(l+1) − w(l)] = [w∗∗ − w∗]
[∆wn] = [w(p) − wn] 2@ /69 7rP\N[]+#%2:;\"0:, "[2/69#%T¥;CU69r&(69.`#B69¥e\ &:7r&2i(69&\2@/69&2;269;P\NQ]¢"!/&01,24^TP<"71i2"@ &:71&2M!869&bWV;&2Q*Y,(i&2!"$69i2@WV4 && &~9g2,WC /69 71R:!"/8^ 27r-!"4o9g2K /69 71#%2:;p¬&. sM;C,2z2!"#$R,/69&$9g2ML69p86!/,W>¥/'69&_3V^|!/A/'#B69 &2p71s\!/("69Rt^ 2%!"&:71r&2!"%/69%¥#$R/7r#%&:TR7:;";W:svP<N[] ^| . ._WVp6"/";G,4\6@±&~J[,#B69&&<69&(69. sC| &~2M&2'eO!/(69R8^
! &" "$# & # (Xx&v2i#$;O 4:"83(Pb6!#B6!/GRRV4";H,2i.U6e69&~"!/&0:,2"z &H/;| &2!86M2B!&2!s9g| #$R. ! @869#%&: c [3 c4· ^vq`@~(69TRV:;D3o2B #$R. ! n MLkRV/69 ^56r&2/69. . s< &:7r";\,2 &G2TR7O,2. suR"&:";bN[]¢MNQjiX5gm6! ´69&k#%2:;D^ 2@,[email protected]&u#%2:;0^56zR#%";~WOsG2S #T,. /69&2",2[;7r.R#%&:[9g #$R. ! !/2#%"^|!/)&2,";2B;CU69r&(69.5;#$ &(69&2!"d9g|2BsC#[71&2,4)#%B,2;C"p(6871%R71";W.:!/:> #$R. !' |!"&:";v!/2#%"9g n 6eO"A&2;C<xs:RVQ!",.;v69.pWVQB! &O. sB. 71";B,2 &T24/69#%S. &2'>.U6e269&\"!/&0:,2" Zd Z a ^zL: &2!"p(69 #%43(#B69&:s~,2;C"(6871SWV"&<;";C!869";<2"z#%2:;83O69&2;D31#%z"!"&:. s131±K&469" ^569/69& Z [ !869";B,6 ^|;,2;Cs@/p69&(69. s:´82MRV-gh#B69&2!"T!869R(69W . [9go #$R. ! ,RM^| &2;u!/2#%". 71;HW:sd2"/i"!/&0:,2" d _[b[ ^Xx&H"!& ^M &::;C,2!"S2SR &2! R."Q9g2M.U6e269&u#%2:;[69&2;H2& ^S;869.)^| u2#%4zRVR,.U69Q9go2#~^
cbn
@)@ )%A #f# .6# 3I ; ! > 3 #&)3i; #b7# ; !N[p69. ^568sC83t,T/69 &uRV &:pS2$. &2869TsC# 6/:!U69";L^| A2% #$R. ! p,RM^| &2;/!/2#%;"! WV";~ &H2MR7O,2!/(69Rp
M · [∆wn] = [Rn] ,
Nr&2!S.U6e269&~"!/&0:,2S#B68sGWVM'eCR"";HW:sd2igm. .^| &BR
M ′ · [∆w(l+1)] = [Rn] − (M −M ′) · [∆w(l)] , [C^_[C^ Z
^|2∆w(l) = w(l) − wn 69&2;
M ′; 71"g¨# 6\R(69!,.U69BR. &A9gi2v#B69¥e
M#%
R"!. s13M ′
M!/24&kH6iv&2,2$69WV71@. &2869SsC# ^| . .`WV86Mv. 7r@(69&k2 & U69.5&2<@xsOR!869.¥. s13
M ′ @!/24&66~;CU69r&(69.#B69¥e &2!"G6~;CU69r&(69.|sO# !869&WV
. 71";\ghp6G";C,2!"";!"4M,2 &<6GR(69!,.U69 n Zgm6!¥´69&O&2^|&6 2#B6T69. r # ·/Z `Mg ^MXx&b;iG¥&(!'/"6/@2@ &: &2!B!&2!s\9g.U6e69&k#%2:;83D [i!,2#B69suvR. M
&H;C D&:]^568sC83C!""RV&2;C &$gh5 &2/69&2!"i$69. &(69M. &2i @69&2;vs:#$#%!M ^"R28Ogh &2/69&2!"[email protected]&~"!/&0:,2M#B69O &%,2S9go ^% ^"R2i^,.;H86;A
M ′1 · [∆w∗] = [Rn] − (M −M ′
1) · [∆w(l)] , [C^_[C^_[rM ′
2 · [∆w(l+1)] = [Rn] − (M −M ′2) · [∆w∗] ,
^|2∆w∗
6@R7O&(69._7469. ,2p6g¨z2ic2|.U6e269&< ^"R_^LO &M ′′ = M ′
1 +M ′2 −M
32"M ^%R2z!869&~WVS!"#TW &2";<69&2;'^| &<6p
M ′1 ·M ′′−1 ·M ′
2 · [w(l+1) − w(l)] = [Rn] −M · [w(l) − wn] , [C^_[C^ ·
^|!/\M69&< /69 7169RR/8eO #B69Tg>6! ´69&b71&b9gK2T. &2869MsO#~^QIeCR"& [C^_[C^ · #B691"Q!.869Q2p!.4/p!"&&2"!&\WV ^"&\.U6e69&\#%2:;i69&2;u /69 7r@69R2R8eO #B69pg>6!- ´69&b"!/&0:,2"83_6[2^|&\ & Z [ ^iX&b"01,(69&+ [C^_[C^ · '3
lR"&:S6 ,W p /69&A!",&:
^|!/$!""RV&2;Q#@,. R.5 ^"R29g(2. &2869`sO/'#~3$^| .n
;&2"2&2&C>. &2869K!",&:9gD2 #%'>#B69!/ &2@R:!"";C,4^KXx&BR/6!!"3
w(0) & U69. ´8";v6w(0) = wn 69&2;
p,WC> /69&2
69[!869";G,8^V2&d2Q#%2:;d(65!&O7rr";D3w(l+1) = w(l) 69&2;d2[ML$9gt"0:,(69&k [C^_[C^ ·
M69-c(";Dwn+1 i2&A/M"0:,(69.`
w(p) 69&2;k2$. ,&kp6;C769&2!"";kv2&2'eCi #%%R_^ 2pB!&2!su9g`2S.U6e69&<#%2:;u. "Q&u2p&:,#TWV[9g2T,WC> /69&2Q(69[69p&2"!'"/69sHd69RR/r6!/\2Te26
cbd
^|2M2M.^Q69&2;H,RRV#B69!""Q6/:!U69";'^| u86!/uR(6!"S;C "!&\69M4 7r&~W:sb
L1 =
· · · · · · ·x · · · · · ·· x · · · · ·· · C−
1 · · · ·· · · x · · ·· · · · x · ·· · · · · x ·
, U1 =
· x · · · · ·· · x · · · ·· · · x · · ·· · · · C+
1 · ·· · · · · x ·· · · · · · x
· · · · · · ·
,
L2 =
· · · · · · ·· · · · · · ·x · · · · · ·· C−
2 · · · · ·· · x · · · ·· · · x · · ·· · · · x · ·
, U2 =
· · x · · · ·· · · x · · ·· · · · x · ·· · · · · C+
2 ·· · · · · · x
· · · · · · ·· · · · · · ·
.
[C^_[C^ `:
@)@ )@ 7b; #f7b; 0# ";V./& 5# #; 7# & 3 #&?./; > 2 #; i^. .¥>:&2^|&u.U6e269&~#%(1;~. "&~2igh. .^| &dR. &%#B69!""T
M ′1 = L1 +D + U1 , [C^_[C^ c
M ′2 = L2 +D + U2 ,^|!/!""RV&2;"RV"! 71. s<2G.U6e69&+9gQ'eO/6;CU69r&(69.|,&:&2^z&) &2!#%&:
∆w &
2y
;C "!& ^|2&b6%sO#. 71";'^| M ′
1
6 n MLv#B69¥eu69&2;u%2S.U6e69&\9gK'eC/6;CU69r&(69.K,&O&2^|&k &2!#%&:
∆w &k2
x;¥/!&W^|2&k2$sC# S. 71; ^|
M ′2
^ 269. &(69'>. &2 6!"Wo69. r #~3(;&2S286g¨
ALJ(p)3z'eOR"/";~65gm. .^zS
[∆w(0)] = 0
l = 0, p
M ′1 · [∆w∗] = [Rn] − (L2 + U2) · [∆w(l)] [C^_[C^_ar
M ′2 · [∆w(l+1)] = [Rn] − (L1 + U1) · [∆w∗]
[∆wn] = [∆w(p)] 2vc2%R¢!""RV&2;%A6b. ¥&(> #$R. !x
^"R_3^| .H2~"!"&2;+R+!""RV&2;BA6. &2'> #$R. !
y ^"R_^ 2d!"4pRV@,WC> /69&9gzp#%2:;p&2@u'eCRV&2 71B &2!"d&2
#$R. su&2"";iG &:71S6d #$R.@;CU69r&(69.KW.:!/<sC# &k86!/kR(6!";C "!&_3^|!/A!869&bWV86 . su6!/71";uW:sH2T,2p9g 2#B6M69. r #69&2;u(6QBRV-gm#%";\&. s~&2!"pRVQ&2&C>. &2869 /69&<6` &%2!"6/z9gV /69 71!""!&Bgh69RR8eC #B69|g>6! ´69&_3:&2.¥s2zML9g [C^_[C^_ar(687rQpWVQ"!"#$R,";H69586!/
l,WC> /69&_^Q^7183:2Q4.W(69. !"459g_2Q#%2:;
^| . .o#B69 &\.^&. s~¥gK2p/69.o&:,#TWVp
9g,WC> /69&2i&2[:B.U69r4^ [RV &: ^| . .oWV!869'gh,. . s\69&(69. s:´8";\ &k!/(69R · ^MXi!869&A69. 86;CsuWV/69";<(69
ALJ(p)[r&2/69. . s<&2iR¥#B69.
&)#%$9g4.W(69.zB!&2!s%.U6e69&#%2:;$9giM69,(-YLO;.5xs:RV43`(69#B691d,2G9gQ2/69"@69. 86;Csk7469. ,(69";L^| &6H ^"R_369% &Ar&2/69.#%BB!&:83`6 ^| . .WV%2^|&A &2&2'eCz!/(69R,4u2i±&~J[,#B69&&<69&(69. sCz9g2p69#$R. ¥c(!869&ugm6!8^
abe
@)@ )*% 7b; #f7b; 0# "; *25!$! j; # >"; #; % 7#%& 3 #&?./; >2 #; 2p69. &(69'>. &2@M69,2/-YLO;._#%2:;HRV-gm#%zsO#$#%!
x ^"R25gm. .^";HW:sGsO#$#%!
y
^"R28^`j[, &$86!/v ^"R_3:([/69"I^|!/v(6871i69. 86;CsBWV"&v7469. ,(69";H69&2;,^|24Q&2h^¦769. ,2"69T687469 .U69W.@69S,2/";~B!"#$R,p2p&2'eOQ/69"8^ .86;Q%2T69. &(69'>. &($M69,2YLO ;2'.69. r # hN n SL
p59g2igh#
[∆w(0)] = 0
l = 0, p
(M ′1 + L2) · [∆w∗] = [Rn] − U2 · [∆w(l)]
(M ′1 + U2) · [∆w∗∗] = [Rn] − L2 · [∆w∗] [C^_[C^ g4
(M ′2 + L1) · [∆w∗∗∗] = [Rn] − U1 · [∆w∗∗]
(M ′2 + U1) · [∆w(l+1)] = [Rn] − L1 · [∆w∗∗∗]
[∆wn] = [∆w(p)]
N[ ^| . .oWV@2^|&\ &\2T&2'eCi!/(69R83 iN n SL<69. r #R7:;"S6BWVi!"&:71r&2!"/69(69&2BN n_ 69. r #~^HQ^71836T69. 86;CsAW271";A & Z [ 3¬ ^| . .569.HWVB2^|&(69p269WV71[R:!"";C,M|&. sG!"&2;C &(69. . sH/69W.[?^|!/H#B68sG"#/692z,R &% &2!"
ALGS(p)G6^|;. s:>,2";¢. &2'>.U6e269&R:!"";C,<W, ^ ^| . .Q!"#$#%&:v& BRV &:d.U69G&_^ 2ALGS(p)
#%2:;\69. .^zz/$"!"71'2S.7r.¬9g &: &2!pB!&2!sHR7O;";~W:sH6%;C "!'[. ,&9gQ2v #$R. ! B/69rdgm%6<7rs#B69. .|&:,#TWV
p9gi,WC> /69&2852GR!"v\R(68sgh$,2!/+6
4\ &:¥&(!@B!&2!s<Q2@&2"";<d &:714
;CU69r&(69.KsC#%Q & [C^_[C^ g4[69i86!/\&2&C>. &2869 /69&_^<NQ&)69. &(69¥71G694s^,.;WVBu,2G6H/'.U6e269&)#%2:;(69;:"T&2T"0:, 69&:sdsC#® &:7r&_32692i'eCRV&2[9g6pRW(69W. sd#T,2!/v.^| &: &2!iB!&2!s137r&Ggh.U69r7469. ,2"59g
p^ 2iW(69.U69&2!"[WV ^"&H5RV:5 &: &2!MB!&2!sv69&2;G2[R(69!,.U69. sG.^¦,& |!"4
9g2S69RRr6!/u#B68sG.86;[email protected](69. . sHB!&:Q/694s1^@)@ ) &V#%V7 V./& I# #f; 7b#%& 3 #f&?./; >2 #; 2Sq &: 6!"W¬q .U6e69&u#%2:;u|R"!/.¥sH2S #$R."Q!&(! 7469W.p.U6e69&u"!/C&0:,2z &2!"| K/69 &2&. s$;CU69r&(69.(#%69K2|&2h^¢.7r.
(l+1) &%2 n iLVr25gm. .^| &T!/2!"
gm|#B69¥eM ′
M ′ = D , [C^_[C^ n
.86;z(iRV &: 6!"W¬69. r #hq pM
[∆w(0)] = 0
l = 0, p
D · [∆w(l+1)] = [Rn] − (L2 + L1 + U1 + U2) · [∆w(l)] [C^_[C^ d
[∆wn] = [∆w(p)]
Ji69,/69. . sr3(2SRV &: 6!"W69. r # 'eORV"!";u%(687rT6#@,2!/uRV:[!&O7rr&2!"p/69Sa69&ALGS(p)
t2501,(69&:¥c(!869&B9g( &: &2!5B!&2!sT.4j^| . .CWVRV-gh#%"; &2&2'eC!/(69R,4¢](,d69&(69. sO^)P\"7r83±&+JQ,#B69&& 69&(69. sC ^| . .z7r869.|a69
PJ(p) &g>6!
,&2/69W.5^|2&@ ¬¬,2/";p &@!"&*-,&2!& ^z¥6W.:!/S #$R. ! `!/2#%4^oJ[712."83969. r #^| . ._R71i$WVp6@7rsv!/2869R~#%2:;~%. 7ri2M#B69¥eOmg¨"S #$R. ! Q!/2#%4^
a Z
@)@ ) V"1 14;67#f#%. &#%V7 *25!$! j; # >"; #; 7#%& 3 #&"./; >2 #; 2@qo &: 6!"WK69. r #!869&k69. /BWVT #$R7r";\,2 &H6GM69,2-YLC';.xs:RVTR:!"";C,v|/69"7469. ,(69";;C, &\6~ ^"R)69$ #$#%";CU69. sA,2";A~!"#$R,B2%&2'eOT&2"8o#%"71@6v71 ^"RRV-gh#%";\ &2!86v2HB!&2!s9gQ2v#%2:;D^&2/"01,2&:. s132v!/2!"H9g[2#B69!""[6:!U69";'^| u,2!/<6@R:!"";C,Sz2[gm. .^| &B&2%
M ′1 = L2 + L1 +D , [C^_[C^ Z er
M ′2 = D + U1 + U2 ,
^|!/~.86;z$2Ss:#$#%!SRV &:[M69,2-YLO;.o69. r # >LSL pi
[∆w(0)] = 0
l = 0, p
M ′1 · [∆w∗] = [Rn] − (U1 + U2) · [∆w(l)] [C^_[C^ ZbZ
M ′2 · [∆w(l+1)] = [Rn] − (L1 + L2) · [∆w∗]
[∆wn] = [∆w(p)]
Xx5#T,2|WV ^. .D,&2;::;H(69z2p69WV71S69. r #;:"z&2| &:/1;,2!"p69&Osv6!,(69._ &O7r&u9g;CU69r&(69.¬sO#%Sµ`Xxz!",.;u69.$WV ^| /'&\6S
[∆w(0)] = 0
l = 0, p
D · [∆w∗] = [Rn] − (L1 + L2) · [∆w∗] − (U1 + U2) · [∆w(l)] [C^_[C^ Z [rD · [∆w(l+1)] = [Rn] − (U1 + U2) · [∆w(l+1)] − (L1 + L2) · [∆w∗]
[∆wn] = [∆w(p)]
& ;dA#B69ru!.869d2Hg>6!B(69B2u0:,(69&: xs(L1 + L2) · [∆w∗]
&¢2Hc2B ^"R !869&WVp!"#$R,";<,2 &B2p&2h^/69/
∆w∗W:sH #$R. sH/69O &G6;C7469&:/69rT9gK #$#%";CU696!,(69. ´69&
^| &u2S ^"R_22S/69#%S!"#$#%&:2.;Q!"&2!"& &B2p01,(69&: xs(U1 + U2) · [∆w(l+1)]
&~2"!"&2;~ ^"R_^L: &2!"i2[c25#B69¥ev,2";G & [C^_[C^ ZbZ z6T.^|U69&4,.U69z#B69¥e~69&2;G2M"!"&2;H&2iz69&v,RRVU69&4,.U69[#B69¥e 3([69. r # #% #%"Q'gm";~d6 n ZYLSLV^aZQ &d'eCR"&) [C^_[C^ · '32p69WV7rTLSLG.U6e69 &uR:!"";C,T69./86;p
M ′1 ·D−1 ·M ′
2 · [w(l+1) − w(l)] = [Rn] −M · [w(l) − wn]
^| <69&<6:!U69";u69. r #(69!869&~WVS'eCR"";~6p
[∆w(0)] = 0
l = 0, p
M ′1 · [w∗ − w(l)] = [Rn] −M · [w(l) − wn]
[w∗∗ − w∗] = D · [w∗ − w(l)] [C^_[C^ Z8·
M ′2 · [w(l+1) − w(l)] = [w∗∗ − w∗]
[∆wn] = [w(p) − wn]
ab[
.869. s13H^|2&\6$ &4.p,WC> /69&<zRV-gh#%;_322 ^. .¥>:&(^|& n Z69RR8eC #B69Sgm6! ´69&#%2:;uz"!"7r";D^ z#B69CQ69 #%";<69zRV &: &G,Q&2!"T69169 &u2p&B. &vWV ^"&g>6! ´69&#%2:;<69&2;¦.U6e69&R:!"";C,"8^NQ ^| . .MWVk2^|&§ &§2b&2'eO~!/(69Ru&2GW(69gi6<±K&JQ,#B69&&¢69&(69. sC83`2~LSL)69. r # 69. .^zb;C/69#B69!869. . s) #$R7rv2!"&:71r&2!"S/69 ^| ~"RV"!z$2iW(6!Mq #%2:;D^
! ' & (1& (LO71/69.t #$R. ! [869#%&:Q(6871SWV"&<7:h^";D3a(69[!869&uWVS,2/";~d69RR8eC #B69@69[86!/u #%'RT2/. ,&@9g2W.:!/M #$R. ! !/2#%R"/'&O";p &T2`c2¬!/(69R9g¬2"8^¬j[ 7O;C &2#WV' ^"&kgm6! ´69&)69&2;b.U6e269&A"!/&01,2" ^,.;kWV/692T69¥c(!U69. &2!"$/69&2;269;g>6! ´69&"!/&0:,2"T69$&2 &uW,S /69 71g>6! ´69&"!/&2 0:,2d69RR. ";L^| 6H &4. /69&_ "#%T#%d &:" &b<;C &4,2 )WV ^"&)24/B869#%&:(69@#B69rd,2d9g;CU69r&(69.CsC#%t &:7r&25hNQ]|¨Ra'3P<N[]|¨Ra'39N n_ ¨Ra'34N n pL ¨Rat69&2;S24/K(69o;z&2¬"0:, 69&:svsC#¯ &O7r&hq ¨Ra3 n ZYLSL ¨Ra'^ 2i.U69/'869#%&:[69iWV,&2;~R7O;p6@/692RV: &: &2!B!&2!s$W,69|&2&('2. "/ &:" &pghK2 .^¢,& !"4"^|2862|gm#%869#%&:p69T&(69,/69. . s<#%@ #%!"&2,#$ &HW,M M6B#T,2!/bWVM!"&:71r&2!"@/694^pL:,(!/0:,(69. /69 7r!"#$#%&:o69`&(69,/69. . si&2t, B!&:t69. .^A2K."!&T9gC2`WV"_4.W(69. . sMB!&:W.:!/k #$R. ! T869#%&:@2&2!"%2B01,(69&:¥69 7rB±&J[,#B69&&69&(69. sCSRV-gm#%"; &2B&2'eO!/(69R8^
H A)` k` H ``t
#1 +#% " #% ' (P\4K9gD269&(69. sO!869.(:.K,2";%S,2;Cs@2!"&:7rr&2!"RV-gh#B69&2!"Q9g &:,#%!869.V!/2#%"69. #$ ";<B. &2869[RW.#%8^[Q^7183aRV-gh#$ &~6B/69W . xs<69&(69. sOQ&2[86s~7r& ^| b,2!/6\ #$R. ¥c(!869&_^NQ#%&b2G7469,2#%2:;B68769 .U69W.vb!869s),/6V3`±&J[,#B69&&69&(69. sCp@!69¥&2.¥s2B#%4T769. ,(69W.4^uXxppW(6/;&2d](,@/69&2-gh#9g|2B&:,#%!869.!/2#%v69&2; 69. .^zT/u:&2^ ^|22@2d&:,#%!869.|T &::;C,2!"";) &)2d. ,&)W:sA2;C! ´69&k69T69#$R. ¥c(";¨ &2/69W . xs2zQ;269#$RV";)h/69W . xs2 ·/Z `Mg ^5";"832±&uJ[,#B69&&69&(69. sCi!869&\WV@,2";<G" #B69@2^g>6[2@ ^| . .WV@;269#$RV";+hM69#$R. ¥c(";('3DR7O;C &~60:,(69&: /69 7r5 &Cgh#B69&%Q/69&67469xsT9ga #$R. ! K869#%&:K&@(W(69ga2 `!"&:71r&2!"/694^ 0 g2!",432`",. tW/69 &2";SW:sM,2!/T65#%2:;S(6871`|WVK!"#$#%&:";T!869'gh,. . s1^¬X&2;"";D36#%&:&B,RRV831±K&BJQ,#B69&&B2"s@"0:, "6[. &2869KRW.# ^|!/B#%"69&(K(69K2z!"1B!&:9g2@"0:,(69&2S69@!"&2/69&:8D#%"7183 RV:;C!@WV,&2;269s\!"&2;C &2S69$6,#%";< &b;i69. .^2S,2p9gK](,Q/"8^5X&uR p9g`2"S. #$ /69&283a#B69&:sH"869!/283a,2!/<6QP\:.4=?!"9 ^1i &\2@ZMLON Z `23 ZbZ 3 n/Z 69&2; n /698=?i869# &b]2/69&2!" ·bd 3 ` · 3 `b` 3V(687r@#$R.s1";\"!/&0:,2S%,2;Csv2S/69W . xs~69&2;~B!&2!sHRRV"[9g&:,#%!869.o!/2#%"&2&. s~69RR. ";M. &2869!869.U69"0:,(69&2KW,69. ^z2&%,2";S. 71525g¨,. .¬¨. &2869 ´8";(`IK,.J[67:-YL:1"8^ 2v/69W . xsm69&2;B!&2!s2T",. W/69 &2"; &@.U69%!86v69G &)r&2/69.5S69&2;) &R 9gS2b69WV71'>#%&:&2";¦"!&2v&2</!"RV<9gp±K&J[,#B69&&§69&(69. sCB#%869& &g¨,.[ghR/6!!869.D!"#$R,/69&2^ 2'gh43C!/(69R ^| . .VWVgh:!,2/";d&G2Q±K&vJQ,#B69&&~69&(69. sC9g2z #$R. ! 869#%&:R"&:";B &d!/(69R [C3^z2&B,2";%p. 71|2zW.:!/:> #$R. ! 5/69r9gD2:,RM^| &2;G!/2#%QR"&:";d &d2zc2!/(69R59g_2Q2" &d2Q!86Q9g_2. &2869 ´8";vI`,."0:,(69&28^ 2rr69.`9giR. #$ &(69sb,2;Cs<[v;&:¥g¨s<2&:,#%!869.K869#%&:MR7O;C &2~WV"B4.W(69.QB!&2!sgmd69. .¥RV""; \(^z83169;C."G9gi#%#%s:/69ru/,2vghB2H¥#%WV &2^ 2v."!W.:!/:> #$R. ! $869#%&:+^| . .|69&2;6$6<'gh&2!"vRV &:gh2G#B69¥eOmg¨" #$R. ! [869#%&:[ &::;C,2!"";\ &<2p&2'eO[!/(69R8^Xx&u2pR"&:[!/(69R283a2pR &2! R."i9gK2±&$JQ,#B69&&B69&(69. sCK69c2W \2s@,. &2";D1K69&(69. sC2&%69R2R. ";$[25W.:!/1> #$R. ! :'>xsORVz,RM^| &2;G!/2#%[69&2;B4.W(69. 69&2;B.:!869.V/69W . xs MB!&2!s%",. 569zR"/&:";BWVBgh2M'gh&2!"p;C "!. 7169&2;H2M /69 71M#%2:;z &::;C,2!"";~ &H2MR7O,2!/(69R8^
a ·
a2`
& " "$#%& # & # & ( n @,2@!"&2;@2d #$R. ! @,RM^| &2;!/2#%%gmT.^5Pb6!/),&286;Cs\(^zH Z ^ `2^ d S(69 ^56@;'! WV";)69p2d&2;9gz2$c2T!/(69R8^~N. .2@!86"Gh86;Csk &:7O!;\(^z69&2;/~&a'3`69.;"! WV";¢ & !/(69R Z 3|!869&¢WVu; 71";gh# d!/2#%<6ghd;C &:G #$R. ¥c(!869&28^ !/2#%M86;83,2 &%2M&2/69&2 &::;C,2!"";~ &u!/(69R Z
H · ∆wn,mi,j = −∆τ · Rn,m
i,j ,
^|
∆wn,mi,j = wn,m+1
i,j − wn,mi,j ,
Rn,mi,j =
3wn,mi,j − 4wn
i,j + wn−1i,j
2∆t+
(δ1(F
E − F V )
δx+δ2(G
E −GV )
δy
)n,m
i,j
,
H =
(P−1 +
3
2λId+ σ1δ1A
Eµ1 + σ2δ2BEµ2 − σ1δ1(D1 +
AV1
δx)δ1 − σ2δ2(D2 +
BV2
δy)δ2
)n,m
,
^|2
λ =∆τ
∆t, σ1 =
∆τ
δx, σ2 =
∆τ
δy.
Xx&+;%A69RR. s±K&¢JQ,#B69&&69&(69. sO83^H(687rHb. &2869 ´8~BsC#~^ %%;&2HW:sgh"´"¥&2 M696@!"&2/69&:z/69i2 6!"WU69&~#B69!""9g¬2i &:7O!;H69&2;v7O!",2I\2,Ce"z6 ^. .t62$;C R(69&k#B69!""83tG(69p69. .`2"@#B69!""p;G&2S;RV&2;A&kR(6!"&2M #%%69&:s:#%4^Xx&~,2!/u6$!8643C2MR:sC!869._ &:7:/!'; \2,Ce"z #$R. sG86;A
(fE)n,m = AE · wn,m , · ^_[C^ Z (gE)n,m = BE · wn,m .
&2; &B 42-;['eC&2&_32M&:,#%!869.t &:7O /!;,\2,Ce"69M. &2869 ´8";<65gm. .^zp
δ1(FE)n,m
i,j =[AE(δ1µ1 −
ε
4(1 − κ)δ3
1µ1
)+ D1
((1 − ε)δ21 − ε
4(1 − κ)δ4
1
)]· wn,m
i,j , · ^_[C^_[r
δ2(GE)n,m
i,j =[BE
(δ2µ2 −
ε
4(1 − κ)δ3
2µ2
)+ D2
((1 − ε)δ22 − ε
4(1 − κ)δ4
2
)]· wn,m
i,j ,
^| .M2M&:,#%!869.t7O!",2 \2,Ce"|86;A
δ1(FV )n,m
i,j ≈[AV
1 · δ21
δx+AV
2 · δ1δ2µ1µ2
δy
]· wn,m
i,j ,
· ^_[C^ · δ2(G
V ),mi,j ≈
[BV
1 · δ2δ1µ2µ1
δx+BV
2 · δ22
δy
]· wn,m
i,j .
Z[ &$2"S'eOR"/&283C2M. &2869 ´8";usC#!869&~WV ^| &~,&2;2[gm. .^| &$gm#
H · ∆wn,mi,j = −K · wn,m
i,j +3λ
2wn
i,j +λ
2∆wn−1
i,j . · ^_[C^ `:
a c
^|
K = KIII =3
2λId+ AE(δ1µ1 −
ε
4(1 − κ)δ3
1µ1) − ((1 − ε) ˙D1 + AV1 )δ21 − ˙D1
ε
4(1 − κ)δ4
1
+BE(δ2µ2 −ε
4(1 − κ)δ3
2µ2) − ((1 − ε) ˙D2 + BV2 )δ22 − ˙D2
ε
4(1 − κ)δ4
2
−SV δ1µ1δ2µ2
H = P−1 + 32λId+ AEδ1µ1 + BEδ2µ2 − ( ˙D1 + AV
1 )δ21 − ( ˙D2 + BV2 )δ22 · ^_[C^ c
^|2M2igm. .^| &%&2/69&20^M,2";A
AE = σ1AE , AV
1 = σ1AV
1
δx, AV
2 = σ1AV
2
δy,
BE = σ2BE , BV
1 = σ2BV
1
δx, BV
2 = σ2BV
2
δy,
SV = AV2 + BV
1 .
Xg(2c2-;K,RM^| &2;$/!/2#%5,2";D342. &2869 ´8";B'eOR. ! KRV/69 K /69r"25 #$R.gm#
K = KI =3
2λId+ AEδ1µ1 + BEδ2µ2 − ( ˙D1 + AV
1 )δ21 − ( ˙D2 + BV2 )δ22 − SV δ1µ1δ2µ2
Xx&$2|#B69 &2;9g K!/(69R283f^]^| . .2#B69 &. s@gm:!',(K&%2| ;O;'eCR. ! /69r KIII &2!"
2i&2Q,2";d &GR/6!!"4C2^7183O2Q #$R. ! |RV/69 H ^| . . WVQ2[&2[;";C,2!"";dgh#®2c2-;,RM^| &2;d/!/2#%@h" · ^_[C^ c '31gm!"486&v6'eOR2.69 &2";d &G!/(69R Z ^`N ghh^ ^;5^| . .WV%69;69WV,p2% #$R(6!T9g|p!/2!"43V^| 'gm'/'&(!BH2B!86+^|2B2$ #$R. ! @/69r$!"&2&:i^| \6Tc2-;Q'eOR. ! Q/69rM;'c2&2";~W:s KIII
^
& " ( Xx&;@<RV-gh#³6u±K&)J[,#B69&&69&(69. sC@9gz2dr&2/69.. &2869/!/2#%\ · ^_[C^ `:'3` T#T,2TWV/69&2-gm#%"; &Ok2~](,BRa6!4^ 2u69RRV&2;C¥e¢69%2~&2;+9gi2H2"%R7:;"%#%;/69 .M&\2@](,i/69&2gh# gh[2T &O"";\86;8^ 2@R"&:M"!&br:"i/69 4:i2SRV &:Q9g`4 7O &B2p](,/69&2gh#9g`2Sr&2/69.o!/2#%d · ^_[C^ `: · ^_[C^ c z(690^| . .tWVS,2";286gh5¥&v2Q±&GJ[,#B69&&~69&(69. sC8^NQ&H69&4.[9gtR(6/[";C,2!"";,^56871&:,#TWV5|6/1!U69";^| u86!/~r"#%!p;C "!&
ξ =2π
λx· δx , η =
2π
λy· δy , · ^ · ^ Z
^|2λx
69&2;λy
;&2Q20^56871'.&4( &G86!/G;C "!&_^ 2&v69RR. ";d/p2QR(6!"Q;C D&2!"RV/698322S](,z/69&2-gm#B69&u.86;zH
δ1µ1 =7 s1 , δ2µ2 =
7 s2 , δ12
= −2z1 , δ22
= −2z2 ,
aba
^|2
s1 = sin(ξ) , s2 = sin(η) , z1 = (1 − cos(ξ)) , z2 = (1 − cos(η)) .
L: &2!"<!"&2/69&:Bgh,&2!&2v69u&2dR"&:"; & 2<](,vR(6!"4352<!"&: W,&§9gp/69"wn 69&2;
wn−1 36^|!/~#B69 &<!"&2/69&:;, &%2S!"&:71r&2!"p9go(M;C,(69.¥> #%TRR &B!/2#%437469&_^`sd69&2-gh#$ &d86!/u;S9g"0:,(69& · ^_[C^ `: ^M71&:,(69. . sHW/69 &
H · ∆wn,mi,j = −K · wn,m
i,j , · ^ · ^_[r
^|2H
3K
69&2;wn,m ;&2p2S69&2-gh#%";~gh,&2!&2Q9g H 3 K 69&2;
wn,m "RV"! 7r. s1^|X&u2R"&:!86S9go2S;C,(69.¥> #%S #$R. ! Q!/2#%43(RV/69
H69&2;
K86;A
K =3
2λId + 2
((1 − ε) ˙D1 + AV
1
)z1 + 2
((1 − ε) ˙D2 + BV
2
)z2
+ ε (1 − κ) ˙D1 z21 + ε (1 − κ) ˙D2 z
22 − SV s1s2
+7 (AE s1
(1 +
ε
2(1 − κ) z1
)+ BE s2
(1 +
ε
2(1 − κ) z2
))
H = P−1 +3
2λId + 2
(˙D1 + AV
1
)z1 + 2
(˙D2 + BV
2
)z2 +
7 (AEs1 + BEs2
)
· ^ · ^ · Xgz2Bc2-;,RM^| &2;!/2#%GT,2";D3`2G](,/69&2-gm#9g2v'eOR. ! $RV/69
KI 4 71&HW:sb
K =3
2λId+ 2 ( ˙D1 + AV
1 ) z1 + 2 ( ˙D2 + BV2 ) z2 − SV s1s2 +
7 (AEs1 + BEs2)
& " #%& " #% Xx&b;iv,2;Csu2/69W . xsuR/RV'p9g2@ #$R. ! M,RM^| &2;k!/2#%43-^@(6871Tv;'c2&2269#$R. ¥c(!869&)g>6!8^u]269&:sAR2,2;9> #%d.71.
(n,m)3¬2%gh. .^| &\.U69&)WV ^"&
wn,m+169&2;wn,m !869&~WV ^| /'&
wn,m+1(ξ, η) = G(ξ, η) · wn,m(ξ, η) · ^ `2^ Z
^|2G
;&2"|2M69#$R. ¥c(!869&~#B69¥e ^|!/~;RV&2;|& ^56871&:,#TWVξ
3η
6 ^. ._65&v2 6!"WU69&d#B69!""69&2;B;C R(69&B#B69!""8^ 2z. ¥&("69/69W . xs$9g 2z!/('#%zKO&2^|&$gh#l269#$R. ¥c(!869&Agm6! ^|!/ki2RV"!/69.`/6;C ,2
ρ(G)9gS69#$R. ¥c(!869&#B69¥e
GQ&(69#%. sr3
¥gρ(G) < 1
2%S &k2%sC# 69$;269#$RV";L^|286S¥gρ(G) > 1
2%T69B69#$R. ¥c(";.86;C &Ak &2/69W . xs1^P\v #$RV/69&:. s132v#B694& ,2;u9gi2~69#$R. ¥c(!869&¢g>6!B69. .^z%" #B69p2p!"&:71r&2!"T/69T9g`2p&:,#%!869.!/2#%4^zXx&2;"";D3(¥g
ρ(G) → 03V6g>6Q;269#$R &v9g
69. .2G@#%:;"!869&WVd'eCRV"!";69&2;)!"&2"01,('&O. s6Hgm6T!"&:71r&2!"43?^| .d¥gρ(G) → 1
36uRV:%;269#$R &69&2;)2&2!"H6<RV1%!"&:7rr&2!"v/69GWV,&2;)\WVGW/69 &2";D^X&)R6!' !"4369&(69. sO´8i2iRV-gh#B69&2!""9go2M!/2#%4326Tc2& i&:,#TWV9go769. ,2"5gh|2 ^|687ri&:,#TWV|ce";¨ &+2l^;$2 ^56871v&:,#TWV$R.U69&2H$;C/! ´8; ^| ¢6\4 7r&.71.z9g['c2&2#%&:@2&
ag
2@69#$R. ¥c(!869&\gm6!QQ!"#$R,";<ghQ86!/\!"#TW &(69&k9gK2"p769. ,2"8^ [!"#$R,/69&\&2RV-gm#%";H69&(69. sO!869. . sQ^|2&H;869. & ^| ~6p. &2869 ´8";HsO#®9g¬"01,a69&GW2,,2 &,|^|&](,69&(69. sO!":;4^Xx&@`!/(69R831,69&:&+^| . .WV"!";[2|!8659ga2. &2869 ´8";¨R"!"&2;C &28;(|I`,.Q0:,(69&2zWV"!869,2S,z,2;C"z(687rM2^|&~(69|2S!"&2!. ,2&2;C/6^|&~ &(69S!86169;C &u2#%4SB!&:S #$R. ! S869#%&:p!",.;kWV$'eO&2;";^| A!"&Cc(;&2!"%2S!86M9g7O!",269&2;H,&286;Cs \(^z8^
" (1( 4# # ( 2c2o#%2:;@QWV|69&(69. s:´8";@o2&2ghj^|!/25sC#6:!U69"; ^| 2 #$R. ! K/69rK. 7r";$;C "!. s13 ^| 2,69RR8eO #B69&_^ K9!869. .";%;C "!. 71K;:K&2` &::;C,2!"Q69&:sTm!&O/69 . svB69RR8eC #B69Mg>6! ´69&u#%(1;Hgmz &2/69&2!"'3( z'eORV"!";~$R7O;2WV"69#$R. ¥c(!869&Ggm6!8^`Q&2!"43: ^| . .V/69&2;G6,RV &:9g_'gh&2!"43C6Kg>696!"&:71r&2!"/695o &: &2!5B!&2!sT!"&2!"&2";D34 &2#B69 &2;K9g(25!/(69R8^ 2569#$R2.¥¥c(!8694&$#B69¥e6:!U69";~!/2#%M86;p
G∗ = Id−H−1 ·K · ^ c ^ Z ](~64 7r&¦ #$R. ! u69&(;¦'eOR. ! ~/69r43Q2k69#$R. ¥c(!869&§g>6!H9gT2b;C "!~. 71khu69&:s2| #$R. ! |869#%&:zWV";"569RR. ";v@2[. &2869 ´8";~I`,.z"0:,(69&2z;RV&2;z&~6T. #$ ";&:,#TWV9gzR(69/69#%83,4)2d;'c2& &9g
H69&2;
K@2Q^56871'>&:,#TWV9g!",432
/69σ1
3σ2
3D2 6!"WU69&#B69!""AE 69&2;
BE 69&2;k2%;C R(69&A#B69!""T!"RV&2;C &u2B:%,RM^| &2;/!/2#%43¬(69pS4 71&W:s Z ^_[C^ n igmp!"#$R" W.l\(^z@69&2;AW:s+ Z ^ · ^ · erM &2.^5Pb6!/~&:,#TWVR"!"&2;C &2";<!864^`](z2i. &2869 ´8";uIK,."0:,(69&283
AE 3BE 3
D169&2;
D269G!"&2/69&:#B69!""(69"/'&OU69. .¥s;RV&2;&)2GPb6!/)&:,#TWVM
69&2;2Q\(^;C "!&v/u
¨2RV"";\9g,&2;c
69&2;<2;&2 xsρ
!869&\WVT1Ri!"&2/69&:S69M6B'gh&2!"@7469. ,2D^| 2,.459gtr&2/69. xs ^| .[2Q/699g¬RV"!¥c(
a n
&:,#TWV|69&2;G6RV"!/69δy/δx
:2Q;C R(69&d#B69 !""|69[69.Sgh,&2!&29g_2"Q/69#%R(69/69#8^`Xx&G2[R"!"&2;C &(";~!8643
P3 ˙D1 = σ1D1
69&2; ˙D2 = σ2D2!869&~69./TWVi'eCR"";G &v#%
9gt2"/QR(69/69#%z &2!"[2QR"!"&2;C &2 &$Ra69/69#%β
3O'eCR"";G,2 &@2[4.W(69.D;'c2& & Z ^ · ^_[r5;RV&2;z'e!. ,2 71. sG&H2%£ceC";(5P\6!/H&:,#TWV8^L:,#$#$ &% &r|,R_3¥||:,2|RV4 W.+^| %
G∗(ξ, η;M,CFL, v/u,AR) = Id−H−1(ξ, η;M,CFL, v/u,AR) ·K(ξ, η;M,CFL, v/u,AR)
q`. #$ &(69sB,2;C"(687r2^|&B(692z#%4 &/\2,2&:U69.VR(69/69#% &B2!869gD2z. &2869 ´8";I`,.T"0:,(69&2@69$2%Pb6!/A&:,#TWVSgmS2$R:sC!"p69&2;A2
CFL&:,#TWVSgmM2%&:,#%!"8^
Xx&b2#B69 &2;p9gM!/(69R83_ ^| . .`2'gmWV%6,#%";b(69M2%6RV"!M/69vS"01,(69.`1
F^|!/T6v86&(69W.d6,#$R&kgmS2$4;S,2/";b & &:7:!; \(^!"#$R,/69&2p69&2;A2\(^ ;C "!&G;CU69r&(69.^| B"RV!S2[569"U69&B4 ;\
v/u = 1^|!/d#B69r"2i\(^¢,. s
2D'^LO71/69._xsORV"z9g`69&(69. sCz!869&~WVMRV-gm#%";~&~2p69#$R. ¥c(!869&ugm6!p
• .:!869.t/69W . xsH69&2;~B!&2!s~69&(69. sCSghp6G4 7r&M9gR:sC!869.
M3v/u
i69&2;b&:,#%!869.[CFL
3AR
QR(69/69#%3_2%RV"!/69./6;C ,2
ρ(G∗)(ξ, η)z!"#$R,2";:^| \6$ ^"Ru&~2 ^56871'>&:,#TWV
ξ ∈ [0, π]3η ∈ [0, π]
¥g2T'eC[69Q.86i6$7469. ,2T9g
(ξ, η)gm0^|!/
ρ(G∗) > 12T!/2#%pQ,&2/69W.SgmQ2
ceC";<7469. ,2"i9g2TR:sC!869.`69&2;<&:,#%!869.`R(69/69#%M!"&2;";D^SNR(69Qgh# 2@!",R.(ξ, η) = (0, 0)
gm ^|!/ρ(G∗) = 1
gh[!"&2&2!s~86&_3D6!/7O &G2@#B69. ."[RV4 W.769. ,2T9g
ρ(G∗)ghi69. .
(ξ, η) 6= (0, 0)&2,"i6$gm6Q;269#$R &khi6$2¥4\ &: &2!@B! &2!s2
9go2S!/2#%M,&2;Q,2;Cs1^
• 4.W(69.t/69W . xs~69&(69. sOp2p0:,(69&: xs
ρmax∗ (M,CFL) = maxξ∈[0,π],η∈[0,π](ρ(G∗(ξ, η;M,CFL)))
!"#$R,"; ^| b6 ^"R&(GR:sC!869.5R(69/69#%
M69&2;2G&:,#%!869.zR(69/69#%
CFL^X>gQ2v!/2#%
G/69W2. <ghH64 7r&(M,CFL)
!",R.43G#B6eO #B69.i7469. ,2<G"0:,(69.[1
hW/69 &2"; gmξ = η = 0
'^MLOd(6983V¥g6d!/2#%T[4.W(69.¥. s</69W.pghQ2D^|2.@/69&r@9gP\6!/k69&2;k] n&:,#TWV`!"&2;"; &2,(;Csr3
ρmax∗ (M,CFL) = 1 ∀M, ∀CFL ^oXg225!/2#%o,&2/69W.
ghz/#%(M,CFL)
!",R."3769. ,2"z.U69rz(69&1
gmρmax∗ (M,CFL)
69RRV869 &H2S,(;Csr^
• 4.W(69.tB!&2!s~69&(69. sCS2[01,(69&O xs
ρ∗(M,CFL)(Q#%869&G7469. ,2[9g
ρ(G∗(ξ, η;M,CFL))!"#$R,"; ^|2&
ξ69&2;
η7469suWV ^"&
069&2;
π^pXx[M!"#$R,";^| A6G ^"RA&b2@R:sC!869.R(69/69#%
M69&2;
2<&:,#%!869.iR(69/69#%CFL
^ G#%869&7469. ,2<#B68s¢9gT!,2<WV<.^G(69&1
71&2,4~2M!/2#%i5,&2/69W.S69&2;H'eC W
ρ(G(ξ, η)) > 1ghz#%i769. ,2"z9g
(ξ, η)C/@2
01,(69&: xsρ∗(M,CFL)
;:"|&25R7O;S69&:sB &Cgm#B69&u&v2M!/2#%M/69W . xs1^`[^7183 i!"#$R."S7rsu,2'gh,. . s~2@R7:,2i4.W(69.K/69W . xsb69&(69. sOiW(6";b&
ρmax∗ (M,CFL)
^X&2;"";D3¥g[2H!/2#%v$/69W.Ggm%6<4 71&
(M,CFL)!",R.
ρmax∗ (M,CFL) ≤ 1
'3K2#%869&7469. ,2%R7O;"@6H0:,(69&: /69 71% &Cgm#B69&)&2d;269#$R &\p &: &2!dB! &2!s9g2!/2#%vz(@.^i2T7469. ,29g
ρ∗(M,CFL)3 2TWV/'M2;269#$R &~D";\W:s~(
&:,#%!869.t869#%&:zgm|z!",R.p9gPb6!/<69&2;<] n &:,#TWV8^Ji69,/69. . sr3o2%.:!869.5T4. Wa69.569&(69. sCT!869";),pgmp2d;C "!@/. 7r@!869&WVd69./~RV-gm#%";D3^| 2v/69#%G &:/'R/69&gm@2~69RR8eC¥#B69/H. ,&#%2:;%!"&2;"; &2GR"/&:^ ^ n K,2/69"^| d6M,2;Cs9g 2z;C "!. 7169#$R. ¥c(!869&dgm6!
ρ(G∗)^|2&
M69&2;
CFL69ig¨"MR(69/69#%@F^| .v/u = 1
69&2;AR = 1
'^
a d
%) )%A i; .7!'&)(D#%5./&5!2# !7$; . ];67 ;H; #%143I%# .6#87'> ; 35%# .6# 7:!7b3); ! 2d "!T&)B!&2!sA9g!,2R. ¥&2b6~ #$R.%c2-;T #$R. ! RV/69 HI = I + KI ^z¥)6 ;O;eCR. ! TRV/69 KIII (6T69. 86;CskWV"&)69&(69. sO´8";A &2$. /69,\¨ghS &2/69&2!"%W:sq`,. . U69# & _a2` '^L: &2!"iz #$R.i&2&C!"&2&:!/2!"M |/69 &2";v,4(,z ^ 3 "#% &:" &@@,#$#B69 ´8Q2[!"&2"0:,2&2!""59gt,2!/v6T!/2!"gh2 ^|2.Q/69&r[9gtPb6!/&:,#TWV8^%]¬ M9gz69. .>3DWV
III/III69&2;
I/III/!/2#%"p69@gm,&2;bvWV$,&2!"&2;C &(69. . sA/69W.
^|2&H'e26!. sd. 71";D3OWVQ 5 &G2 |&2&C>R"!"&2;C &(";~.^5Pb6!/HR"!"&2;C &2";H7r&_^ 2B!&2!s~9gWV<!/2#%" &~2S&2&C>R"!"&2;C 4&2";\!86MQ69&(69. s:´8";~,4u2SR.9g
ρ∗ &
gh,&2!&u9gM
69&2;CFL
;CR.U68sr";H &Hc24, · ^ c ^ Z • ^|2&.U69r
CFL&:,#TWV$69B,2";D32d!"&2&:!/2!"
III/IIIR7:;"$69&)'eC!". .&:
;269#$R &H6.&v6Q2TP\6!/<&:,#TWVi;:"Q&2QWV"!"#%p:B.^S^) 2&M → 0
3(2T;269#$R &r:"5
1m69. #%45&2@;269#$R &B69|69. .V2&2!"M6p71s%.^¦!"&:71r&2!"i/69"^z(697r
257469. ,2|9g 2] n &:,#TWV8^ ?^. .¥>O&2^|&%WV(687OK`xsOR!869.29g 25.^5Pb6!/%&:,#TWV,2 ^| ~,RM^| &2;u!"#$R" W. \(^. 718^
• 2G&2&C!"&2&:%!/2!"I/III
;CR2.6sO2v/69#% ^5"69O&2"@gh.^¯769. ,2"$9gQ2vP\6!/&:,#TWV8^oP\"71'3gmt.U69r769. ,2"¬9g22
CFL&:,#@WV'392#%869&T769. ,29g269#$R. ¥c(!8694&
gm6!T;:"S&2S&2;AH´8<69&:s\.&rpW,pu6Gc2& %769. ,2<¨xs:R!869. . sA69WV710.5
gmS2^|2.%/69&r%9g|Pb6!/A&:,#TWV$SS.4S9gz'B!&2!skT6H;C "!p!"&2/"01,2&2!"B9g52$gm6!HI = I + KI 6= I + KIII ^
0
0.25
0.5
0.75
1
Ave
rage
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL0
0.25
0.5
0.75
1
Ave
rage
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL
]o 4, · ^ c ^ Z oN¬7r/69r|69#$R. ¥c(!8694&$gm6!gh25j[ "!K!/2#%4^ n 'gh|t!"&2&:`/!/2#%5XXX 9XXX'^z 4:p`Xx&2!"&2&:!/2#%SX 9XXX'^&2; &M2WV(687O`9g(256871/69r7469. ,29ga2|69#$R. ¥c(!869&$gm6!834 !869&$"#,&&2"!""/69sH,2/$ 4)] n &:,#TWVpghS2% &2!"&2&:T/!/2#%%X 9XXX'^ Xx&;pH &:71" 169%SRV &:836B.:!869.`,2;Cs13V(69iM6d,2;Cs<9g2$69#$R. ¥c(!869&bg>6!i7r,(Q2 ^5687r&O,#TWV
ξ69&2;
ηghM6
RV"!¥c(!pPb6!/~&:,#TWV[69&2;Hghz ^B;C &:i] n &:,#TWV832zRV-gm#%";D^N,W2&!S!86BM =
0.5M."!";¨/H687r;k69&:s< &:-gh&2!"+^| b.^5Pb6!/AD"!i(69 ^| . .WV$;869. ^| kWV.^Q
g$e
69&2;<6$!"#$R(69&< z#B6;SWV ^"&<2p69#$R. ¥c(!869&<g>6!'/9g2pj[ "![LC!/2#%pW/69 &2";:^z¥CFL = 100
69&2;CFL = 106 ^ 2S",. [69SR"/&:";u &Hc24,/ · ^ c ^_[C^|NQ'eCRV"!";D3(2S4.W(69.
(69RV59ga2|69#$R. ¥c(!869&$g>6!`01, 25/69#%gmCFL = 100
69&2;CFL = 106 ^oJQ712."83
&d2.^5mgh"01,2&2!s%4&_3O(69Kgm^56871'>&:,#TWV5!.4p´83:2Q6S&2&C>&24. 4 W.i169 &9g¬B!&2!sH6!/'71";GW:sB,2 &%6p71s% 4u] n ^CXx&H69RRV&2;C¥ev[3^i'eOR.U69 & ^|:sB2iB!&2!sG9g6%#%2:;< &<2p.^5mg¨"0:,2&2!s~´8&2T &ugm6!["&OU69.tghQ2T!"&:7rr&2!"@RV-gh#B69&2!"4^ :,283 &HR i9g`69&~6871/69r[7469. ,2
ρ∗69.¥#%45 &2;RV&2;&:9g
CFL&2!"i5R(69/69#%z5.U69rM&2,4_3
2Mz . .t6@WV&2'c2$,&u69z &Cc2& @] n &:,#TWV8^
0.4
0.5
0.6
0.50.4
0.4
0.3
ξ / π
η/π
0 0.5 10
0.5
1
0.4
0.5
0.6
0.50.4
0.4
0.30.2
0.1
ξ / π
η/π
0 0.5 10
0.5
1
]o 4, · ^ c ^_[CNQ#$R. ¥c(!869&dgm6!gh2Qj[ "!|LO!/2#%[69M = 0.5
^ n 'g¨[CFL = 100
^z 4:QCFL = 106 ^
%) )@ i& 9<.f= 251]; # ; i; ./7b!Xx(6WV"&G&2";v&Bc24, · ^ c ^ Z (692i687r/69ri69#$R. ¥c(!8694&Gg>6!59g_2[j[ "!zLO!/2#%[r:"u,& xsL^|2&2dPb6!/&:,#TWV@r:"Tu´83.86;C &\\6~RV:@;269#$R &b9gz2G8^ RW.##% 7469";@Q,29g2.^5Pb6!/R"!"&2;C & &2M#%2:; &;oQ #$R712B!&2!s9g2S!"#$R"/ W.p. 71| &~z4 #%4^ 2pj[ "!
I/III!/2#%S,2 & ,1._R"!"&2;C &2
"!"71Q671sGr::;uB!'&(!'svgm#B69. .tPb6!/~&:,#TWV@ c24, · ^ c ^ · z69&2;H#B69 &2Q/69W.Mgh69. .5] n &:,#TWV^vJQ%2%",. TW/69 &2";AgmS2Bj[ "!
III/III!/2#%B;#%&2/69B(69
.^5Pb6!/R"!"&2;C & &<p"&:U69.KHR7O;$,2.¥sk'e!".¥.&:T;269#$R &b69&2;bg>6S!"&:71r&2!"gm2 ^|2.|/69&rz9g P\6!/$&:,#TWV"^|2&%.U69r
CFL&:,#TWV695,2/;~¨W,3169.U683r2|!"&2&:
III/III!/2#%TQ!.869. sH:d'eCRV&2 71SWV< &<#%[9g`,& [!"#$R,/69 &a69.`!"4[69&2;u#%#%s
/69r'^MXx&<2T#B69&&2M9g2TR7:,2M"!&_3D6B.:!869.K69&(69. sC[!869&bWV@!869";b,[ &b;i69&(69. sO´8#%|!.4. sp25169 &$9gVB!&2!sW/69 &2"; ^| (R"!"&2;C &2";%j[ "!
I/IIILC!/2#%
^| k"RV"!Mv2&2&C>R"!"&2; &(";A71'/&_^@N.^5Pb6!/!86~M = 10−3 M&2^l."!";
69&2;G6S 4H] n &:,#TWVMCFL = 106 `/69 &2";G &2!" (6WV"&dR7O,2. sB2^|&d(69,2!/v6
!/2!"[&(69W."T &2!86Q2[;269#$R &$7r2.^5mgh"0:,2&2!s%R(6959g_2QRV"!,#~^ 2Q",. 69MR"&:";~ &vc24, · ^ c ^ `<
• gh2p&2&C>R"!"&2;C 4&2";b!/2#%43(2pQ . .o6$r::;<;269#$R &G &u2p 4C ^5687r&:,2#TWV
g Z
4&2%W,$(~69#$R. ¥c(!869&¢gm6!% &(v.^ ^56871&:,#TWVd6986bWV"!"#%"B7rs)!.4/v,& xsr3.86;C &B2'ghSRV:Q;269#$R &d69&(;H!"&:71r&2!"d¨.^ &: &2!pB!'&2!s2'^
• &S22¬(69&2;D32KR"!"&2;C &28;Tj[ "!LO!/2#%R7O;"o6571s[r::;T;269#$R &[7rt2&: ^56871&:,#TWV5;#B69 &_^KXx&2;"";D3:2Q",. 569Q7rs%!.4/p2[ 4 &(69. !/2#%Q,&2;/69&2;269; !"#$R" W.+\(^ !"&2;C &2p4 7r&k &bc24, · ^ c ^_[C^BL: #$ .U69T69&(69. sO´8"p69#B6;W:sAP<O.G69&2;±&69" ^569/69& & n/Z 69&2;2B/"6;2'!869&)'ghTuD^kghT &2/69&2!"ghdg¨,2v,2;C"v69WV,vR"!"&2;C & &"!/&0:,2"k¨7O!",2Q\(^S3z 4Cx6RV"!v/69),&286;Cs \(^§gm| &2/69&2!"'^
L: &2!"p2Tj[ "!MLO. 71[B!&2!s~RRV"[R"&:M69&u;869.od6!/71@69Q2S.^"QRV4 W.!"4T,2 &b69&69RR8eC #B69d /69 71G. ,&/!/2&01,243`&. sA2BR"!"&2;C &2"; #$R. ! $/69r^| . ._WVS!"&2;";~¥&~2M#B69 &2;Q9gz!/(69R8^
0
0.25
0.5
0.75
1
Ave
rage
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL0
0.25
0.5
0.75
1
Ave
rage
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL
]o 4, · ^ c ^ · kP\869&¢7469. ,2<9gS2\69#$R. ¥c(!869&gm6!BgmB2<j[ "!v!/2#%u,2 &2 ,1.R"!"&2;C &2^ n 'g¨p?!"&2&:/!/2#%SXXX 9XXX^O 4:S`Xx&2!"&2&:!/2#%SX 9XXX'^
g$[
0.9
0.4
0.5
0.6
0.7
0.8
0.4
ξ / π
η/π
0 0.5 10
0.5
1
0.4
0.5
0.6
0.50.4
0.4
0.30.2
0.1
ξ / π
η/π
0 0.5 10
0.5
1
]o 4, · ^ c ^ `2zN#$R. ¥c(!869&\g>6!gm2Tj[ "!SLO!/2#%69M = 10−3 ^|
CFL = 106 ^ R)J[$q`"!"&2;C &2 &2^5# ,1.tq`"!"&2;C &2 &2^
g ·
" (1( " " #% #%"$#% ' & # #1 "$# #Xx&¢!/(69R [C35/7r/69.[69RR8eC #B69u &:71&¢#%2:;d(6871~WV"&;"! WV";¢. 7r~2~. &2869sC# Z ^_[C^ Zn '^ 2MR"&:/69&<9go2"i#%2:;z(6|WV"&~R"!. sGW(6";~&u6@. &2869sC#7Oh^|RV &:83 &::;C,2! &~#B69!""
M ′1
3M ′
2
69&2;D
vR. [[email protected]'>W(69&2;";k#B69¥eM
6:!U69";^| Z ^_[C^ Zn @6D^5.¥.z6@.^$69&2;),RRV#B69!""
L13L2
3U1
69&2;U2
^ 2dR"&:$"!&);71";<G2TR"&:69&k9g6d,&¥g¨sO &G7Oh^|RV &:M&k69. .2"T#%2:;i(69 ^| . .#B69r862](,569&(69. sO9gD2"z769,2 #$R. ! 869#%&:^ ,&¥g¨sO &Sgm#T,.U69&dKW(6";B&G6;C D&2!"RV/697:h^|RV &:/692o(69&$6. ¥&("69`sC#7:h^|RV &:z¨2,4@WV$69"01,2¥7469.&:'^n M,2[:,2[c2i!"&2;M2@ #$R. ! S/!/2#%Tgm[.^5Pb6!/k&:,#TWVM,&286;Cs \(^z% Z ^ `2^ d i69&2;.z,25gh:!,2&H2M #$R. ! QRV/69 H 3/^|!/<!869&~WVMR2.¥ z &:~
H = D0 + T1 + T2 · ^_aC^ Z
^|2D0
3 T169&2; T2
69S;C &2!"pRV/694 7r&~W:sb
D0 = P−1 +3
2λId ,
T1 = σ1
(δ1A
Eµ1 − δ1(D1 +AV
1
δx)δ1
),
T2 = σ2
(δ2B
Eµ2 − δ2(D2 +BV
2
δy)δ2
).
N[,#$ &%!"&2/69&:!":B!&:832(69zz!"&2 ;2'¥&2d6. &2869zRW.#~322SRV/69 TpWV"!"#%B
T1 = AEδ1µ1 − ( ˙D1 + AV1 )δ21 ,
T2 = BEδ2µ2 − ( ˙D2 + BV2 )δ22 .NQ. &(69. sr322i #$R. ! QRV/69 H !869&~WVp69.%R. 65gm. .^zp
H = D + S1 + S2 · ^_aC^_[r
^|2D
3 S169&2; S2
69S;'c2&2";<6S
D = C0 ,
S1 · ∆wn,mi,j = C+
1 · ∆wn,mi+1,j + C−
1 · ∆wn,mi−1,j ,
S2 · ∆wn,mi,j = C+
2 · ∆wn,mi,j+1 + C−
2 · ∆wn,mi,j−1 ,^|
C+1 = (
1
2AE − ( ˙D1 + AV
1 ))i+ 1
2,j = A−
i+ 1
2,j,
C−
1 = −(1
2AE + ( ˙D1 + AV
1 ))i− 1
2,j = −A+
i− 1
2,j,
C+2 = (
1
2BE − ( ˙D2 + BV
2 ))i,j+ 1
2
= B−
i,j+ 1
2
,
C−
2 = −(1
2BE + ( ˙D2 + BV
2 ))i,j− 1
2
= −B+i,j− 1
2
,
C0 = P−1 +3
2λId+ (A+
i+ 1
2,j− A−
i− 1
2,j) + (B+
i,j+ 1
2
− B−
i,j− 1
2
) .
g`
](6. &2869zRW.#~322SRV/69 S 69&2;H2S!":B!&:D
!869&HWVS #$R. ¥c(";~ &:~
D = P−1 +3
2λId+ 2( ˙D1 + AV
1 ) + 2( ˙D2 + BV2 ) ,
S1 = A−E+1 − A+E−
1 ,
S2 = B−E+2 − B+E−
2 ,
^|2M2SRV/69 Ep69S¥ghzRV/69Q7r&2i4;~!". .t,2!/H(69p
E±
1 · ∆wi,j = ∆wi±1,j ,
E±
2 · ∆wi,j = ∆wi,j±1 .
25 ^MR. &rQ · ^_aC^ Z 69&2;~ · ^ aC^_[rj^| . .WV5,2'gh,.CM!/(69/6! ´8Q69RR8eO #B69|gm6! ´69&G69&2;.U6e269&B869#%&:&2!"Q69. .(2"|869#%&:/R,,&2;2|gm#9gt6ir&2/69.( /69 71!/2#%436@'eOR2.69 &2";)WV.^S^ 2G](,@/69&2-gm#
H9gz2B #$R. ! $RV/69 H (6TWV"&;'c2&2"; &
· ^ · ^ · '(69. &(69 71. sr3¥gT1
3T2
;&2i2M](,|/69&2-gm#%z9g T169&2; T2
3269&2;H #$ .U69. sr3S1
69&2;S2
69M2S](,z/69&2-gm#%9g S169&2; S2
32Ms:#TWV.H
!869&<69.$WVM'eCR"";u6S
H = D0 + T1 + T2 = D + S1 + S2 · ^_aC^ ·
Pb69: &%,2i &~R(69!,.U69Q9go2M](,z/69&(-gh#B69&<9g2S;C &2!"pRV/69 E±
E±
1 = e±7 ξ ,
E±
2 = e±7 η .
.86;z([gm. .^| &B;'c2& &<9g2SsO#TWV.Tp
69&2;Sp
T1 = 2 ( ˙D1 + AV1 ))z1 +
7 AEs1 ,
T2 = 2 ( ˙D2 + BV2 ))z2 +
7 BEs2 ,
S1 = −( ˙D1 + AV1 )) cos(ξ) +
7 AEs1 ,
S2 = −( ˙D2 + BV2 )) cos(η) +
7 BEs2 .
^|
z1 = (1 − cos(ξ)) , z2 = (1 − cos(η)) , s1 = sin(ξ) , s2 = sin(η) .
«M"R &H &b#$ &2;b2"@769,2M;'c2& &2Mgh[g¨,,@,2/43^$69&2^r &vG &::;C,2!"B6dr&C/69.gm# gmi2 /69 71$!/2#%"M,&2;S,2;Csr^@Nlr&2/69.K /69 71$!/2#% ^|24/Tgm#!7rWVu /69 71S.U6e269&u#%2:;h,2!/\6
ALJ(p)3ALGS(p)
3PJ(p)
SGS(p)
|69&2;~ /69 71g>6! ´69&u#%2:;Th,2!/u6
AF (p)3MAF (p)
86;p
Hα · ∆w(l+1)i,j = −∆τRn,m
i,j − (H−Hα) · ∆w(l)i,j
· ^_aC^ `:^|2
∆w(l+1) = w(l+1) − wn,m 69&2; HαS6v;C &2!"%RV/69M(69S;RV&2;p&b2$!/2!"$9g
2% /69 71%#%2:;D^vXx&kg>6!83o6pRV &:";,p &!/(69R [C3¬2BRV/69 HαS&2p&2"!""/69 . s
g c
2S/69#%p6986!/u/69rS9g2S,WC> /69&uR:!""3/(69Q6#%Mr&2/69._gh#9g2M /69 7r!/2#%MS
α = 1, N
Hα · ∆w(lN+α)i,j = −∆τ · Rn,m
i,j − (H−Hα) · ∆w(lN+α−1)i,j
· ^_aC^ c
2ig¨,2.¥.¬69. r #|2&~4 71&HWOs\
∆w(0)i,j = 0
l = 0, p− 1
α = 1, N
Hα · ∆w(lN+α)i,j = −∆τ · Rn,m
i,j − (H−Hα) · ∆w(lN+α−1)i,j
∆wn,mi,j = ∆w
(pN)i,j
· ^_aC^_ar
2i'eCR"&H9g¬RV/69 Hαgm2i;C &:| /69 7ri#%2:;5R7O,2. sB &::;C,2!""; ^| . . WV
;/69 .";u &~2M&(eCzR(69/694/69R(8^]o 832p](,/69&2-gm#Q69RR. ";u$zr&2/69.¬ /69 7r69. r # &u;z%W/69 &<6!.4/";Omgh#'eOR"&vgm|2p6:!U69";u69#$R. ¥c(!869&ug>6!p
∆τ · Rn,m = K · wn,m
∆w(0) = 0
l = 0, p− 1
α = 1, N
Hα · ∆w(lN+α) = −K · wn,m − (H −Hα) · ∆w(lN+α−1)
∆wn,m = ∆w(pN)
· ^_aC^ g4
N|z86!/uR~9g2S,WC> /69 71MR:!""326R7:&(69.o69#$R. ¥c(!869&<#B69¥eH!869&~WVS;'c2&2";A
∀i ∈ [|0 , pN |] w(i) = Gi · wn,m
0 W:7O,2. sr3C2M!/2!"∆w(0) = 0
s:.;G0 = Id
69&2;G2S69#$R. ¥c(!869&u#B69¥eG",. &$g¨#24.W(69.¬69. r #S
G = GpN
i^569&:T'eCR"#B69¥eG656Mg¨,&2!&v9g_2[]2,5sO#TWV.H
3K
69&2;Hα
^HA0^| . .V&2^"!869. .D2M69&(69. sOR"&:";v & ·bd 3 Zd ^ n 5,2 ^| [2[ /69 71M!/2#%[,2 &2iR7:&(69.#B69¥e
Gi
Hα (GlN+α − Id) = −K − (H −Hα) (GlN+α−1 − Id)
J[^ .,2§ &::;C,2!"l &eCR" &269#$R. ¥c(!869& #B69¥e9g)2lj[ "!LO!/2#%G∗ = Id − H−1 ·K C #$#%";CU69p!869.!,.U69&2Qs:.;A
GlN+α = GlN+α−1 −H−1α H GlN+α−1 +H−1
α H G∗
g$a
L:,W/6! &G∗
g¨#2 n MLv69&2;~QiLd.86;zH(GlN+α −G∗) = (Id−H−1
α H) (GlN+α−1 −G∗)2&α
769"|g¨# Z N
36^MW/69 &
(G(l+1)N −G∗) =N∏
α=1
(Id−H−1α H) (GlN −G∗)
I`71&:,(69. . sr3H^|2&~2S,WC> /69 71MR2:!"/|z!"#$R.";\6g¨p
/69&283H^M(687r%
(GpN −G∗) =
[N∏
α=1
(Id−H−1α H)
]p
(Id−G∗)
2p69#$R. ¥c(!869&<#B69¥eH9g2Mr&2/69.t¥/'69 7rS/!/2#%M|2'ghp;'c2&2";HW:sb
G = G∗ + V p (Id−G∗)
V =N∏
α=1
(Id−H−1α H)
· ^_aC^ n
Tgm#T,.U6\&(69W.",2@\01,(69&O¥g¨sA2G #$R(6!$&) &: &2!vB!&2!s9gQ2d ^\.7r. ^|!/;C 71[2iB!&2!sv9g69&v /69¥71i#%2:;koc2. s13O2M'e6!5 #$R. ! !/2#%M2,.;GR7O;i2WV"p;269#$R &23t(69SM2$WV"T69#$R. ¥c(!869&#B69¥e
G∗
¬"!"&2;C. s13_2$ /69 7r%#%2:;A2,.;!"&:71rT6zgm6Q6RV4 W.S%Q'e26!'Q/. 7r8^Nz!"!";C &d$gh#T,.U6< · ^_aC^ n '32gmQ6$4 7r&<'e26! #$R. ! |69rh,2!/H62QW.:!/d&2[,2;C";G &G2QR7O,25/!&a3O2[!"&:71r&2!"[/69i9go69&69RR8eC #B69$. 7ri;RV&2;S&\2!":B!&:
V^|!/b[ .¥g56%g¨,2&2!&k9g2'e6!i #$R. !
/69r|](,K/69&2-gm#H
69&2;$9gV2|](,K/69&2-gm#uhHα
RV"!¥c(!5i2|!/24/&B69RR8eC #B69869#%&:8^ i!":B!&:
V2,.;uWV@6[#B69. .6QRV4 W.T/d6QBR7:;T2pWV"i;269#$R &
gm6~/;,2!";)&:,2#TWV9gQ,WC> /69&2^bXgHα = H
2&V = 0
69&2;2v69#$R. ¥c(!869&gm6!9g2j[ "!pLO!/2#% ^|!/bR7O;"i2TWV"M;269#$R &v[&(69,/69. . s<"!"71";D^SX&\2Tgh. .^| &"!&283Cgh#T,.U6~ · ^_aC^ n I^| . .DWVi,2/";G%69&(69. sO´8i2S69#$R. ¥c(!869&~g>6!z9go2i /69 7rM #$R. ! 869#%&:Q;"! WV";~ &~!/(69R [C^
# "$#% +#% & # # 2N[jiXoN[]b#%2:;B #$R71";%W:s%6S,W>¥/'69 7rR:!""69 #%";G69!869&2!". &@2|gm6! ´69&G!869&WV<!.869. s+!86B &:2ur&2/69.Qgh# · ^_aC^_ar%,2 &2< &4.u #$R. !¥HRV/69 Hα
(α = 1)
HADI = (D0 + T1) ·D−10 · (D0 + T2)
· ^ gO^ Z (69z
HADI = H + T1 ·D−10 · T2
· ^ gO^_[r 2p69#$R. ¥c(!869&<#B69¥e~6/:!U69";'^| ~z#%2:;H| #$#%";CU69. s~;";C,2!"";Hgh# · ^_aC^_arM
GADI = G∗ + V pADI (Id−G∗)
VADI = (Id−H−1ADI H)
· ^ gO^ ·
gbg
^| HADI = (D0 + T1) ·D−1
0 · (D0 + T2) = H + T1 ·D−10 · T2
· ^ gO^ `:0 &2Q!869&d&2!"Q(695^|2&v6p &4.Q,WC> /69&GRV-gh#%";b
p = 1'31( 4 &(69. N[jiX!/2#%
"!"71"; ^| G(Q!""RV&2;C &%69#$R. ¥c(!869&vg>6!GADI = Id−H−1
ADI ·K^ n ,2&2^¦,2;Cs
2MRRV"Q9go2ADI(p)
AF (p)
#%2:;H,2 &$±&~J[,#B69&&<69&(69. sC8^%) )%A %& 5- - "!$#%!
0
0.25
0.5
0.75
1
Ave
rage
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL1
1.005
1.01
1.015
1.02
1.025
Max
imum
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL
]o 4, · ^ gO^ Z |N¬7r/69rG¨.'gh69&2;<#B6eO #T,# ¨ 4:Q69#$R. ¥c(!869&\g>6![9gK2T&2&C> /69 7rN[jiX!/2#%4^ 2~687r/69r
ρADI69&2;#B6eC #T,#
ρmaxADI
69#$R. ¥c(!869&+gm6!gh2v&2&C> /69 7r~N[jiXS/!/2#%69pR.";\¥&<c24,v · ^ gO^ Z [6[6$gh,&2!&b9gK2TPb6!/<&:,#TWVM69&2;<2] n &:,#TWV8^ 2p71s^. .¥>O&2^|&WV(687OT9g|S#%2:;D3o&(69#%. sA6H7rsbRV:T;269#$R &ughS 4] n &:,#TWV@;C,2\6<;#$ &(69&:Tg>6! ´69&83T#B6;G!.869@W:sA2"BR.^ AdW271B(69@2GR #B69.] n 769. ,2$S69,&(;
5^|!/Ap69. ^. .¥"/69W. (;W:s\71/69.K,2;C" _a2` ^ ($,WC> /69 71
R:!""B(69Q^56BR7O,2. s+;"! WV";ADI(p)
AF (p)
#$R71"B2\69#$R. ¥c(!869& g>6!8^ 2[N[jiX`!/2#% ^z¥
20,WC> /69&2|R7:;"|6pWV|;269#$R &$9gt2[S c24, · ^ gO^_[r'^
[^7r83%;269#$R & . .z;/69"$gm 4] n &:,#TWVd69&2; ^H7r&&2H &2/69W . "gm
CFL > 100^ 2M . .69&kR #T,# ] n &:,#TWV ^|24769. ,2S69WV,
20^,A$2^";
R7O,2. s+(69BgmG,2!/ 769. ,2\9gS2b] n &:,#TWV83z2\j[ "!'~LO!/2#%b;:"d&2dR7O;<2WV"%;269#$R &23/\2HB!&2!s9gQ(HNQjiXS#%(1;_37r& ^| ¢6b,W>¥/69¥71HR:!""83$&2/69-g>6!sH &2!"S ;:"&2[69. .^69&:sO2^%"!"7r2p!"&:71r&2!"p/69pD";~W:sG2p'e26!. 718^%) )@ i&?./ " "!$#%!L:,2;CsO &G#%TR"!. s~2pRRV"M9gK2@NQjiX|!/2#%T"01, "M6%.:!869.`69&(69. sO8^)p!/2:4/6<.^5Pb6!/+!86/k
M = 10−3 69&2; ^G,2v2vN[jiXp!/2#% ^| 20
,WC> /69&2^L: &2!"H2
g n
0
0.25
0.5
0.75
1
Ave
rage
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL1
1.025
1.05
1.075
1.1
Max
imum
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL
]o 4, · ^ gO^_[C|N¬7r/69rG¨.'gh69&2;<#B6eO #T,# ¨ 4:Q69#$R. ¥c(!869&\g>6![9gK2T&2&C> /69 7rN[jiX!/2#%4^
4.W(69.69&(69. sO2^";)(692HR #T,# ] n &:,#TWV+^5620
&(69%!8643?^H!#$Ra69v#%2:; ^| 2uj[ "!HLO!/2#%b69v] n
20^ 2u",. G69uR"&:";¢ &¢c24, · ^ gO^ · ^ 2
69#$R. ¥c(!869&\g>6![9gK2@NQjiX|!/2#%p[!.4pB2@jQ "!SLO!/2#%pgm2T.^ ^56871&:,#TWV8^J[7r2."3^[&2[(69
20,WC> /69&2Q69[&2|, B!&:|@W/69 &H2i/69#%iB!&2!sd(69&
2pjQ "!iLC!/2#%3a"RV"!U69.¥. svgmz2M 4~gh"01,2&2!sv4&2^]2,2[69&(69. sO´8"(687rp2^|&u(6950
,WC> /69&2(6871pWV#B6;4369.8683:p"!"71B!&2!sd712[&: ^5687r&:,#@WV';#B69 &_^ ",. Q!"&Cc2#%Q(69[NQjiX/!/2#%43(71& ^| \6%,WC> /69&\R:!""83(z &2B!&:8^]o 835 d"0:, "v6kc2& u #%'R¢gmQ^|2 !/2<v;269#$R & ^| . .QWV\69&:s^568s+,WCR #B69.F^| "RV"!$\2H&2G(69%!869&WVv'eCRV"!";)g¨#³2v'e26!869#%&:'z"!"&2;D3K $&2"";%6.U69rp69#%,&:Q9g`,WC> /69&2$WVS6B!&:[6|2Sj[ "
g d
0.50.6
0.4
0.5
0.4
0.50.5
ξ / π
η/π
0 0.5 10
0.5
1
0.4
0.6
0.4
0.5
0.4
0.5
0.5
ξ / π
η/π
0 0.5 10
0.5
1
]o 4, · ^ gO^ · NQ#$R. ¥c(!869&g>6!@69M = 10−3 69&2;
CFL = 20^ n 'ghG@NQjiXi!/('#%Q^| [be
,WC> /69&28^ 4:SKjQ "!iLO!/2#%4^
n e
"$#%" #% "$#%& # 2[jijzN[jiX`P\N[]k#%2:;G¥#$R27r";dWOsB6p,WC> /69 7r[R:!""569 #%";H69!869&2!". &2g>6!- ´69&ki!869&\WV@!.869. su!86Q &:d2Tr&2/69.ogm# · ^_aC^_ar,2 &G2@ &4.T #$R. ! MRV/69Hα (α = 1)
HMAF = (D + S1) ·D−1 · (D + S2)
· ^ n ^ Z (69z
HMAF = H + S1 ·D−1 · S2 · ^ n ^_[r
2p69#$R. ¥c(!869&<#B69¥e~6/:!U69";'^| ~z#%2:;H|4 71&~W:s\
GMAF = G∗ + V pMAF (Id−G∗)
VMAF = (Id−H−1MAF H)
· ^ n ^ ·
^| HMAF = (D + S1) ·D−1 · (D + S2) = H + S1 ·D−1 · S2
· ^ n ^ `:N[o#%&:&2";gh2|NQjiX_#%2:;D31&25!869&$&2!"5(69gh`&2|,WC> /69&<
p = 1'34&25"!"71
2 4 &(69.CP\N[]~!/2#%43r69&2;T2'gh2 4 &(69.269#$R. ¥c(!869&gm6!GMAF = Id−H−1
MAF ·K ^n z,2z&2^,2;Csd2MRRV"Q9g
MAF (p),2 &%±&~J[,#B69&&<69&(69. sC8^
%) )%A %& 5- - "!$#%!
0
0.25
0.5
0.75
1
Ave
rage
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL1
1.25
1.5
1.75
2
2.25M
axim
um(
ρ(G
))
0
1
2 Mach
0
50
100
150
200
CFL
]o 4, · ^ n ^ Z `N¬71/69r$¨.'gh69&2;v#B6eO #T,# ¨¥4O569#$R. ¥c(!869&Hg>6!z9g¬2[&2&C> /69 71SP\NQ]!/2#%4^ZQ&. r%2BN[j[Xi!/2#%43¬2d6871/69rd69#$R. ¥c(!8694&gm6!T9g|2BP\N[]¦!/2#%BR7:;"6H0:, r::;u;269#$R &BgmQ69. .] n &:,#TWV[69&2;HgmQ69. .¬P\6!/u&:,#TWV c24, · ^ n ^ Z '^ZQ&Cgm,&(69. sr3
n/Z
^B69./H&2%(69S2%#B6eO #T,2# 769. ,2B9g52d69#$R. ca!"69 4&g>6!SS4869pa69&A,& xs\gmS2/69#%M!"&2;C &28^X5#%869&2|(69z2iP\N[]#%2:;H5,&2/69W.M¥g¬&2$,WC> /69&uR:!""| |,2";D^ 2G69#$R. ¥c(!869&g>6!@9g|2%#T,. ¥ ^"R)P\N[]§!/2#%B,2 & [beu,WC> /69&2@pR."; &c24, · ^ n ^_[C^[ZQ &d2T,WC> /69&bR:!""/Q #$R71"[;C/69#B69!869. . s<2TB!&2!s<9g2T/!/2#% &2!"Q "!"7r2Q#B69Rv9g_2Qj[ "!zLO!/2#%$ c24, · ^ c ^ · '^K5";"83C62#B6eC #T,#®7469. ,29g2v69#$R. ¥c(!869&gm6!@#B69 &2$"0:,(69.<,& xs132G/69W . xs@&2,";D^b[&2!"432d#T,. ¥
0
0.25
0.5
0.75
1
Ave
rage
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL0
0.25
0.5
0.75
1
Max
imum
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL
]o 4, · ^ n ^_[C|Nt71/69rd¨.'gh69&2;u#B6eO #T,#³¨ 4:Q69#$R. ¥c(!869&\gm6![9gK2p#T,. ¥ ^"RbP\NQ]!/2#% ^| [be$,WC> /69&28^ ^"R\P\N[]+!/2#%pi6%7rs~!"#$RV 71T#%2:;b!"#$R(69TB2TN[jiX5/!/2#%4^ Q"!/&2 0:,2&(69W."|2[,2i9gt &Cc2&2¥/[¥#%'Ru69&2;G 5"01, "z6T. .S69#%,&:z9go ^"R25W/69 &H2i/69#%B!&2!sv(69&~2Sj[ "![LO!/2#%4^
%) )@ i&?./ " "!$#%!n M,2i&2^.86;A6d.:!869.K,2;Cs<9g2P\NQ]!/2#%@ghS6d.^5Pb6!/A!86H
M = 10−3 '^TXx&<c24,· ^ n ^ · 3^;Ch^2$69#$R. ¥c(!869&kg>6!M9g2¥4 &(69.P<N[] !/2#%$69
CFL = 1069&2;\2&2
.U69";%M2z#T,. ¥ ^"RGP<N[]b!/2#%Q69CFL = 106 ^|
20,WC> /69&2^`NQ'eORV"!";$gh#
2~4.W(69.[69&(69. sC83 &2/69W . ¥Ys+69"%gh%2u 4 &(69.[!/2#%~ &+2~.^ ^5687r&:,#TWVd4&28^ 2dR.$69./<2^z@(6983` &)2G/69#% ^5687r&:,#TWV$4&283`2G;269#$R &k9gQ,2!/6<!/2#%d0:, iRV:8^ 0 &~2S2z(69&2;D322M#T,. ¥ ^"R\!/2#%MR7O;"Q69&u'e!"'. .&:Q;269#$R &d71z2&: ^5687r&:,#TWV;#B69 &_^KJ[7r2."3C!"#$R(69i@2ij[ "!LO!/2#%436^[&2M6T. .i.459gB!&2!sv &~2M.^§g¨"0:,2&2!sH6986C^ z.4z!869&~WVS! !,#T7r&:/;uW:sG¥&(!'/"6 &%2M&:,#TWVQ9g2S ^"R28^NQRR8eO #B69]a6! ´69& #%2:;,2 & 6,WC> /69& R:!""R7:;"71s;C &:WV(687OT6!"!";C &<H2%!/2!"B9g|]a6! ´69&¦¨W(6!%N[jiXQS #$R71";P<N[]K'^DXx&2;"";D3t2 /69 71v$#T,. ¥ ^"R+694s71869. .¥gQ\WVH69. #%4,2."/Tgm@2HNQjiXp!/2#%,^| . G;C/69#B69!869. . s¢ &2!86"d2<B!&2!s 9gM2<P\N[]#%2:;D^L:,2!/ ",. d(6871uWV"&¢gh,2
n [
0.5
0.6
0.4
0.7
0.4
0.8
1.05 0.9
1
0.3
1.05
0.3
ξ / π
η/π
0 0.5 10
0.5
1
0.5
0.6
0.40.4
0.4
0.5
0.4
0.20.3
ξ / πη
/π0 0.5 10
0.5
1
]o 4, · ^ n ^ · 5N#$R. ¥c(!869&<g>6![69M = 10−3 69&2;
CFL = 106 ^ n 'ghT 0 4 &(69.oP\NQ]!/2#%4^z 4:p`P\NQ]!/2#% ^| [be$,WC> /69&28^
&:71" 169";H & _a2` 69&2;v(MW27469&2z#B6;M &~z"!&u;$&2;C |gh#2S!"&2!. ,2&2R7O,2. sv;C/6^|&~W:sG2"p69,28^
# "$#% & " 6 &" " #%& 2TN n_ .U6e269&<R:!"";C,T #$R. "Q ^ds:#$#%!T ^5"'R(RVQ,WC> /69 &t^z[&2!"3V&2S(6%;'c2&2M ^%RV/69[65gh. .^zp
H1 = (D + S1)
H2 = (D + S2)
· ^ d ^ Z
2p69#$R. ¥c(!869&<#B69¥e~6/:!U69";'^| ~z#%2:;H|4 71&~W:s\
GALJ = G∗ + V pALJ (Id−G∗)
VALJ = (Id−H−12 H) · (Id−H−1
1 H)
· ^ d ^_[r
^|
H1 = (D + S1)
H2 = (D + S2)
· ^ d ^ ·
Xxt¬ &:" &[QRV &:o,2¬(69o &T2 ^9;C #%&2&(69.!864392N nt !/2#%839 &pg>6!839;&:!869.$2MP\N[])!/2#%4^X&2;"";D32 #$R.p69. rW/6$4 71"S
VALJ = Id− (H−12 D H−1
1 ) ·H · ^ d ^ `:
n4·
69&2;~W:7O,2. s,^i(6871$
H−12 D H−1
1 = H−1MAF
· ^ d ^ c &2"0:,2&:. sr3f^|;i&2K &:7r" 169gh,2K`#%2:;B69&2;% #$R. s@'gh`M25R7O,2K"!&_^[^7r839.¬,2¬#%&:&
MAF (p)69&2;
ALJ(p)69&2.&ro;&:!869.O &T2
3D!86437r&T2,4
2M ^$#%2:;z1"R~0:, S #$ .U69zRRV"@ _[b[ Z g '^
#%"$#% & " (%( 6 &" " #%& 2pN n pLv.U6e69&<R:!"";C,T"0:, "zgm, &&2- ^"R2QRVQ,WC> /69& !/a69R [r'^5Xx!869&~WVM!86| &:$2Mr&2/69._gh# · ^_aC^_arR7O;";
4RV/69 Hα
69S;'c2&2";<65gm. .^zS
H1 = (D + S1 − B+E−12 )
H2 = (D + S1 − B−E+12 )
H3 = (D + S2 − A+E−11 )
H4 = (D + S2 − A−E+11 )
· ^ Z eC^ Z
2p69#$R. ¥c(!869&<#B69¥e~6/:!U69";'^| ~z#%2:;H|2&~4 71&~W:s\
GALGS = G∗ + V pALGS (Id−G∗)
VALGS = (Id−H−14 H) · (Id−H−1
3 H) · (Id−H−12 H) · (Id−H−1
1 H)
· ^ Z eC^_[r
^| ~2S](,sO#TWV.S
H1 = (D + S1 − B+e−7 η)
H2 = (D + S1 − B−e+7 η)
H3 = (D + S2 − A+e−7 ξ)
H4 = (D + S2 − A−e+7 ξ)
· ^ Z eC^ ·
n z,2z&2^,2;Csd2SB!&2!sH9g!/2#%M,2 &%±K&uJQ,#B69&&\69&(69. sO^%)%A )%A L & 5 - "!$#%! 2T687r/69r
ρALGS69&2;H#B6eC #T,#
ρmaxALGS
69#$R. ¥c(!869&ug>6!9g`2SN n SLv#%2:;<69SR."; &bc24, · ^ Z eC^ Z ^BN&~ &2/69W . xs\M&2!"869W.$gmM2$ 42"@] n &O,#TWV8^%I`7r&A2,4!/2#%MR7O;"Q67rsGr::;H4.W(69.¬;269#$R &%gm69. .o] n &:,#TWVQ69&2;u69. .tPb6!/~&:,#TWV832 !869&&2TWV$,2/";bgm@] n 4869T(69&
30^GXx&2!86 &u2%&:,#TWV@9g| ^5"'R(p;~&2T! !,#T71&:
2"5 &2/69W . "831&$25!"&:69sT K;/69"7r&$#%|;C/69#B69!869. . s2|69#$R. ¥c(!8694&$g>6! c24, · ^ Z eC^_[r'^ 2%gh#T,.U6 · ^_aC^ n MR7:;"6~!.869@'eOR.U69&(69&9g|TWV(687:83K69p.86^|2&<6@/!869.U696;C7r"!&~RW.#z!"&2;|KW,|2i#%"!/(69&#'eC&2;|2MsO#!"6/[z2p.U69&2 R |G −G∗| = |V |p|1 −G∗|
!869&<WVp #$#%";CU69. s<;";C,2!"";ug¨# · ^_aC^ n 'V¥g |V |p > 13
n `
0
0.25
0.5
0.75
1
Ave
rage
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL1
2
3
4
Max
imum
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL
]o 4, · ^ Z eC^ Z oNt71/69rM¨.'gho69&2;#B6eO #T,#¨ 4:69#$R. ¥c(!869&$g>6!K9ga25N n SL@!/2#%I^z¥&2S,WC> /69&_^
^|!/u,&(z%WVM2S!86 ^|2&<N n SLvQ69RR. ";u$2Mc2-;Q,RM^| &2;~ #$R. ! [/69r4322& &2!86 &p2|&:,#TWV9g ,WC> /69&2
p&2;Kig¨,2K &2!86/|2z;C D&2!"|WV ^"&
G∗
69&2;G
,RBT6S!"#$R.Q;C 71r&2!"Q9gD2z /69 71/. 7r8^L:,2!/G6MWV(687:,5^56,. &2";BW:s$,2.^ & Z [ ^N&a69.¥sO´8"%9g[2HX 9XpN n SL)!/2#%HRV-gh#%"; &+^69.k2^,&2/69W.WV(687Ozgm 4b] n &:,#TWV8^ #%869&2Q(69 &2/69W . xsu;:"&2[69Sg¨#2S&(&C!"&2&:z!/2!"M9g6pc2-;| #$R. ! z/69r S ;O;z'eCR. ! z/69r[W,5|;C "!. sd.U69";i2c2-;2'K #$R. ! /69r4^ `RV &:Kg¨,2!"&Cc2#%";$W:s@/69W . xs@",. KW/69 &2";+^z¥N n SL~69RR. ";<d6 n 6e:"&2;Cxs:RVS #$R. ! i!/2#%Sgm0^|!/<&2% &2/69W . xs~Q;"!"; Zn ^Xx&kgm6!83¬S &2/69W . xsA9g
ALGS(p)"#%pHWVB!.4. sk!"&&2"!"; ^| 2%#T,. ¥;C #%&2&(6.
,RM^| &2;B #$R. ! /69rpoN n SL &&2 /69&dR:!""!869&dWVR. &:S ^5S#B69 &d ^"R28312|c2&2B &2
x;C "!&_3`2B/"!"&2;&2B &2
y;C "!&_K&2^S3o,2 &b6~ ^"R &)&2B;¥/!&
&. s687r;T2G:!"!',2&2!"G9g¥&(/69W . "@W,T.86;@\6~RV:@;269#$R &k9gz2d ^| .d2!"#TW &(69&\9goWVu ^"R2z!869&~R7O;p6@71sdr::;~;269#$R &B¥g2M #%'R~|"!";D^
%)%A )@ i&?./ " ?!$# !Xx&vc24, · ^ Z eC^ · 32p69#$R. ¥c(!869&ugm6!9g2MN n pL Z 5!/2#%M|R"&:";vgm
M = 10−3 69&2;gmp ^\;C &:%] n &:,#TWV8^<NQpR7O,2. s2^|&)&2d4.W(69.|69&(69. sO3tgm
CFL = 106 32%'eOS6G. .$RV"69<gm ^|2 !/A2B69#$R. ¥c(!869&Ag>6!MM4869S(69&A,& xsr^ MRV869<.:!869";k &k27rs<.^g¨"0:,2&2!sb6986C3t!"&2"0:,2&:. sr3_ &kR/6!!869.!"#$R,/69&283t¥g2@#%"b&2@c2&2v&2,24_3K 4 ] n &:,#TWV%!869&WVG,2";69&:sO2^ ^| 2,$ 4r &A &2/69W . xs1^ #B68sAWVd2B86/&gmT2G!"&:71&:&(69. ^|;# ^|!/!"&2;BM69,2-YLO;.#%2:;69RR. ";v6%W.:!/1> #$R. ! M,RM^| &2;\/!/2#%6Q,&2!"&2;C &(69. . s\/69W.4^ 0 &\2T2i(69&2;D3V2TN n SL Z !/2#%|69
CFL = 20`6B!&:625jQ "!LO!/2#%M c24, · ^ gO^ · o69&2;@#B69 &2K/69W.571
2%&: l^5687r&:,2#TWVT;#B69 &_^ :,23_,&."S2$ #%'R(6SHWV/!";D3t2%N n SL Z !/2#%|R7O;"6ir:1;B;269#$R &231#%"71K K"0:, "&. s&2z,WC> /69&Q^|!/B`71s@!/2869R
nbc
0
0.25
0.5
0.75
1
Ave
rage
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL1
1E+07
2E+07
3E+07
Max
imum
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL
]o 4, · ^ Z eC^_[CoNt71/69rM¨.'gho69&2;#B6eO #T,#¨ 4:69#$R. ¥c(!869&$g>6!K9ga25N n SL@!/2#%I^z¥[be$,WC> /69&28^
gm|R/6!!869.¬!"#$R,/69&28^
0.4
0.6
0.4
0.5
0.4
0.5
0.5
ξ / π
η/π
0 0.5 10
0.5
1
0.4
0.6
0.4
0.5
0.4
0.3
0.5
0.40.2
0.3
1.15
ξ / π
η/π
0 0.5 10
0.5
1
]o 4, · ^ Z eC^ · N#$R. ¥c(!869&gm6!@9g5(BN n SL Z M!/2#%G69M = 10−3 ^ n 'ghd
CFL = 20^
z 4:pCFL = 106 ^
n a
& # " &" "$#%&' q !/2#%d@!"/69 &. s2G #$R."@.U6e69&#%2:;D^bXx&2;"";2v6/:!U69";) &4.GRV/69Hα (α = 1)
|4 7r&~W:sb
HPJ = D · ^ ZbZ ^ Z
2p69#$R. ¥c(!869&<#B69¥evz2'ghp;'c2&2";u65gm. .^zp
GPJ = G∗ + V pPJ (Id−G∗)
VPJ = (Id−D−1 H)
· ^ ZbZ ^_[r
ZQ&Cgh,&(69. s13O,2!/v6p #$R. ! xs% &:7r. 71"69&d #$RV/69&:.U6!/B9g_B!&2!sQ^|!/ ^| . . 69RRV869 &d2±&~J[,#B69&&<69&(69. sCz;"! WV";~ &H2M&2'eC|R(69/694/69R_^%)%AA*)%A L & 5 - "!$#%! 2iq /!/2#% ^| 2,5 &&25 /69&
p = 1K5&4. sB,&2/69W.Qgm|69. ._5] n &:,#@WV'/z69&2;dgh
69. .OPb6!/T&:,#TWV5 c24, · ^ ZbZ ^ Z '^ ¬ &2/69W . xsStR(69U69. . sp! !,#T71&:";@¥g c e,WC> /69&269M#B6;%
PJ(50)!/2#%M|2&</69W.igh[] n &:,#TWV#B69. .(69&
30^ (p687r/69rS9g2
69#$R. ¥c(!869&ugm6!"#%z$2^6@/692zr::;~;269#$R &B &u,2!/~!"&2;C &28^ n z,2|&(^§#B6916G.:!869.69&(69. sCM &A;Sv &:71" 169B6GW M#%/$;"R. sb2R(69!,.U69pWV(687Op9g2
PJ(p)!/2#%4^
0
0.25
0.5
0.75
1
Ave
rage
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL1
1.1
1.2
1.3
1.4
1.5
1.6
Max
imum
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL
]o 4, · ^ ZbZ ^ Z Nt71/69r%¨.'gh|69&2;H#B6eO #T,# ¨ 4:|69#$R. ¥c(!869&ug>6!9go2Sq Z |!/2#%4^
%)%AA*)@ i&?./ " ?!$# ! 2B.:!869.569&(69. sCppRV-gm#%";gh
M = 10−3 69&2;CFL = 20
h/69W.d!"&2;C &2'p69&2;",. 69MR"&:";~ &vc24, · ^ ZbZ ^ · ^Z[ &%q !/2#% ^| c e$,WC> /69&2Q&(69W."z%W/69 &<6@r::;
n g
0
0.25
0.5
0.75
1
Ave
rage
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL1
1.1
1.2
1.3
1.4
1.5
1.6
Max
imum
(ρ(
G)
)
01
2 Mach
0
50
100
150
200
CFL
]o 4, · ^ ZbZ ^_[CMN¬71/69rH¨.'ghi69&2;b#B6eC #T,# ¨ 4:i69#$R. ¥c(!8694&Agm6!S9g2$q !/2#% ^z¥c e$,WC> /69&28^
;269#$R &<7rp6G.U69r$R(69p9g2 ^56871&:,#TWVT;#B69 &_^ 2B69#$R. ¥c(!869&Ag>6!M#B69RAWV"!"#%"!.4i25j[ "!LO!/2#%4=?K&2M c24, · ^ ZbZ ^ · '12^7183rR,,2`4! . .U69&2K:!"!,831RV"!U69. . sgm2S2¥4<g¨"0:,2&2!"83V69.&d2T;CU69r&(69.o. &2
ξ = η^ d#$R(6 ´8pQWV(687Oi6%!,. &2@9g
2B69#$R. ¥c(!869&Ag>6!p69.&ξ = η
S69./vR.";bgmM2$q !/2#%B69&2;b2%j[ "!@LO!/2#%4^";C,2! &2\&:,#TWVH9g@,WC> /69&2~s:.;v."v4! . .U69&2HW,H9gp.U69ru69#$R. ,2; ^|!/!869&\2'[email protected];[B &2/69W . "8^ 0 &\2T2i(69&2;D3V &2!86 &G(p&:,#TWVM9g,WC> /69&2sO.;v64869H&:,#TWVH9gT4! . .U69&283Q86!/§9gT#B69. .~69#$R. ,2;4^lZQ &
100,WC> /69&2
69CFL = 20
69. .^z<#B69rv69. .2v4! . .U69&2769&83`#%dR"!. s13`s:.;,2!/#B69. .¥69#$R. ,2;u4! . .U69&2%(69%2s)!869&+&2b.&r%WV~;"!";D^j[, &A2~!",H9g[B,2;Csr3 ^56&2";)(692 n &2' 6!"4Wz/!/2#%v;CR.U68sO$6\ #$ .U69WV(687O@i&2 n &2' 6!"4WT n_ #%869&2.U6e269&dKRV-gm#%";G69.&p2
x;C "!&G2
y;C "!&v69&2;%&2,2!"!"" 71. s$69.&
WVu;C "!&2Q6| &H2MNQ. &(69' n &2 6!"W_#%2:;D^ 2 nt #%'2:;u69RR. ";H &H2[c2/px
R(6!"';C "!&H#B691",2[9gt2[RV/69 H1 = D+S1
69&2;d |6:!U69";VLJ
!":B!&:|86;i
VLJ = Id−H−11 H = −(D + S1)
−1S2 · ^ ZbZ ^ ·
2[69#$R. ¥c(!869&v#B69RG9g_#%2:;dR"&:";d &%c24, · ^ ZbZ ^ `2^ 2QB!&2!s%!.869. sB!.4Q25j[ "!LO!/2#%4=?`&2W,j^&2(69o2|69169 &$4/!' . .U69&269RRV86983169.&i2. &2
ξ = 0 &
z!"6/4^ 2iR.z9g¬2S69#$R. ¥c(!869&~gm6!69.&$|. &2M2^zz6 &2i!,71[R2&2#%&2&_^](WV\2Tq 69&2; nt #%2:;83a2 [4! . .U69&\RW.# 4. su;C,W2Q2T;269#$R &H69Q 4] n &:,#TWVQ &2!"T6.U69rT69#%,&:Q9gK,WC> /69&2Qz&2"";";~%!869&2!".¬2 &2p!,71 SD"!83#B69O &%2"M#%'2:;zRV&:U69. . sH'eCRV&2 714^
nbn
0.5
0.4
0.5
0.4
0.6
0.5
0.5
0.4
0.5
0.6
ξ / π
η/π
0 0.5 10
0.5
1
η / π = ξ / πρ(
G)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
PJ(50)Direct Scheme
CFL = 20 and M = 10-3
]o 4, · ^ ZbZ ^ · iNQ#$R. ca!"69 &Agm6!S9g2$q c eri!/2#%%69M = 10−3 69&2;
CFL = 20^ n 'g¨%
Pb69R~71|2 ^|2. ^56871&:,#TWVQ;#B69 &_^z 4:p`j[U69r&(69._R/9c2.p69.&ξ = η
^
0.5
0.4
0.5
0.4
0.6
0.5
0.4
0.6
ξ / π
η/π
0 0.5 10
0.5
1
η / π (ξ = 0)
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
LJ(20)Direct Scheme
CFL = 20 and M = 10-3
]o 4, · ^ ZbZ ^ `2MN#$R. ¥c(!869&Agm6!S9g2 n_ [beri!/2#%%69M = 10−3 69&2;
CFL = 20^ n 'gh%
Pb69R~71|2 ^|2. ^56871&:,#TWVQ;#B69 &_^z 4:p`j[U69r&(69._R/9c2.p69.&ξ = η
^
#1 "' (1( ! &" "$# & 2pLSLB.U6e269&HR:!"";C,M"0:, "5 ^ &&2- ^"R2zRV'z,WC> /69& !/(69R/' [r'C2" ^%R269S;'c2&2";~W:sd2M ^gm. .^| &BRV/69[65gm. .^zS
H1 = (D − A+E−11 − B+E−1
2 )
H2 = (D + A−E+11 + B−E+1
2 )
· ^ Z [C^ Z
nbd
2p69#$R. ¥c(!869&<#B69¥e~6/:!U69";'^| ~z#%2:;H|2&~4 71&~W:s\
GSGS = G∗ + V pSGS (Id−G∗)
VSGS = (Id−H−12 H) · (Id−H−1
1 H)
· ^ Z [C^_[r
^|
H1 = (D − A+e−7 ξ − B+e−
7 η)H2 = (D + A−e+
7 ξ + B−e+7 η)
· ^ Z [C^ ·
2&H&. sG&2i,WC> /69&H5RV-gm#%";D3aLSLd!/2#%iz69.!869. ."; n Z69RR8eC #B69[g>6! ´6&B#%2:;D3C69&2;+^| . .(WV;&2"; n ZzYLpL@ &B2z#B69 &2;9g 2"!&\¨/692(69&
SGS(1)'^
J[^S3C.z,2z,2;CsG2SB!&2!sH9g n ZzYLpLH69&2;SGS(p)
,2 &%2M±K&uJQ,#B69&&\69&(69. sO8^
%)%A @)%A L & 5 - "!$#%! 2#B6eO #T,#¯69&2;B2Q6871/69rz7469. ,2"9gD2[69#$R. ¥c(!869&dg>6!71,2] n 69&2;BPb6!/B&:,#TWV69MR"/&:";~ &Hc24," · ^ Z [C^ Z 69&2; · ^ Z [C^_[C^2AM&2!"S(69 n ZYLSLv/!/2#%Mz,&2/69W.MgmzPb6!/&:,#TWV!.4Bu,& xs1371&¥gz2v] n T#B69. .>^uXxpT2'ghd&2"!""69skuRV-gm# /7r/69. &&2i¥/'69&2i¥&b;iG! !,#T7r&:i2@ &2/69W . xsr^iN[[2^|&\ &<c24, · ^ Z [C^_[C3V,2 &
20,WC
/69&2z&(69W."5T"!"7152[B!&2!sG9g¬2ijQ "!LO!/2#%4^X&2!"869W.i(6983O,&2.¥ 1[2N n SLG/!/2#%432pLpLG!/2#%M#B69 &2/69W.S71&u69| 4\] n &:,#TWV^KQ^5'71832,('.^ & Z [ 2^¦(69QLSL
(p).4"6T4869|R(69z9go |B!&2!sdgh| 4u6RV"!5694K5RV &:
^| . .oWVpg¨,2[ &:71" 169";\ &<2@"!"&2;<R(69i9g[2"83 ^|2&ALpLHi69RR. ";<gm[. 7O &v6#B69¥eOmg¨"5 #$R. ! K/69r4^oJ[712."832LSLp#%2:;#B69 &2K9g(4869` &:"K &2!" `!869&WV86 . sd69RR. ";v&G,&2,(!',";G4;83O. r[q 9g¬!",43^|!/G&22[!86gm25#%2:;8^
0
0.25
0.5
0.75
1
Ave
rage
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL1
1.1
1.2
1.3
1.4
1.5
Max
imum
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL
]¬ 4, · ^ Z [C^ Z KN¬7r/69rB¨.'gh569&2;H#B6eC #T,# ¨ 4:z69#$R. ¥c(!869&ugm6!9g2 n ZzYLpLv!/2#%4^
d e
0
0.25
0.5
0.75
1
Ave
rage
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL0
0.25
0.5
0.75
1
Max
imum
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL
]o 4, · ^ Z [C^_[CKN¬71'69r%¨.'g¨|69&2;H#B6eO #T,# ¨ 4:|69#$R. ¥c(!869&ugm6!9go2TLSLv!/2#% ^z¥[be$,WC> /69&28^
%)%A @)@ i&?./ " ?!$# ! 2d.:!869.|69&(69. sCTT#B6;Bgm6u. ^5P\6!/&:,#TWV!86b
M = 10−3 p69&2;69CFL = 106 ^<Xx&
,2!/!"&2;C &2832 n ZzYLpL!/2#%v/69W.vW, $B!&2!sRV:83"RV"!U69. . s &2v.^^56871&:,#TWV$4&2~ c24, · ^ Z [C^ · '^Z[ &[be< &&2-> /69U&2%R7:;"$'e!". .&:%;269#$R &71v2b&: ^56871&:,#TWVu;#B69 &_3Q2bB!&2!sv7rs !.4/\)2kj[¥/!<LO!/2#%4=?u&2#B69O &S2QLSL!/2#%69&d69/6! 71|#%2:;D3: &$R(69 !,.U69gh!"#$R, & \(^z&%,&2,2!,2";4;832 &2!"M z!869&~WVS86 . sG #$R.#%&:";u&u,2!/~4;8^
& (%& ( 2R"&:`!/(69R`!"&2!. ,2;"2c2oR(69`9g(2";7r";@[2569&(69. sC9ga#%!.U6/!869. #$R. ! z#%2:;836^| H2MW*-"! 71i$."!|2i#%4zB!&:z869#%&:|gm52iW.:!/G #$R. ! !/2#% ^| :^|!/ ^MR.U69&~%!"#$R(69p2M#B69¥e:mgh"S869#%&:Q;71.RV";~ &~2M&2'eCzR(699g2d2"8^ n @,2T"!869.¥.5#%#%s/69rd(6T&2@s1TWV"'&)/691& &:b6!"!",&: &),$69&(69. sO9gpB!&2!sr3i^|!/(6BWV"&#B69 &. sgh:!,2/";¢& &: &2!<B!&2!s1^NQ#%&2<.¥ &2< #$R. ! 869#%&:83 24M(69#B691M,2p69Q#%S/69rS9g`;CU69r&(69.osC# &:71&th32N n SL
(p)^| p
7rsk#B69. .[¨xs:R2 !869. . sp = 1
2S69RRV8696S2%#%4TB!&:T869#%&:8371&2,4
#% &2/69W . "#B68s%:!"!,5&d71s@c2&2Q4;8^ 2QRV &: #$R. ! 869#%&:|,2!/G6q (p)
69&2;LSL
(p)D56p#T,2!/G.^,& 5!"4 &2!"Q2sB&. s%"0:, (.:!869.V &:71&v9g¬6
4× 4¨ &
2D
W.:!/ughM86!/k&2&C>. &2869M /69&AW,M &2!"@2s\69.G&2"";A6d.U69rM&:,#TWVS9g &&2M /69&2h"RV"!U69. . s\q
(p)WV"!869,29g24! . .U69suWV(687OM9g2$69#$R. ¥c(!869&kgm6!v6!/71@2
j[ "
d/Z
0.9
0.4
0.5
0.40.7
0.4
0.6
0.8
ξ / π
η/π
0 0.5 10
0.5
1
0.2
0.4
0.5
0.40.5
0.4
0.6
0.40.30.2
ξ / πη
/π0 0.5 10
0.5
1
]o 4, · ^ Z [C^ · NQ#$R. ¥c(!869&gm6!@9gz2GLSLA!/2#%G69M = 10−3 69&2;
CFL = 106 ^ n 'ghdLSL Z '^z 4:SLSL [ber'^
2 \(^l. 718^TXx&b ^ 3t69. .(@W.:!/\ #$R. !' S869#%&:SR"/&:";b,RkG&2^(6871@WV"& #$R.#%&:"; ^| &6 '#%:;. u,2!,"; \(^. 7183;";C!869";+k2~. ,& 9gi2~I`,.69&2;J[687O-YL:1"T"0:,(69&2@&569"U69&4;8^~J,#%!869.5'eORV #%&:!869";,D^| !":;%(687r%2^|&N n SL
(1)^56S &AR/6!!"B2$#%4pB!&:T869#%&:G¨169;C."/p9g|#%#%s
,2'D!"&2"0:,2&:. s13 N n SL(1)
^| . .oWVT/69 &2";b &\2"!"&2;<R(69M9g ^\6[2T'gm&2!"W.:!/k #$R. ! T869#%&:Hm6Sgm69@6T!"#$R,/69&(69.5B!&2!sAp!"&2!"&2";('^~XxS ^M^| .dRV &:i,MLSL
(p)3a2,4<gh,&2;<."Q!"#$R,/69&(6. . s<B!&:i(69&\N n SL
(1)^|2&\,2";u &\,
#%:;.t,2!,";O>4;:\(^. 71|$. 71i2iW.:!/:> #$R. ! Q:i,RM^| &2;~!/2#%43|R(69!,.U69. s &:" &v &\2@RVRV"! 7r9g69&k69RR. !869&k &\2@!"&:'eCi9g,&2,2!,";\4;8^ \RRVxsB &G#$ &2;D36pR(69!,.U6969&:& ^| . .VWV[;71";dTRV &:5869#%&:5 &G2Q&2'eCR(699g|p2" ^|2&.:: &ugmT69&69RR8eO #B69G. ,&)9g|2%#B69¥eOmg¨"B #$R. ! @/69rB &2!"2~R.U69&&2"; ¥#$R2. #%&:69&¦9gM2u#B69¥e:mgh"u869#%&:d &¦NML · P (6%WV~869. ´8";+gh,&2,2!,";~4;~!"#$R,/69&2@h&2p69.2Sq
(p)^| . ._WVi"!"&2;";~WV"!869,2M zRRV"
^|2&69RR. ";bG2#B69¥e:mgh" #$R. ! S/69r ^| . .WV$01, $;C &:igh# 24W2/7r"; ^z¥2MW.:!/G #$R. ! Q/69r'^
u H ``t ¯
d4·
A) H ``t
#% ' (
2c2MR(69p9g(2"M(6SWV"&;";C!869";Av2%;"! R&9g5 #$R. ! S869#%&O ^|!/R7O;@6$ 4\¥&O &2!TB!&2!s13a&(69#%. s<6g>6Q;269#$R &v9g69. .o'Q#%:;"8^z[^7183V6#%&C&2"; &2v &::;C,2!&¢9g[ ^ 3,2!/+69&B!&2!s),2,(69. . s)W/69 &2";+69$2v'eCRV&29gi6<.U69 7r. s 4,& $!"#$R,/69&(6.!4$69&2;&b#%#%s"0:, #%&:8^ 2v'e!"" 71#%#%s~;#B69&2;< &::;C,2!"";<W:s~69&u #$R. ! i/69rS#B69 &. sH.U69";uB2S,2p9g`W.:!/ 6!"WU69&#B69!"" ^|2&A. &2869 ´" &u2$&:,#%!869."\2,Ce"S/~6M~; 7r$2$ #$R. ! pR(69p9g2%!/2#%4^[&2!"4326 ^568sG";C,2!"M#%#%sv"0:, #%&:|$;71.R~ #$R. ! z869#%&Oz(69;$&2z. s&dgh,. . 6!"WU69&H#B69!""z69&2;d#B68sBWVi6!/7r";GW:s% &::;C,2! &% #$R. ¥c(!869&2| &G2['eOR. ! /69rB. &2869 ´694&_^ 2BR!"duR(68skgh@,2!/6u #$R. ¥c(";)!/2!"G9gz #$R. ! $/69rB6~.4@9g &: &2!vB!&2!s &2712."83`¥g2G,& $!"#$R,/69&(69.z!"4@$69.<";C,2!"";D36<!"#$RV 714.W(69.5B!&2!s!869&)WVdR"71";D^\NQ69. 86;CsARV &O";),@ &2dr&2/69.5 &::;C,2!&+9gz^ 3o6v.^5!"4 .^5>#%"#%s .^® &: &2!BB!&2!sk #$R. ! T869#%&:Tp&2B9g52%/6;'. ,&2p2#T,. ¥WC*"! 7r[R #$ ´69&vRW.#®R"&:/;GW:s2Q; 7469&v9gt6S4.W(69. . sB!&:d #$R. ! G!/2#%4^ 2<"!"&2;+R(69d9gMB2"BB&: . s+;71";+A2~7Oh^9g
Cost per pointper iteration
(to minimize)
Intrinsic Efficiency(to maximize)
Second Part
First Part
]o 4, `2^ Z ^ Z q69BR #B69.t/5ghQ69&~B!&: #$R. ! 869#%&:8^
dbc
d a
,2!/.^5!"4T869#%&:$6T69. &a69 7rd/. ,&2p~2B 4C!"4 .U69r'>#B#%s 4) &: &2!B!&2!sv869#%&:Q;/69 .";~ &~2[c2|R(699g2M2"ph"p69.@c24, `2^ Z '^LO71/69..^5!"4 #$R. ! $869#%&:(6871dWV"&;7r.RV"; &2!"d2G869. s: 4:"8^ 2G;CU69r9&(69. ´8"; #$R. ! %869#%&:$c2. s &::;C,2!"";W:s)q`,. . U69# 69&2;¢(69,2E" _a c &2v!"&:'eC%9g!"#$R" W.D\(^zM69&2;\"!"'&O. s<'eO&2;";\d.^5Pb6!/\,&2/"6;s'\(^z Z e 69. .^d";C,2!"@,& !"4MW,S2sb;v&2S&2,$&2,4A/;,2!&A &A#%#%s\"0:, #%&: ^|2&A.U69r%!869.RWC.#%@69B!"#$R,";D^~XpS2'gm%&2"!""69sbuRSgmp71&#%B¥#$R2.¥¥c(";#%2:;@,2!/6Pb69¥e:]2"T #$R. ! i869#%&:^ 2"T#%2:;M!"&2[ &<R.U6! &H269W2. ,T7469. ,2"i9g 6!"WU69&d#B69!""(69569RRV869 &B2 #$R. ! 5/69rQ6gh2. &2869 ´69&v9g¬69&B,R^| &2;d!/2#%43:W:s2 zRV"!/69. 6;C ,2 ·bn ^ 2&_3269RR8eO #B69 &%2iR:;C,2!|WV ^"& 6!"WU69&~#B69¥eH69&2;G #%' &2!#%&:9g 2!"&271";%769U69W."W:s2z #%'> &2!#%&:59g 2]\2,CeO>7r"! g Z 3r RV4 W.W/69 &~6T,. sdP\69¥eO]2"[ #$R. ! z!/2#% ` n ^[^7183^|2&v.^5Pb6!/v,&286;Csl\(^zz69!"#$R,";\,2 &v6B;C,(69.¥> #%RR &H69RRr6!/_3 2T!/('#%T.4"Q Q#B69¥eOmg¨"TRRV"S69&2;WV<,& Q!"4[69&2;~#%#%sH"0:, #%&:Q &2!86/p 4&¥c(!869&:. sr^ 2pWC*"! 7rT9g`Q!/(69RQd,. &2@69&< #$R. ! [869#%&: ^|!/\R"71"2T #$R. ! xsu9gK2TP\69¥eO]2"@69RRr6!/<gh.^5Pb6!/d,&286;Cs \(^z8^ 2Q;CXB!,. ".U69";BpO &2;B9g\(^z^| . .aWVQ;/69 .";G69&(;% 5^| . .WVp2^|&<2^%869R<2SWV&2'c2[9g`2 ,r.>=?R"!"&2;C &(8=?[RRV"[/B6%R:;C,2!"@6,. sbP\69¥eO]2"$!/2#%ghp69. .KRV""; \(^z^ 2@gm. . ^| &~!/a69RB c 3VaH69&2;Wg4 ^| . .`2&AWV;71";~%2psC#B69!T69&(69. sOQ9g`2S &: &2!pB!&2!su9g`2S&2h^ #$R. ! Q869#%&:gmQ6^|;S/69&rT9gK!"&Cc24,/69&2$¨ &:7:/! ;<z7O!",283(86;CsH,&286;Cs,\(^z'( &uR(69!,.U6983a2 ±&vJ[',2#B69&&~69&(69. sOI^z¥. . 4 71Q,22[RRV,& xsdT,&2[RRV. sd2[!,-\;'c2& &(69p#T,2pWVB."!";W^|2&!"#$R, &\. ^5P\6!/&:,#TWV \(^z ^| )6HR"!"&2;C &2";,RM^| &2;!/2#%$o2i&2"";G$6;269R25|;'c2& &~@2ixs:RVi9g¬RW.#,&2;z,2;Cs\¨ &:7O!; 7:/!",28386;Cs ,&286;Cs2j^| . .(WV|,&2;. &2";D^ 2|.U6!/(69R9g "!"&2;$R(69^| . .(71&:,(69. . sR7O;#%v!"#$R(69/&2$WV ^"&¢69&B!&:% #$R. ! $W.:!/869#%&:d69&2;2vPb6¥e:]2"H!/2#%"71[R"&O/69 7r"-!86/"83D!"#$R,";k&k,2!,";b4;8^ 2@RV4¥W2¥. xs<vW/69 &A6G/69-g>6!sA4.W(69.5B!&2!sAgh#6u.^5!"4@#B69¥eOmg¨"d869#%&: ^| . .WVd;#%&2/69/; &6 ^568s!"&:7O &2! &<&2,4$*Y,2¥ghsugh,2T;7r.R#%&:pghS2% #$R.#%&:/69&9g5S"!/&0:,2% &2M&:,#%!869.tR.U69-gm# NML · P^
" #1 & # #1 " # # )@ )%A ; "; # 3 #f#%5.6# 3I ; !n 5,2c2;"/! WVQ2i;" 4&GR &2! R."|9gt2[Pb69¥eO]2"Q #$R. ! 5869#%&:5gh2 [jI`,."0:,(69&28^KX&v2i#B69&&29go2[c2|!/(69R832. |,2z!"&2;|2M!.U6z9g¬,RM^| &2;~!/2#%"]^|!/69S;'c2&2";u6|gh. .^zp
Fni+ 1
2,j
=((µ1f
E)n −D1δ1wn)i+ 1
2,j
Gni,j+ 1
2
=((µ2g
E)n −D2δ2wn)i,j+ 1
2
^|2D1
69&2;D2
69M#B69!""9g&:,#%!869.¬;C R(69&_^ 2M #$R. ! Q/69ri86;p
wn+1i,j − wn
i,j
∆t+
(δ1F
δx+δ2G
δy
)n+1
i,j
= 0F`2^_[C^ Z
d g
2T; 7469&b9g6BPb69¥e:]2"p #$R. ! M/69rpQ!869";b,Q &< ^dR28^]o 83a&. s~2T;C¥R(69 71iR(69/z9g2M&:,#%!869.)\2,Ce"69M. &2869 ´8";<69,&2;H #%S.7r.
(n+ 1)
[D1δ1w]n+1 = Dn1 δ1w
n + ∆ [D1δ1w]n ≈ Dn1 δ1w
n +Dn1 δ1∆w
n ,
[D2δ2w]n+1 = Dn2 δ2w
n + ∆ [D2δ2w]n ≈ Dn2 δ2w
n +Dn2 δ2∆w
n ,
F`2^_[C^_[r
^| .$2%!"&:";AR(69p9g2$&:,#%!869."\2,Ce"p69%!"#$R,";A,2 &H2%;'c2& &9g52¥#% &2!#%&:
∆φn = φn+1 − φn µ1(f
E)n+1 = µ1(fE)n + µ1∆(fE)n , µ2(g
E)n+1 = µ2(gE)n + µ2∆(gE)n .
F`2^_[C^ · [&2!"432i&:,#%!869.\2,CeC"Q69| #%M.71.
(n+ 1)&2^86;A
Fn+1 ≈ Fn + µ1∆(fE)n −Dn1 δ1∆w
n ,
Gn+1 ≈ Gn + µ2∆(gE)n −Dn2 δ2∆w
n .
F`2^_[C^ `:
2z"!"&2;%R%!"&2` &%R.U6! &p2z;C R(69&B#B69!""D1
69&2;D2
WOsT2|&:,#%!869.V!869.U69;C R(69&<9go2Sz,269&27G!/2#%%
D1 −→ 12ρ(A
E)Id = 12ρ
E1 Id
D2 −→ 12ρ(B
E)Id = 12ρ
E2 Id
F`2^_[C^ c
J[b,2!/6) #$R. ¥c(!869&!869&§WVbRV-gh#%";S^| 2,<69. &+2b/69W . xs9g@2\&:,#%!869.869#%&:pWV"!869,2/$2%;C R(69&#B69!""@69%;'c2& %RV4 7r4^ 2$&:,#%!869."\(,CeC"T69M¥#%.71.
(n+ 1)69ic2&(69. . sH69RR28eO #B69";<6|gh. .^zp
Fn+1 ≈ Fn + µ1∆(fE)n − 12(ρE
1 )nδ1∆wn ,
Gn+1 ≈ Gn + µ2∆(gE)n − 12(ρE
2 )nδ2∆wn .
F`2^_[C^_ar
Xx&2 &B'eOR"&2pF`2^_[C^_ar &O(i #$R. ! Q/69rBF`2^_[C^ Z 5s:.;SC0∆w
ni,j + C−
1 ∆wni−1,j + C+
1 ∆wni+1,j + C−
2 ∆wni,j−1 + C+
2 ∆wni,j+1
= −∆t · Rni,j − σ1δ1µ1∆(fE)n
i,j − σ2δ2µ2∆(gE)ni,j
F`2^_[C^ g4
^|2M2S'eCR. ! z";C,(69. Rni,j
z;'c2&2";<65gm. .^zS
Rni,j =
(δ1F
δx+δ2G
δy
)n
i,j
,
69&2;'^|2M2S!":B!&:C
69M&2^4 7r&~W:sb
C−
1 = −12σ1
(ρE1
)ni− 1
2,j
C+1 = −1
2σ1
(ρE1
)ni+ 1
2,j
C−
2 = −12σ2
(ρE2
)ni,j− 1
2
C+2 = −1
2σ2
(ρE2
)ni,j+ 1
2
C0 = 1 − C−
1 − C+1 − C−
2 − C+2
F`2^_[C^ n
dbn
I`71&:,(69. . sr3 2@ #$R. !¥S!"&: W,&2M &\2 n ML<69&2;\QiL~9g/!/2#%vF`2^_[C^ g4[69T,. s<P\69 eO]2"~69&2; $$&2!"869W.u(69$(H #$R. ¥c(!869&2B(69%(6871vWV"&+RV-gm#%";+;k&2d6 "!%2'eCR. !¥/69rS6!"!,/6!sr^ :,283C2p6!"!,/6!sv9go(M!/2#%M#B69 &2z,&2!/(69&r";~R7:;";H(69z2#%2:;k!"&:7rr"iH6d86;Cs:/694^SP<"7183 .i,2i#$R(6 ´8@(69iMP\69 eO]2"@ #$R. ! 869#%&:`;:"&. sS"01, #$R. ¥c(!869&2K69&2;69RR8eC #B69&2`71¬2;C R(69 71#%; 71";gh#2T. &2869 ´69&_^ 2@!/2#%vF`2^_[C^ g4 &:71. 7: &G &2!#%&:i71"!
∆wn 69&2; \2,Ceu &2!#%&:71"!
∆(fE)n 69&2;∆(gE)n (6|&2^$WVS. 7r";D^
,UMJLE6N&:,#TWV%9gQ7:h^|RV &:/%69v&2!",&O"; &2G. /69, ^|2& $!"#%"\2v; 7469&+9g2iR7O,25#B69¥e:mgh"i #$R. ! /69r4C2sv69iW \2sG7O h^5;H2[gm52M/691[9g`6!"#$R.'gm&2!"i'eO &+^O^](. .^| &%2[7O h^|RV &O6;R";vW:svL:(697v69&2;HJ[69469(6 g Z 69&2;.U69z,2";vW:s n &2 & ` n 326/69 &%RV &:|gh|2i#B69¥eOmg¨/M869#%&:Q; 769 & ^| . .WVM2S!'.U6/!869._. &2869 ´869 4&<9g2M&:,#%!869.)\2,Ce"69,&2;~ #%M.7r.
(n+ 1)
Fn+1 ≈ Fn + (AE)nµ1∆wn −Dn
1 δ1∆wn ,
Gn+1 ≈ Gn + (BE)nµ2∆wn −Dn
2 δ2∆wn .
2&_3RV-gm#$ &G6RV"!/69.t/6;C ,2 #$R. ¥c(!869&<71|2S;C R(69 7rM#%z.86;zH
Fn+1 ≈ Fn + (AE)nµ1∆wn − 1
2(ρE1 )nδ1∆w
n ,
Gn+1 ≈ Gn + (BE)nµ2∆wn − 1
2(ρE2 )nδ2∆w
n .
Xx&2 &%2"M'eCR"&2z &:$2M #$R. !' Q/69risO.;S
∆wni,j + σ1δ1((A
E)nµ1∆wn)i,j − 1
2σ1δ1((ρE
1 )nδ1∆wn)i,j
+ σ2δ2((BE)nµ2∆w
n)i,j − 12σ2δ2
((ρE
2 )nδ2∆wn)i,j
= −∆t · Rni,j
F`2^_[C^ d
N|5RV &O83C 55&2!"869W.M(6983269. 2,4u6@RV"!/69.D/6;C ,2z #$R. ¥c(!869&:^565 &::;C,2!"";D322n ML~ . .o!"&:/69 &2 6!"WU69&b#B69!""
AE 69&2;BE ^Q[&2!"43V2p//. ,&b9g2TsO/'##B69 &2
'eCRV&2 7rM69&2;G2[,2".U69";v@#%#%sd"01,2¥/'#%&O69Q&(|. 7r";D^`Xx&v;5! !,#T71&:Q;CXB!,. xs13(2SR:;C,2!i9g`2 6!"WU69&\#B69¥e<69&2;~ &2!#%&:/69.o71"!Q!869&uWV@69RR8eO #B69";W:sG2i &2!#%&:[9go2 \2,Cev71"!p
AE · µ1∆wn ≈ µ1∆(fE)n = µ1
((fE)n+1 − (fE)n
)
BE · µ2∆wn ≈ µ2∆(gE)n = µ2
((gE)n+1 − (gE)n
)F`2^_[C^ Z er
Xx&2 &%2"i.U69&2 &F`2^_[C^ d sO.;S
C0∆wni,j + C−
1 ∆wni−1,j + C+
1 ∆wni+1,j + C−
2 ∆wni,j−1 + C+
2 ∆wni,j+1
= −∆t · Rni,j − σ1δ1µ1∆(fE)n
i,j − σ2δ2µ2∆(gE)ni,j
^|2BF`2^_[C^ g45|"!"71";D36^| ~2S!":B!&:C±
p
. .t;'c2&2";HW:skF`2^_[C^ n '^
dbd
L: &2!"#%&:,#%!869. \2,Ce"¦,2!/6§N|ZMLOP ZMLOq !869&&2§WV86 . s ^| &,&2;2~gm# 9gT6!"&:"; \2,e 69,4#%&:";4^| ¦6A#B69¥e v!869.U69G;C R(69 & \2,CeD3|srv69&227Oh^|RV &:gm2;" 4&d9g_2Pb69¥eO]2"z #$R. ! 5!/2#%Q!869&dWVz4 71&_^ n ,2 &2;"";d!"&2;2M&:,#%!869.)\2,Ce"
Fn+1 69&2;Gn+1 69&2;H.z,2z'eCR(69&2;H2" \2,CeC"Q65gm. .^zS
Fn+1 = Fn + ∆Fn ,
Gn+1 = Gn + ∆Gn .
Xx&d/;TW/69 &~6pPb69¥e:]2"Q #$R. ! |/69r43C69&G &2!"&2&:|769. ,(69&H9gt2 #%'> &2!#%"&:\2,Ce7r"!vvRV-gm#%";D3 2\.U69u69b!"#$R,";§,2 &2<c2H;~R(6!"A6!"!,/69z,269&27G!/2#%%
∆FRus = µ1(∆(fE)n) − 12ρ
E1 δ1(∆w
n) ,
∆GRus = µ2(∆(gE)n) − 12ρ
E2 δ2(∆w
n) .
T"!869. .`a69M&2!"286;Cs<. ,&ki86!/2";D3_2"@ &2!#%&:S7469&A69&2;D3D2'gh43_2!/2!"9g 2z,2/69&27!/2#%|S7469. ,(69z2#;21"&26 D"!'26!"!,/6!s$9gD2z. ,&_^`JQ^ &2 &i2"'eOR"&2¬ &:Q2 #$R. ! `/69rs:.;`69169 &@2R7O,2. spW/69 &2";@#B69¥eOmg¨" #$R. ! Q/69r%
∆wni,j + σ1δ1µ1∆(fE)n
i,j − 12σ1δ1
((ρE
1 )nδ1∆wn)i,j
+ σ2δ2µ2∆(gE)ni,j − 1
2σ2δ2((ρE
2 )nδ2∆wn)i,j
= −∆t · Rni,j
2#B69¥e:mgh"869#%&:K!869&:,2oWV5"&$6`6&2&C!"&2&:¨ &@2r&2/69.!86t #$R. ! /69&9gS6k4 7r&¢,RM^| &2;¢'eCR. ! d/69rHp2~ #$R. !¥v69&2;¢'eCR. ! d/69r"%WV &!"&2&:Q^|2&¢2z,269&27/!/2#%z69.S,2";$gmK2'eOR. ! R(6!"; /! ´69&_3^|!/B/69. s%;&2 &BR/6!!"WV"!869,[email protected]@&:,#%!869.;C R(69&A6/:!U69"; ^| bM!/2#%4^ 2@;86v9g!",R. & !"#TW & &T769,2 #$R. ! 569&2;%'eCR. ! /69r"K &B;KS #$R7rz2|4.W(69.aB!&2!s$RRV"9gi6 \(^. 71@(6WV"& &:71" 169";)gm@ &2/69&2!"v & Z 3KW, &2v!"&:'eC$9g[W.:!/1> #$R. ! !/2#%"8^ 2|"!"&:"^5 c/Z 3^|2|269WV71|,2/69&27:>xs:RV5 #$R. ! /69r|`&2^¢!",R.";+^z¥69& n_n >W(6/";'eOR. ! T/69rd6g¨S(687O &uWV"&A,2/;A &!"&*Y,&2!& ^| 2p,RM^| &2;'eOR. ! /69r"834 . . ,2/69"2I\('eO W . xs@9gV,2!/%6Q&2&C!"&2&:K!/( !"4^`J[69,/69. . s13424.W(69.2B!'&(!'s9g2p#B69¥eOmg¨"T869#%&:i;RV&2;i&<2p,& [!"4(69[!869&\WV@6!/7r"; ^|2&b6!,(69. . su/.¥7O &2d #$R. ! $/69rbF`2^_[C^ g4F`2^_[C^ n G$@RV &:@;/69 .";)286g¨83K &)R(69!,.U69 &)2G!86G9g69. .¥RV"";'\(^z836^|!/~ &::;C,2!""[6@R"!"&2;C &2 &d#B69¥ev &~2M #$R. ! /69r4^
)@ )@ ; !2& 2"7# & &)( 7=; 9<7#M# #;H; !2.M="; 14;Xx&%Ra69/694/69R_3^i^569&:p,. &22^ p. 71z2Pb69¥eO]" #$R. ! 5/69ri^|!/G!"&:/69 &27469U69W.[ &2!#%&:|7r"!
∆wn 69&2; \2,Ced &2!#%&:|71"!∆(fE)n 69&2;
∆(gE)n ^ 6!/71[/6 3269&H /69 71M"!/&0:,2S|,2";~69&2;H2 \2,Cev &2!#%&:71"!Q69[c2|.U6e";u65gh. .^zp
C0∆w(l+1)i,j + C−
1 ∆w(l+1)i−1,j + C+
1 ∆w(l+1)i+1,j + C−
2 ∆w(l+1)i,j−1 + C+
2 ∆w(l+1)i,j+1
= −∆t · Rni,j − σ1δ1µ1∆(fE)
(l)i,j − σ2δ2µ2∆(gE)
(l)i,j
Z ebe
^|2u2~'eCRV&2&:(l)
;&2"d2u!",&:G9gi(~¥/'69 7ru#%2:;D^¦Xx&¢;BA";C,2!"u2#%#%s~/69rp9gK2S#%2:;D3V&2p!869&\#$R.sH2Tqo &: 6!"Wo.U6e69&\R:!"";C,D^|!/u"RV"!U69. . sb86s<v #$R.#%&:T&k,&2,2!,";A4;8^N. .K'eO/6;CU69r&(69.#%T69%69.v.U6eC";D3:,2832SP\69¥eO]2"Sqo &: 6!"WhP<]¬q |!/2#%M86;p
∆w(l+1)i,j =
1
C0·[−∆t · Rn
i,j − σ1δ1µ1∆(fE)(l)i,j − σ2δ2µ2∆(gE)
(l)i,j
+ C−
1 ∆w(l)i−1,j + C+
1 ∆w(l)i+1,j + C−
2 ∆w(l)i,j−1 + C+
2 ∆w(l)i,j+1
]F`2^_[C^ ZbZ
TPb69¥e:]2"%gh#T,.U69&;:"p&2p"0:, d69&:sb#B69¥eA &:71& &2!"%2B!":B!&:C0
R,. sv!869.U698^ 2i#%2:;H52'gmM7rsG!/2869R~ &v#%9g¬WVu!"4|RV| /69&<69&2;v#%#s)/69r4^ 0 gi!",432Hqo &: 6!"WzR:!"";C,~ ^. .|O&2^|&k;CR.U68s6\RV:$ &: &2!B!&2!s+T6v2,24)69&(69. sCSS2'gm%"0:, ";A~;#$ &2 ^|22p2%.^5!"4T9g|2869#%&:p!869&b&2&22."/MW(69.U69&2!"iRV:SB!&2!sV¬,2!/6G,2;Csr3 . sO &~&b±&bJ[,#B69&&69&(69. sCgh57469. ,(69 & &: &2!iB!&2!sv69&2;v&G"->RW.#%| &v;5T/691Q &:6!"!",&:|26!,(69.t!"4z9g2S69R2Rr6!/_3H^| . ._WViRV-gm#%";H &~2igh. .^| &d!/(69R8^ 2P\69¥eO]2"| #$R. ! !/2#%z!",.;%WVz #$R7r";D3O6`g>696` &: &2!'B!&2!s$!"&2!"&2";D3:W:s" &%$2. ,&~#%2:;832,2!/u652M. &2'>869#%&:Q;"! WV";H &H2iR7O,2z!/(69RC8^S[^7r83D &2!" ^$69.G"01, @2@#%2:;bG#B69 &A"6s~G #$R.#%&:S&b,&2,2!,2";4;83)^/692S,&A^569;k2BL:sO#$#%!dM69,2YLO;.`.U6e269&AR:!"";C,43^|!/"#%p6r::;!869&2;C;269v\&2,v,2!/)"01, #%&:3`(69@@< #$R71v7rqo &: 6!"W|!"&2!"& & &: &2!@B!&2!s:^|2¥.T#B69 & &H86sHB #$R.#%&:i&<,2&2,2!,";<4;^zXx&2;"";D3 ^pR7O¥,2. sG2^";v(695|R:!"";C,S&(69W."|2[ #$R. ! .:!/G!/2#%i@/!"71|2ij[ "!QLO. 71B!&2!s ^| +6<&:,#TWV$9gQ &&2-> /69&($#T,2!/";C,2!""; ^| "RV"!bqo &: 6!"W>^ 2&2'eCK!/a69R831 &2!. ,2;C &p±&%J[,#B69&&G69&(69. sC6"^. .a6K769. ;269&B"-!86"83^| . .2WVz;";C!869";~2B,2;Csk9g|pR:!"";C,G!",R.";W^| 2dP\69¥eO]2"% #$R. ! @/69r%(69T(6z*Y,2SWV"&;"! WV";D^iP\869&M^| .43V.[,2[;"! WVT(^d#%:;C¥ghs~2Tq 69. r #/G6QG6;269Ri Qd2LSLv/694s1^ 2SP\]¬q 69. r #86;S
∆w(0)i,j = 0 , l = 0, p
∆w(l+1)i,j =
1
C0·[−∆t · Rn
i,j + σ11
2∆(fE)
(l)i−1,j + C−
1 ∆w(l)i−1,j + σ2
1
2∆(gE)
(l)i,j−1 + C−
2 ∆w(l)i,j−1
−σ11
2∆(fE)
(l)i+1,j + C+
1 ∆w(l)i+1,j − σ2
1
2∆(gE)
(l)i,j+1 + C+
2 ∆w(l)i,j+1
]
wn+1i,j = wn
i,j + ∆w(p)i,j F`2^_[C^ Z [r
2~LSL)69. r #³#B69r" #$#%";CU69. s,2/G9g2v/69"@(69%69GWV &k7469. ,(69";;C, &k2gm ^569;A ^"R¢¨"R_^2W(6!/^569;A ^"RaQ &;SH!"#$R,%2$&2'eOS&(^ :,283_2$&2h^
Z e Z
69. r #86;p
∆w(0)i,j = 0 , l = 0, p
]2 ^569;u ^"R
∆w(∗)i,j =
1
C0·[−∆t · Rn
i,j + σ11
2∆(fE)
(∗)i−1,j + C−
1 ∆w(∗)i−1,j + σ2
1
2∆(gE)
(∗)i,j−1 + C−
2 ∆w(∗)i,j−1
−σ11
2∆(fE)
(l)i+1,j + C+
1 ∆w(l)i+1,j − σ2
1
2∆(gE)
(l)i,j+1 + C+
2 ∆w(l)i,j+1
]
56!/^569;~ ^"R
∆w(l+1)i,j =
1
C0·[−∆t · Rn
i,j + σ11
2∆(fE)
(∗)i−1,j + C−
1 ∆w(∗)i−1,j + σ2
1
2∆(gE)
(∗)i,j−1 + C−
2 ∆w(∗)i,j−1
−σ11
2∆(fE)
(l+1)i+1,j + C+
1 ∆w(l+1)i+1,j − σ2
1
2∆(gE)
(l+1)i,j+1 + C+
2 ∆w(l+1)i,j+1
]
wn+1i,j = wn
i,j + ∆w(p)i,j F`2^_[C^ Z8·
XxK!.869(692[69WV7r[69. r #¯;:"&256;;d!"#$R.'eO xs%S2#%2:;d &2!"z &. s$"0:, "2G &:71&+9gQ2v!":B!&:
C0^P\"71 $ . .|;:"$&2 &:71. 7rH69&:s#B69¥e/69r4^Xx&
2T"i9g2T2"83V ^| . .WV@;&2";bP\]¬YLSLV^VN[['eOR2.69 &2";k869. 83 2@P\]¬YLSLu!/2#%T'eCRV"!";v@R27:;S6@WV'zB!&2!sG(69&H2MP<]¬q !/2#%4^ ^| . ._,2;CsB2iRRV"9g2""z/!/2#%z &H2M&2'eO!/(69Ri^|2&u,2!"!"" 71. sv69RR. ";H$ &:7:!;D37:/!",2z69&2;H,&286;Cs\(^RW.#%8^ 2'gm43¬.S,2p;"! WV% &A2%&2'eOp/"!' &2B'eO&2&9g52%Pb69¥eO]2"gh/69#%h^G$7O /!",2 \(^z8^
)@ )*% 7$; !$# & 7$& -"#%!$./& 25! 55& !Xx&2vc2B!/(69R83I^H/69 &2"; 6b #$R. ¥c("; #$R. ! d/69rvgh$(v7O!",2+\2,CeC"% &+/;%R"71i2S;CU69r&(69.¬;#$ &(69&2!"p9go2SX 9XXXsO#~^ 2M&:,#%!869.t7:/!",2I\(,CeC"69z #%i.71.(n+ 1)
69M2'ghS. &2869 ´8";<65gm. .^zS
(F V )n+1i+1/2,j ≈ (F V )n
i+1/2,j + (AV1 )n
i+1/2,j
(δ1∆w)ni+1/2,j
δx
(GV )n+1i,j−1/2 ≈ (GV )n
i,j−1/2 + (BV2 )n
i,j−1/2
(δ2∆w)ni,j−1/2
δy
^|2 ^"!869. .K(69AV
1 =∂fV
∂wx
69&2;BV
2 =∂gV
∂wy
^LO~6ivR"71@(Pb69¥eO]"$RRVxsr3#B69!""
AV1
69&2;BV
2
[email protected]!"";bW:s~2 i7O!",2iRV"!/69.`/6;C ,28^SX&k69RRV&2;C¥ebN@3a [M2^|&gm|2S/69rM9g`!"#$R.&2"(69zWV~#B69!""RV4"|2M69#%MRV"!/69.t/6;C ,2p
ρ(AV1 ) = ρ(BV
2 ) = ρV =γ µ
Pr Re ρ
F`2^_[C^ Z `:
Z eb[
[^7r83C &H2i"z9g¬2i2"3:gm52M/691i9g!.U69 xsr36^ ^| . ._;C &4,H2i7O!",2|RV"!/69./6;C ,2z &u86!/u;C "!&uW:sG/ &u
ρV1 = ρ
(AV
1
δx
) 69&2;ρV
2 = ρ
(BV
2
δy
) F`2^_[C^ Zc [&2!"432M #$R. ¥c(";H&:,#%!869.)\2,Ce"|86;A
(F V )n+1i+1/2,j ≈ (F V )n
i+1/2,j + (ρV1 )n
i+1/2,j(δ1∆w)ni+1/2,j
(GV )n+1i,j−1/2 ≈ (GV )n
i,j−1/2 + (ρV2 )n
i,j−1/2(δ2∆w)ni,j−1/2Xx&2 &%2"M'eCR"&2z &:$2M #$R. !' Q/69risO.;S
C0∆wni,j + C−
1 ∆wni−1,j + C+
1 ∆wni+1,j + C−
2 ∆wni,j−1 + C+
2 ∆wni,j+1
= −∆t · Rni,j − σ1δ1µ1∆(fE)n
i,j − σ2δ2µ2∆(gE)ni,j^|2M2S!":B!&:
C/691i &:B6!"!",&:2M7:/!",2z!"&: W,&
C−
1 = −σ1
(12ρ
E1 + ρV
1
)ni− 1
2,j
C+1 = −σ1
(12ρ
E1 + ρV
1
)ni+ 1
2,j
C−
2 = −σ2
(12ρ
E2 + ρV
2
)ni,j− 1
2
C+2 = −σ2
(12ρ
E2 + ρV
2
)ni,j+ 1
2
C0 = 1 − C−
1 − C+1 − C−
2 − C+2
F`2^_[C^ Z ar
5";"832M'eCR. ! /69rM86;p
Rni,j =
(δ1(F
E − F V )
δx+δ2(G
E −GV )
δy
)n
i,j
.
[&2!"43¬2BP\69¥eO]2"% #$R. ! T869#%&:'eO&2;T/69 4:-gh ^569;C. sA~7:/!",2 \(^z8^vXx&2;"";D3(69&C[257O!",2`RV"!/69./6;C ,2K #$R. ¥c(!869&_3125&2[&2"";i527O!",2 6!"WU69&#B69!""@69&2;k2%¥#$R2.¥! xsk9g52%#%'2:;kSR"71";D^ 2%!/2#%B!869&A2&AWV%/.¥71";k,2 &LSLbR:!"";C,dTq #%2:;6 ^. .>^GXx&!/(69R aC3t2% #$R(6!T9g|2%7O!",2pRV"!/69./6;C ,2 #$R. ¥c(!869&$&p2B!&2!sp9g22!/2#%^| . .:WV,2;C";D^oXx&p2K&2'eC¬R(69/694/69R_32'eO&2&9g2i#%2:;~$2M". ,&u9g,&286;Cs,\(^z||R2"&:";D^ )@ ) 7$; !$# & 7$& 25!7$; ->" 55& !Xx&2Bc2@!/(69R83K2G;C,(69.¥> #%vRR &\gh#T,.U69& ^56@ &::;C,2!"";/\6T\!"#$R,G,&C86;CsQ\(^z8^ i2^";v(69||#%2:;~&(69W."|,2|@,2/[2M!"&:71r&2!"SRV"";C,R~/69/'4";C &(69 . su69RR. ";~gm86;Cs,\(^!869.!,.U69&28^5Xx&~R(69/694/69R_3 ^ ^| . .¬,. &2p2^2S;C,(69.¥ #%SRR &d!/2#%S!869&~WVS69./$86 . sH6;269R";~2SPb69¥e:]2"igh/69#%h^ ^n z,2|c2|"!869. .t2M'eCR"&u9go2M;C,(69.¥> #%pRR &B;C! ´69&)
wn,m+1i,j − wn
i,j
∆τ+
3wn,m+1i,j − 4wn
i,j + wn−1i,j
2∆t+
(δ1(F
E − F V )
δx+δ2(G
E −GV )
δy
)n,m+1
i,j
= 0
Z e ·
^|2∆τ
`2|R2,2;9> #%QRB69&2;wn,m+1 ;&2"K2z/69z69KR2,2;9> #%
(m+1)∆τ &$2
!"&:71r&2!"|R:!""`^569;$25R:sC!869.2/69z69` #%(n+1)∆t
^ 2|Pb69¥eO]"z #$R. ¥c(!869&2!869&kWV$ #$R. sb69RR. ";A;C, &~2. &2869 ´69&)9g2&:,#%!869."\2,CeC"p69,&2;k2R2/,2;9> #%.71.
(n,m)^KNQ5g>696|2M &:7:/!;v&:,#%!869.)\2,Ce"69S!"&2!"&2";D3 ^i(6871$
(FE)n,m+1 ≈ (FE)n,m + µ1∆(fE)n,m − 12(ρE
1 )n,mδ1∆wn,m ,
(GE)n,m+1 ≈ (GE)n,m + µ2∆(gE)n,m − 12(ρE
2 )n,mδ2∆wn,m .
F`2^_[C^ Z g4
L: #$ .U69. s132M7O!",2zRV"!/69._/6;C ,2 #$R. ca!"69 &<sO'.;p
(F V )n,m+1 ≈ (F V )n,m + (ρV1 )n,mδ1∆w
n,m ,
(GV )n,m+1 ≈ (GV )n,m + (ρV2 )n,mδ2∆w
n,m .
F`2^_[C^ Zn
Xx&2 &%2"M'eCR"&~ &:$2M #$R. ! Q/69risO.;S
C0∆wn,mi,j + C−
1 ∆wn,mi−1,j + C+
1 ∆wn,mi+1,j + C−
2 ∆wn,mi,j−1 + C+
2 ∆wn,mi,j+1
= −∆τ · Rn,mi,j − σ1δ1µ1∆(fE)n,m
i,j − σ2δ2µ2∆(gE)n,mi,j
F`2^_[C^ Zd
^|2M2S!":B!&:C
69M&2^4 71&HW:sb
C−
1 = −σ1
(12ρ
E1 + ρV
1
)n,m
i− 1
2,j
C+1 = −σ1
(12ρ
E1 + ρV
1
)n,m
i+ 1
2,j
C−
2 = −σ2
(12ρ
E2 + ρV
2
)n,m
i,j− 1
2
C+2 = −σ2
(12ρ
E2 + ρV
2
)n,m
i,j+ 1
2
C0 = (1 + 32λ) − C−
1 − C+1 − C−
2 − C+2
F`2^_[C^_[ber
^|
λ =∆τ
∆t, σ1 =
∆τ
δx, σ2 =
∆τ
δy.
2SRV/69 Rn,mi,j
#B69 &2,&2!/(69&r";:^| ~"RV"!z$2MW.:!/1> #$R. ! [!/2#%%
Rn,mi,j =
3wn,mi,j − 4wn
i,j + wn−1i,j
2∆t+
(δ1(F
E − F V )
δx+δ2(G
E −GV )
δy
)n,m
i,j
I`71&:,(69. . sr3QLSL.U6e69&R:!";,\6Q^. .i6dq 69. r # !869& WVu#$R.s1";¢. 71~2Pb69¥e:]2";C,a69. >¥#%%RR &~!/2#%4^ :,283 2'eC&2&k9g2$P\69¥eO]2"@ #$R. ! M869-#%&:BA,&286;Cs \(^zd;:"B&2d6;2;+!"#$R.'eC xsr^ AH&2^ ^569&OBA,2;Cs2~ #$R(6!d9gM2.^5Pb6!/~R"!"&2;C &&G&H2SP\]gm#T,.U69&_^
Z e2`
)@ ) 143.7 &)( 7="; %&9<.f= 3 #; ./&V># 7# &5# 3Xx&$25c2K!/(69R83M^z2^";%(69K2|.^5Pb6!/BR"!"&2;C & &&. s@#%:;C¥c("2|R2/,2;9> #%; 7469 7rG# 69&2;)2v;C R(69&9gQ2G&:,#%!869.| &:7:/!'; \2,Ce"8^ 2GR"!"&2;C & &A(62'gmG69& #$R(6!&2dRV"!/69./6;C ,2 #$R. ¥c(!869&_3` &2;"";D32BR"!"&2;C &2";z,2/69&27!/2#%M86;p
FRusi+ 1
2,j
= (µ1(fE) − 1
2ρ(PAE)P−1 · δ1w)i+ 1
2,j = (µ1(f
E) − 12 ρ
E1 P
−1 · δ1w)i+ 1
2,j
GRusi,j− 1
2
= (µ2(gE) − 1
2ρ(PBE)P−1 · δ2w)i,j− 1
2
= (µ2(gE) − 1
2 ρE2 P
−1 · δ2w)i,j− 1
2
N[6M",. 83C69RR. s: &T2Q,W2 ,&G9g_2z&:,#%!869. ;C R(69&d#B69!"s:.; &dR"!"&C;C &2";u!86%
(FE)n,m+1 ≈ (FE)n,m + µ1(∆fE)n,m − 1
2 ρE1 P
−1 · δ1∆wn,m ,
(GE)n,m+1 ≈ (GE)n,m + µ2(∆gE)n,m − 1
2 ρE2 P
−1 · δ2∆wn,m ,
F`2^_[C^_[ Z
69&2;H &2 &B|. &2869 ´69&< &:$2M #$R. ! Q/69ri.86;z~
C0∆wn,mi,j + C−
1 ∆wn,mi−1,j + C+
1 ∆wn,mi+1,j + C−
2 ∆wn,mi,j−1 + C+
2 ∆wn,mi,j+1
= −∆τ · Rn,mi,j − σ1δ1µ1∆(fE)n,m
i,j − σ2δ2µ2∆(gE)n,mi,j
F`2^_[C^_[b[r
^|
Rn,mi,j =
3wn,mi,j − 4wn
i,j + wn−1i,j
2∆t+
(δ1(F
E − F V )
δx+δ2(G
E −GV )
δy
)n,m
i,j
69&2;
C−
1 = −σ1
(12 ρ
E1 P
−1 + ρV1 Id
)n,m
i− 1
2,j
C+1 = −σ1
(12 ρ
E1 P
−1 + ρV1 Id
)n,m
i+ 1
2,j
C−
2 = −σ2
(12 ρ
E2 P
−1 + ρV2 Id
)n,m
i,j− 1
2
C+2 = −σ2
(12 ρ
E2 P
−1 + ρV2 Id
)n,m
i,j+ 1
2
C0 = P−1 + 32λId− C−
1 − C+1 − C−
2 − C+2
F`2^_[C^_[ ·
:,2832~;CU69r&(69.Q!":B!&:C0
4 7r&+W:sF`2^_[C^ [ · $$&2^ 6\g¨,. .z#B69¥e 3kWVH &:71";D3|69&2;!":B!&:
C±69d69.uW.1!/!"1B!&:3/H(69@/69rB"0:, #%&:$69&2;RV/69&2!",&:@9g
2B #$R. ! 869#%&:@ &2!86/%gmT.^5P\6!/)&:,#TWVD\(^z^ 2B&2'eC@"!&'eOR.U69 &2@2^ GRV4 W.</691b6;C769&:/69rb9gp2\RRV"H9gp2<R"!"&2;C &2 &#B69¥e
P1"R2
R"!"&2;C &2";\69RRr6!/\6 #$R.p6| z!"#$R"¥W2. \(^71&_^
Z e c
(1 & # " #1 1# # # & & )*% )%A ; "; # ( #14; &&# /5'ghp;/69 . &d2pPb69¥e:]2"S#%2:;~gmz.^5Pb6!/u&:,#TWV0\(^z832.,2z &::;C,2!"@6$r&2/69.gh/69#%h^<v!"#$R,2 #%'x6!"!,/69%Ji687O-YL:r"S0:,(69&2S&b#T,. ¥;C #%&2&a69.Q [j®· jS569"U69&G4;[¬4 71",22QRRV,& xsBT2^ 2&:,#%!869.D;7r.R#%&:(69|6969. #%4!"&2/69&:. s%R"&:";d &,[j &d2"gh2Q/69rQ9g_ #$R. ! xs%!869&GWVQ'eO&2;";dp2· j§!86| &d6ir&:, &2. s%/69 4:-gm ^569;$g>6&_^ 2z#T,. ;C #%&2&(69.Vr7r& &T"0:,(69&286;~
P−1∂w
∂τ+∂w
∂t+
d∑
p=1
∂fEp
∂xp=
d∑
p=1
∂fVp
∂xp,
F`2^ · ^ Z
^|2 ^G"!869. .5(69w
@(G7r"!$9gQ2v!"&271";)7469U69W.43τ
@2GR2,2;9> #%43t
2R:sC!869. #%43
fEp
69&2;fV
p
69d"RV"! 71.¥s2G!"&:7r"! 71v69&2;7O!",2 \2,Ce@ &2pth R(6!"'
;C "!&)d
|2S;C #%&2&<9g2MRW.#dz69&2;P
6@R/!"&2;¥ &2 &d#B69¥e ^|2 !/u/6913C &2R//&:^ 342gm#RRV4"; & g c 6;"/! WV"; &$;/69 .` &$"!& Z ^ · ^_[D25RRV"9gQR"!"&2;C &&H#B69¥e ^| . .oWVpg¨,2["!869. .";\ &b"!& `2^ · ^_[C^ d. 71TsC#F`2^ · ^ Z '32igh. .^| &B&:,#%!869.¬!/2#%M|,2/";k
(P−1
c
)n,m
j
∆wn,mj
∆τn,mj
+32(wn,m+1
j − wnj ) − 1
2∆wn−1j
∆t+
d∑
p=1
(δphp
δxp
)n,m+1
j
= 0,F`2^ · ^_[r
^|2m
2R2,2;9> /69&H!",&:83n
2 #%QRd!",&:83∆wn,m = wn,m+1 − wn,m 3
∆wn−1 = wn − wn−1 3j = (j1, j2, . . . , jd)
6¦#T,. ¥> &:r6/:!U69"; ^| ®6RV &:xj = (j1δx1, . . . , jdδxd)
9gp6b4,.U69H569"U69&¢4 ;_^δphp
%2u;C &2!"<RV/69G& 64;~!". .
δp &~2
pth R(6!";¥/!&k69RR. ";~$2M&:,#%!869.)\2,ehp
69RR8eC¥#B69¥&2B2SR:sO¥!869.\2,Ce
(fEp − fV
p )
(δphp)j = (hp)j+ep
2
− (hp)j−ep
2^|2ep
[6$#T,. ¥> &:r ^| <2q><!"#$RV&2&:
epq^|!/<Q"01,(69.¬
0¥gq 6= p
69&2;~1
¥gq = p
^`IK0:,(69&)F`2^ · ^_[r5!869&~WV ^| &~ &~2igh. .^| &%gh#
(P−1
c
)n,m
j
∆wn,mj
∆τn,mj
+3
2
∆wn,mj
∆t+
d∑
p=1
(δp(∆hp)
δxp
)n,m
j
= − (R)n,mj ,
F`2^ · ^ ·
^| <69&u'eOR. ! /69ri4 71&~W:s\
(R)n,mj =
32(wn,m
j − wnj ) − 1
2∆wn−1j
∆t+
d∑
p=1
(δphp
δxp
)n,m
j
.
2&d2Q86;CsB/690^| G"RV"!TR2,2;9> #%τ
86!/2";D3O2 n ML%9gF`2^ · ^ · Kr:"p´832&2!"4322S&:,#%!869.69RRr6!/u,2";~%!"#$R,S2 n MLv9g`"0:,(69&)F`2^ · ^ · |;:"z&2[6 "!2hR(6!"T69&2;~ #%z6!"!,/6!s~9g`2S#%2:;)¨R7:;";u(692p!"&:71r&2!"T&
mz/"6!/(;a'^A
!869&2&69RR. sk2dPb69¥e:]2"d #$R. ¥c(!869&2&2 n MLb#%8^ 0 &2d2T(69&2;D3o &2MLk9gTF`2^ · ^ · '3`2d &:7:!;W\2,CeC"69G!"#$R,";),2 &\!.U6!869. &:7O!;&:,#%!869.I\2,CeHh4^ 2^:43tz,2/69&27bN|ZMLOP + ^"^"^/'3t'eC&2;";kv 42p/;S,2 &u6G7469U69W.$"!"&2,2!&69RRr6!/
Z eba
hP\ZiL n i69&2; ^| 6d&:,#%!869.;C R(69&A/69: &~ &:~6!"!",&:S2R"!"&2;C &2 &<9gMF`2^ · ^ Z '^±[!",2 \2,Ce"69p69RR8eO #B69";\69z"!"&2;O;2'Q,2 &B!"'&O";vgm#T,.U6~ !/(69R Z '^Xx& ;¦69RR. s2Pb69¥eO]"#%2:; (694^|6¦;"! WV"; & 2R7O,2¦/"!&2832¢ &:7O!; \2,Ce")9g~2 n iL69/ 769. ,(69";®,2 &2+c2-;6!"!,/69 R"!"&2;C &(";z,269&27G!/2#%%
hEp = µpf
Ep − 1
2P−1ρ(PAE
p )δpw ,F`2^ · ^ `:
^|2µp
|2p687r/69rpRV/6971Q6@4;~!". ._ &H2pth R(6!"S;C "!&
(µpφp)j =1
2
((φp)j+
ep
2
+ (φp)j−ep
2
),
69&2;ρ(PAE
p ) = ρEp
;&2"52iRV"!/69. /6;C ,2|9gt2[R"!"&2;C &2"; 6!"WU69&H#B69¥e(P ·
∂fEp
∂w)^
Xx&::;C,2! &%gh#T,.U6~F`2^ · ^ `:5 &:$2S'eOR"/&~9g (∆(hE
p )) .86;z~
(∆hE
p
)n,m
j±ep/2= µp
(∆fp
E)n,m
j±ep± 1
2∆((P−1ρE
p
)j±ep/2
(wj − wj±ep))n,m
≈ µp
(∆fp
E)n,m
j±ep± 1
2
(P−1ρE
p
)n,m
j±ep/2∆(wj − wj±ep
)n,m
Xx5gh. .^zz(69p(δp(∆hE
p
))n,m
j= δpµp
(∆fp
E)n,m
j+ep+
1
2
((P−1ρE
p
)n,m
j+ep/2+(P−1ρE
p
)n,m
j−ep/2
)∆wn,m
j
−1
2
(P−1ρE
p
)n,m
j+ep/2∆wn,m
j+ep− 1
2
(P−1ρE
p
)n,m
j−ep/2∆wn,m
j−ep
F`2^ · ^ c
N[gm69652i7O!",2I\(,CeC"z9go2 n MLv69M!"&2!"&2";D3H^i&24."!z2 z;RV&2;&2!" ^| H"RV"!$2i/69&r&:69.o; 769 71"p
(hV
p
)j±ep/2
=
(AV
p
δw
δxp
)
j±ep/2
= (QVp )j±ep/2 (δpw)j±ep/2 ,
2&_36^MR:!""";~2M7O!",2zRV"!/69._/6;C ,2Q #$R. ¥c(!869&\65gm. .^zp(hV
p
)j±ep/2
≈ ρ(QVp )j±ep/2 (δpw)j±ep/2 = (ρV
p )j±ep/2 (δpw)j±ep/2 ,F`2^ · ^_ar
(69p(δp(∆h
Vp ))n,m
j= −
((ρV
p )n,mj+ep/2 + (ρV
p )n,mj−ep/2
)∆wn,m
j
+(ρVp )n,m
j+ep/2∆wn,mj+ep
+ (ρVp )n,m
j−ep/2∆wn,mj−ep
F`2^ · ^ g4
Z eg
Xx&2 &<F`2^ · ^ c |69&2;F`2^ · ^ g45 &:uF`2^ · ^ · sO.;|2igm. .^| &B #$R.M #$R. ! Q!/2#%%
Dn,mj ∆wn,m
j +∑
p
(δpµp
δxp∆(fE
p )n,m
)
j
−(C−
p
)n,m
j∆wn,m
j−ep−(C+
p
)n,m
j∆wn,m
j+ep
= − (R)n,m
j ,
F`2^ · ^ n ^|2
Dn,mj =
(1
∆τP−1
c
)n,m
j
+3
2∆tId +
∑
p
(C+
p
)n,m
j+(C−
p
)n,m
j
,
F`2^ · ^ d
(C±
p
)n,m
j=
1
δxp
(1
2ρE
p P−1 + ρV
p
)n,m
j±ep/2
.F`2^ · ^ Z er
z #$R. ! Q!/2#%Sz. 71";H,2 &d6.U6e69&uR:!"";C,T,2!/<6qo &: 6!"Wh[L:sO#$#%!M69,2YLO;.£I^|!/~.86;z~
Dn,mj ∆w
(l+1)j = − (R)n,m
j −∑
p
(δpµp
δxp∆(fE
p )(l))
j
−(C−
p
)n,m
j∆w
(l)j−ep
−(C+
p
)n,m
j∆w
(l)j+ep
F`2^ · ^ ZbZ [&2!"43a2T!":B!&:
D(6QBWVp &O7r";< &\;[dW/69 &\2p &2!#%&:
∆w(l+1) ^NQ#%&C&2";H &~2iR7O,2|R(69/694/69R_3(2i &:71&H6!/71";~;C "!. s ^|2&~!"#$R// W. \(^zQ69!"#$R,";¢ &2!"43 &%!86432v#B69¥e
D$R,. s!869.U698^Q^7183^|2&+.^5P\6!/+&:,#
WV \(^%69v!"#$R,";D32GR"!"&2;C & &2A,&2";&\(69P
&2\.&r$2v;&: xs#B69¥eb69&2;b!"&2"0:,2&:. s
DWV"!"#%"S6Bg¨,. .o#B69¥eb6 ^. .K6[2!"1B!&:
C±^MJQ712."83
ppRV4 W.B~/691d6;C7469&:/69rd9g|2BRRV"@9gz2%R"!"&2;C &&\#B69¥eP
~r"R2R"!"&2;C &2";\69RRr6!/\6 #$R.p6| z!"#$R"¥W2. S71&_^
)*% )@ ;67b#%0&( 7b=; 3 #; ./&>5# 7# &V#% 3 147#f# K2^";S &p2`c2¬!/(69Rt(69t( ,r.1R"!"&2;C &2!",.;pWV?^| &T,&2;¬2`gh. .^| &gm#
P = Id + (β2 − 1) ·Q^|
Q =γ − 1
c2
q2 = 12(u2 + v2) −u −v 1
uq2 −u2 −uv u
vq2 −uv −v2 v
Hq2 −uH −vH H
=
γ − 1
c2
1
u
v
H
·
(q2 ; −u ; −v ; 1
)
2@#B69¥eQ
^56i2^|&\GWVT;#$RV&:83 Q2 = Q
^ [RRVxs<[7rsu,2'g¨,. &2!"T &(69W."z,2|/ &:71z &u6 #$R. ^568sG2iR"!"&2;C &2 &d#B69¥e
P
P−1 = Id +1
β2 − 1Q
Z e n
J[^S3¥g2S#B69¥eDn,m
j
!869.!,.U69";u,2 &d6$ &4.S7469. ,(69&<9g2S#B69¥eP−1 69zRV &:
j3
!869&~WV ^z &<65gm. .^zpDn,m
j = an,mj ·
(P−1
)n,m
j+ bn,m
j Id^|
an,mj =
1
∆τn,mj
+1
2
d∑
p=1
1
δxp
(ρE
p
)n,m
j+ep/2+(ρE
p
)n,m
j−ep/2
bn,mj =
3
2∆t+
d∑
p=1
1
δxp
(ρV
p
)n,m
j+ep/2+(ρV
p
)n,m
j−ep/2
2&~,2 &%2MRRVxsH9gQ
3C2M &:7r9g2M#B69¥eDn,m
j
!"#$R,";u'eOR. ! . s~65gm. .^zS(D−1
)n,m
j=
(1
a+ b
(Id +
a(β2 − 1
)
a+ bβ2Qc
))n,m
j
F`2^ · ^ Z [r
5";"83O¥g¬ &v2i!":B!&:C±
3O2[#B69!""P−1 69i7469. ,(69";~692[RV &:
j3O 55RV4 W.[
h^| S2M/!/2#%BF`2^ · ^ ZbZ &~2igh#∆w
(l+1)j =
(D−1
)n,m
j·(∆w
(l)1
)j+(D−1
)n,m
j·(P−1
)n,m
j·(∆w
(l)2
)j
^| ~2igh. .^| &d;'c2& &2zgm|2M &2!#%&: (∆w
(l)1
)j
69&2; (∆w
(l)2
)j
(∆w
(l)1
)j
= − (R)n,mj −
∑
p
1
δxp
∆(fE
p )(l)j+ep
−(ρV
p
)n,m
j+ep/2· ∆w(l)
j+ep
−∆(fEp )
(l)j−ep
−(ρV
p
)n,m
j−ep/2· ∆w(l)
j−ep
(∆w
(l)2
)j
=∑
p
1
δxp
(ρE
p
)n,m
j+ep/2· ∆w(l)
j+ep+(ρE
p
)n,m
j−ep/2· ∆w(l)
j−ep
2#B69¥ebR:;C,2! (D−1 · P−1
) (69p69RRV869M &k2$'eOR"&k9g2 #$R. ! p!/2#%$!869&kWV69.%!869.!,.U69";u'eOR. ! . sv(69&C|$2MRRV"9g`86!/~#B69¥eA
(D−1
)n,m
j·(P−1
)n,m
j=
(1
a+ b
(Id +
b(β2 − 1
)
a+ bβ2Qc
))n,m
j
F`2^ · ^ Z8·
2"T'eCR"&2[&(69W.T,2QBW, .;\6%,. suP\69¥eO]2"p #$R. ! [869#%&OQgm[69. .oRV"; \(^z8^L:,2!/<6$!/2#%Mz;"! WV";~ &H2M&2'eC|R(69/694/69R_^ )*% )*% 7# 25 9<7#f# #;H; 7#; 714; V7:( &LJ& !:7 - !3i;H; >!Z[ &~'eCR"&2dF`2^ · ^ Z [rS69&2;¢F`2^ · ^ Z8· i~769. ,(69
∆w(l+1)j
s:.;S2gh. .^| &u #$R. ! p869-#%&:p
∆w(l+1)j =
1
(a+ b)n,mj
·[∆w
(l)1 + ∆w
(l)2
]j
+
(β2)n,m
j− 1
((a+ b) (a+ bβ2 ))n,mj
·Qn,mj ·
[an,m∆w
(l)1 + bn,m∆w
(l)2
]j
F`2^ · ^ Z `:
Z e d
#$R. ! Q869#%&:i&(69W."/%;"!",R.T&!". sH2p/69&2;269;<P\69¥eO]2"S #$R. ! Q869#%&:,2";$gh2!"#$R" W.i\(^¨(69`gm
β = 1M(692"!"&2;B#!869&2!".g¨#2Q6;;";
869#%&:pRV!¥c(!$G.^5Pb6!/A&:,#TWV \(^zB¨ &kS!86β 6= 1
'^T5";"83D869#%&:GF`2^ · ^ Z `:WV"!"#%"l^|2. . sP\69 eO]2"v¥g[2HRV"!¥c(!H'eCR"&+9g[#B69¥e
Q$#$R.s1";k!"#$R,u6
#B69¥eO>7r"!R:;C,2!Q,2!/u6Q ·X ,&2;z2igm#
Q ·X =γ − 1
c2
(q2X(1) − uX(2) − vX(3) +X(4)
)·
1
u
v
H
^| X(m) 2[!"#$RV&2&:|9gt7r"!
X^IK7r&:,(69. . s13
∆w(l+1)j
!869&GWV[!"#$R,";H69|6S.^¦!"4|69&2;^| <6@/;,2!";~#%#%sH/69rM,2 &$2igm. .^| &B'eCR"&
∆w(l+1)j =
1
(a+ b)n,mj
[∆w
(l)1 + ∆w
(l)2
]j+ ∆φ
(l)j ·
1
u
v
H
n,m
j
F`2^ · ^ Zc
^|2M2S!869.U69∆φ
(l)j
|4 71&~W:s\
∆φ(l)j = χn,m
j
[q2(a∆w
(1)1 + b∆w
(1)2
)− u
(a∆w
(2)1 + b∆w
(2)2
)
−v(a∆w
(3)1 + b∆w
(3)2
)+(a∆w
(4)1 + b∆w
(4)2
)](l)j69&2;'^|
χn,mj =
[γ − 1
c2· β2 − 1
(a+ b)(a+ bβ2)
]n,m
j
.
](#T,.U6)F`2^ · ^ Zc T,#$#B69 ´8"2G69. .¥RV"; \(^®.^5!"4@#B69¥eOmg¨"G869#%&: &:7r" 169"; &2$R"&: ^V^$J[igm#T,.U6vMRV!¥c(!869. . s ^| &Aghp6vqo &:- 6!"W869#%&:p9g52 #$R. ! /69r4^ 25'eC&2&$i2LSLpg¨/69#%h^!869&$WVRV-gm#%";%/69 4:-gm ^569;C. sr^`Xx&2;"";D3;C, &~2@gh ^569;A69&2;b2W(6!/^569;k ^"R283D2 &2!#%&:
∆w(l)1
69&2;∆w
(l)2
(6871[*Y,2ivWV!"#$R,";\,2 &d2@/69"Q(69[(687169. 86;CsuWV"'&k769. ,(69";D^M]([/691T9g!.U69 xsr3V.[,2i'eOR"/2M&2h^§gh#T,.U69&<9g`,2!/~ &2!#%&: &H2S!86S9go2igm ^569;~ ^"Ru&. sb
(∆w
(∗,l)1
)j
= − (R)n,mj −
∑
p
1
δxp
∆(fE
p )(l)j+ep
−(ρV
p
)n,m
j+ep/2· ∆w(l)
j+ep
−∆(fEp )
(∗)j−ep
−(ρV
p
)n,m
j−ep/2· ∆w(∗)
j−ep
(∆w
(∗,l)2
)j
=∑
p
1
δxp
(ρE
p
)n,m
j+ep/2· ∆w(l)
j+ep+(ρE
p
)n,m
j−ep/2· ∆w(∗)
j−ep
^|22$'eCRV&(69&:(∗, l) #$R(6 ´8"S2Tg>6!ia69M2@ &2!#%&:p.¥s\&kWVk /69 71.71.
(l)69&2;
(∗) h"S69./BLSLH69. r # F`2^_[C^ Z8· '^
ZbZ e
"$# # # ' " & ( # # "$#%& " "'& ))%A 35%# .6#87 7#;H71 ; V7N #$R. ^|68sd$;869.^| ~2iWV,&2;269"Q9go2M!"#$R,/69&(6.¬;#B69 &~5$/52M &2!#%&:69z2iWV,&2;269 "Q"0:,(69._´8~
∆w(l)B = w
(l)B − wn
B = 0
∆(fE
p
)(l)B
=(fE
p
)(l)B
−(fE
p
)nB
= 0
[&2!"43'eOR. ! 869#%&:Q;:"z&2Q6;;<69&:sv!869.!,.U69&u &:%2M2SPb69¥e:]2"p69. r #~^[^7r83C,2!/H6@!/2! ^| . .VWV[2^|>WV[ &2B!&:| &G2[!86/[9gt 4~6RV"!/69@4;8^`Xx2'gmM&2"!""/69sG%&(69&2!"S2M869#%&:Q9go2MWV,&2;269"^
))@ 1435 #%.6#87 7#f;H7b14; V7Xx&%;M #$R71|2P\69¥eO]2"| #$R. ! #%'2:;D3f^z!869&% #$R.#%&:2|gh. .^| &p869#%&:8^]o 83a.[WV
q2T71"!i9g2D^. .¥, ";\7469U69W."QgmQ2@!"#$R,/69&A9g2TWV,&2;269s\!&
;C &28^ 2&_32v &2!#%&:d69$2GWV,&(;269"%!869&WVH'eOR"/";) &g¨,&2!&¢9gQ2v &: &2!#%&:T
∆q(l)B = A · ∆q(l)I ,
^|2 A ;&2"z(i#B69¥e ^|!/<;'c2&2"z2MWV,&2;269s~!"&2;C &28^I`71&:,(69. . s1322M &2!#%'&O9g2M!"&2/769 71M769U69W."Q69S!"#$R,";<6|gh. .^zp
w(l)B = w(q
(l)B )
(fE
p
)(l)B
= fEp (w
(l)B )
=⇒
∆w(l)B = w
(l)B − wn
B
∆(fE
p
)(l)B
=(fE
p
)(l)B
−(fE
p
)nB
:,283 @*Y,2p&2"!""69sA<7469. ,(69d2d#B69¥e A \6T\!#$R2,d(/B &2!#%&:^\J['eCR(69/694/69R ^| . .t;/69 .t2M7469. ,(69&<9goz#B69¥eGgmzRV"!¥c(!p!"6/"8^
))*% 9<7#f# &)( 7=; i& 25>",#b ./&V># 7# &5!](z86!/u!86/M;/69 .";~ &H([gm. .^| &23(M!"#$R,/69&<7469U69W."Q69ic2|R"!";D^
FE E/F:P EHCFC 2d!"#$R,/69&)7469U69W."69%&(69,69.¥. sA2 '7O!",2%&2"83o&(69#%. s
q = (p, u, v, T )T ^ 2WV,&2;269sH!"&2;C &2[69M4 7r&~W:sb
∂p
∂n= 0 ; u = 0 ; v = 0 ;
∂T
∂n= 0 .
&2"0:,2&:. sr3H^i(6871$
pB = pI ; uB = 0 ; vB = 0 ; TB = TI .
ZbZbZ
∆p(l)B = ∆p
(l)I ; ∆u
(l)B = 0 ; ∆v
(l)B = 0 ; ∆T
(l)B = ∆T
(l)I .
2M#B69¥eH|2&u;'c2&2";u6|gh. .^zp
A =
1
0
0
1
hGHTVUf$JLEHCP EHC8Cir"R~2 '7:!",2 [7469U69W."Q69&2;H2iWV,&2;269sv!"&2;¥ &(z86;
∂p
∂n= 0 ; u = 0 ; v = 0 ; T = T
^.
[&2!" ^M(6871$
∆p(l)B = ∆p
(l)I ; ∆u
(l)B = 0 ; ∆v
(l)B = 0 ; ∆T
(l)B = 0 .
2M#B69¥eH|2&u;'c2&2";u6|gh. .^zp
A =
1
0
0
0
R)CFFK F VGXx&!8643j^G!869&)!/2:4d2dR #$ 7rd7469U69W."
q = (ρ, u, v, p)T /<;'c2&2d2BWV,2&2;269s!"&2;C &2T
∂ρ
∂n= 0 ; Vn = 0 ;
∂Vt
∂n= 0 ;
∂p
∂n= 0 .
^|2Vn
69&2;Vt
;&2M2M&2#B69.o69&2;H2M/69&r&:U69.t7r.:! "8^X5gm. .^z|(69T
∆ρ(l)B = ∆ρ
(l)I ; ∆u
(l)B = n2
y∆u(l)I −nxny∆v
(l)I ; ∆v
(l)B = n2
x∆v(l)I −nxny∆u
(l)I ; ∆p
(l)B = ∆p
(l)I .
2M#B69¥eH|2&u;'c2&2";u6|gh. .^zp
A =
1 0 0 0
0 n2y −nxny 0
0 −nxny n2x 0
0 0 0 1
ZbZ [
R) G jF'8VCFUfXx&vz!"6/43/^[ #$RV4/i2i &.z&:(69. R:s132i &.z&:R:sH69&2;v2 \(^69&4. ^| . ^M'eO69RC.U69M2MR",2[gh#2M &:Q9go2S;#B69 &
HB = H∞ ; SB = S∞ ;(Vt)B
(Vn)B
= α∞ ; pB = pI .
2z71"!9g (z!"#$R,/69 &G7469U69W."K('/gmq = (H, S, α, p)T 69&2;%2z#B69¥e A 86;Q
A =
0
0
0
1
2&_32MR #$ 71M7469U69W."Q69S!"#$R,";<65gm. .^zS
∆p(l)B = ∆p
(l)I
∆ρ(l)B = ρ
(l)B − ρn
B =
(p(l)I
S∞
)1/γ
−(pn
I
S∞
)1/γ
(∆Vn)(l)B =
√2
1 + α2∞
·
√√√√√H∞ − γ
γ − 1
(p(l)I
)(γ−1)/γ
S1/γ∞
−
√√√√H∞ − γ
γ − 1
(pn
I
)(γ−1)/γ
S1/γ∞
(∆Vt)(l)B = α∞ · (∆Vn)
(l)B
f JLKjUMGG CFU:8VCFUf 2M71"!'/z9g2MR #$ 7rS76969W2. S|2MWV"-, ";vgmz,2!/u!"&2;C &2Q &2!" ^i(6871$
ρB = ρ∞ ; uB = u∞ ; vB = v∞ ; pB = pI .
∆ρ(l)B = 0 ; ∆u
(l)B = 0 ; ∆v
(l)B = 0 ; ∆p
(l)B = ∆p
(l)I .
2M#B69¥e A 86;p
A =
0
0
0
1
f JLKjUMGG CFU' CFUM . ._!"&2;z2M7r"!9g2iR2 #$¥¥71S769U69W.%
ρB = ρI ; uB = uI ; vB = vI ; pB = p∞ .
ZbZ8·
∆ρ(l)B = ∆ρ
(l)I ; ∆u
(l)B = ∆u
(l)I ; ∆v
(l)B = ∆v
(l)I ; ∆p
(l)B = 0 .
2M#B69¥e A 86;p
A =
1
1
1
0
"P + ( E6$T48 UfbF Xx&u2@!86@9g6%.^5Pb6!/\ &*"!& ^T #$RV4T2T¥&2. [#$RV/69,243 2T &.[#%#%&:,#~3 2/69&r&:U69.t71.:! xsH69&2;'^S'eO/69RV.U69S2MR",[gh#2M &:Q9go2S;#B69 &
TB = T &* ; mB = m &* ; (Vt)B = 0 ; pB = pI .
Xg¬(i71"!q
,2!/u6q = (T, m, Vt, p)
T 2M#B69¥eH86;p
A =
0
0
0
1
2&_32MR #$ 71M7469U69W."Q69S!"#$R,";<65gm. .^zS
∆p(l)B = ∆p
(l)I ;
∆ρ(l)B =
1
RT &* ∆p(l)I ;
∆u(l)B = nx ·R · T &* · m &* ·
(1
p(l)I
− 1
pnI
);
∆v(l)B = ny ·R · T &* · m &* ·
(1
p(l)I
− 1
pnI
).
`t k` i
! #1 +#% 2uR7:,2G!/(69Rd(6d;"! WV";6k#B69¥eOmg¨"u #$R. ! G869#%&:d(69G!869&¢WV\69RR. ";+gh69. .¥RV""; \(^z ^| 6dR(69!,.U69. sb.^!"4iRVM /69&_^ n M,2S#$R(6 ´8%69169 &A2$;C &:/69r"z &:7r. 71";H &H2S; 7469&u9g| #$R. ! 869#%&:T
• 2dR #B69sgh#³9g2G#B69¥e:mgh"G #$R. ! %!/2#%d@4 71&)W:sF`2^_[C^ g4F`2^_[C^ n $69&2;#B68sWV$7:h^";6p6Gc2-;T,2/69&27:>xs:RV$ #$R. ! T/69r%!",R.";L^| W^|(6971S 4C;,RM^| &2;~'eCR. ! /69rS&2M#B68sv!/2:4M$,2[gm6!"!,/6!svR,RV44^
C0∆wni,j + C−
1 ∆wni−1,j + C+
1 ∆wni+1,j + C−
2 ∆wni,j−1 + C+
2 ∆wni,j+1
= −∆t · Rni,j − σ1δ1µ1∆(fE)n
i,j − σ2δ2µ2∆(gE)ni,j
^| R 2S'eCR. ! z";C,(69.¬69&2;H2S!869.U69z #$R. ! Q!":B!&:4 71&HW:sb
C±
1 = −12σ1
(ρE
1
)ni± 1
2,j
, C±
2 = −12σ2
(ρE2
)ni,j± 1
2
, C0 = 1 − C−
1 − C+1 − C−
2 − C+2 .
NQzgm69[6z(T69#$R. ¥c(!869&<g>6
ZbZ a
• &6~ ;69&2;.U6@R_32Gq P\]69&2;LSL1P\]869#%&:%69d'eCR""; &)2%gm#F`2^ · ^ Zc i~6MH#$¥&2¥#$ ´8%2 S,& p!"#$R,/69&(6.!"4 ^|2&A.^5Pb6!/AR"!"&2;C &&9go2Mr71& &B"0:,(69&2z69RR. ";D^
Xx&S¬!/(69R83$^"^569&:_c2tQ6"tRV"!¥c(!869. . sS2K #$R(6!o&p &: &2!B!&2!sp9g2!/2!"9g6$c2-;i,2/69&27:>xs:RVp #$R. ! M/69r@!",R."; ^| k6% ;O;MQ:T'eOR2.¥! M/69r43a &286;9gM6k!"&2&:$c2-;d:A¨W.:!/:Yx #$R. ! d/69r~6%;&2H &+!/(69R2 Z · 9g[%2"8^Xx&2l^;83/69O &k &:A6!"!",&:%2~69WV7rv#B69C83^H #$R. s ^569&:$A6/"@2H. 4/$9g &: &2!@B!&2!s~ &::;C,2!"";<WOsv2TRV!69.¬/6;C ,2[ #$R. ¥c(!869&k&<2p&:,#%!869.;C R(69& &2<¥#$R2.¥! H/69r4^ A~gm:!,2G&2\86;Cs¢I`,.H"01,a69&28iJi687O-YL:r"H69&2; ,&286;Cs69&(69. sC" ^| . .WV$,2!"!"" 71. suR"&:";b &k!/(69R av69&2;WgO^pJ['eC83^5$69.H69&(69. sO´8@2 #$R. ! qo &: 6!"Wo69&2;<L:s:#$#%!@M69,2/-YLO;._869#%&:[69RR. ";~2M#B69¥eOmg¨"S/!/2#%4^ 2[±K&vJ[,#B69&&~69&(69. sCgm2Q. &2869 ´8";HI`,.z"01,(69&25RV-gm#%";G &G2[#B69&&2|9g¬2 ;T!/(69R8^P<R"!. sr39 ^56o2^|&T(69t &O7:!; 6!"WU69&2`69&2;T;C R(69&T#B69!""o!",.;WVBeCR"/;A,2 &\6v. #$ ";A&O,#TWVT9g|R(69/69#%T,2!/6S2+\(^®69&4.43¬2BP\6!/&:,#TWV832
CFL&:,#TWV[69&2;H2p6RV"!z/69
δy/δx "!& · ^ c '^
! " +# # "$#1 (% "$#% )@ )%A %& 5- - "!$#%!
0
0.25
0.5
0.75
1
Ave
rage
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL0
0.25
0.5
0.75
1
Ave
rage
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL
]o 4, c ^_[C^ Z SN¬7r/69rB69#$R. ¥c(!869&gm6!p9g2%j[ "!p!/2#%4^ n 'ghBM.:!/b/!/2#%4^%z 4:%Pb69¥e:]2"S!/2#%4^Xx&)2G#B69&&2$9gQ2G ;!/(69R83"^v!869s,2$6<4.W(69.z69&(69. sC ^| "RV"!\2GPb6!/&:,#TWVp69&2;<2%] n &:,#TWV8^ 2T",. Qgm[2T&(&C>R"!"&2;C &(";A!/2#%$69TR"/&:";\ &c24, c ^_[C^ Z ^ A&2 #$#%";CU69. sb(69M2$#B69¥e:mgh"$ #$R. ¥c(!869&";C,2!""p;C/69#B69!869. . sb2B!&2!sv9go2i #$R. ! !/2#%432"RV"!U69. . sBgm52MPb6!/H&:,#TWV!.4[,& xsH69&2;Ggh|2i7rs
ZbZ g
.^Pb6!/<&:,2#TWV8^ A69.B#B69u(6983V,&. 1TW.:!/u!/('#%43 4A] n 7469. ,2"[;d&2i&(69W.< #$R71d2H687r/69rv;269#$R &2^ 2d",. Tgm@2dR"!"&2;C &2";+!/2#%v69GR"&:";) &c24, c ^_[C^_[C^ N[%'eCRV"!";D32H.^5Pb6!/¢R"!"&2;C &2"G&(69&2!""B2~ &: &2!uB!&2!s+7r2.^P\6!/k&:,#TWVp;#B69 &_3o69&2;b Mp69.v#B6969W.%(69M2.4S9g5B!&2!sb;C,2$v2 #$R. ¥c(!869&¦B."B #$RV/69&:v(69&¢gmB 42vPb6!/¢&:,#TWV^L:,2!/ ",. d71"69.Q(69d2#%2:;B"01, "6M.U69rQ69#%,&:9g /69&dS86!/B2z86;Cs$/6969&2;D3:!"&2/"01,2&:. s1312!"4RVQ /69&b9g`2TPb69¥e:]2"p#%2:;<(6BWVS71sH!/2869R\ &<;QB"!"71i6%!"#$RV 714.W(69.O!"48^ 2,(;CsM9g2!"4tRV¬ /69&Tt2,WC*"!¬9g2 ;pR(69¬9g2t2"^tXx&p2gm. .^| &GR(69/694/69R_3^D^| . .os~G'eOR.U69 & ^|:s~2T#B69¥eOmg¨" #$R. ¥c(!869&kQ."QRV&(69. ´" &71[2@.^5P\6!/A;#B69 &A69&2; ^D^| . .`;/69 .#%R"! /. su2@ #$R(6!M9g2@R"!"&2;C &&&~2i#B69¥e:mgh"S!/2#%4^
0
0.25
0.5
0.75
1
Ave
rage
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL0
0.25
0.5
0.75
1
Ave
rage
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL
]o 4, c ^_[C^_[CGNt71/69rH69#$R. ¥c(!869&gm6!$9gQ2GR"!"&2;C &28;j[ "!%!/2#%4^ n 'g¨Hd.1!/!/2#%4^K 4:SKP\69¥eO]2"S!/2#%4^
)@ )@ i&?./ " "!$#%!Xx&;%A!869s,d6\R"!u69&(69. sO%9g[2HRRV"B9gi2HPb69¥e:]2"H#%2:;D3I^v,26A.:!869.i69&(69. sC8^ A<2^"; &!/(69R · (69d,2 & 4] n &:,2#TWVG&(69W.<W/69 &§6r::;\;269#$R &v71Q2p.^5mgh"01,2&2!s<;#B69 &_^ S2'gm@!/2:4@,2!/<769. ,2"QBRV-gm# ,69&(69. sC"8^)c2k2+Pb6!/&:,#TWV)69
M = 0.569&2;2 ] n &:,#TWV)69
CFL = 106 32&^%;C/6^l2B69#$R. ¥c(!869&g>6!S#B69RkghS2%P\69¥eO]2"%!/2#%+^| )69&2;^| 2,S.^|P\6!/R"!"&2;C &2^ 2p",. i69T'eO W ";\ &uc24, c ^_[C^ · ^ 269T. 4:. s~;C D&2!""[WV ^"&WVH#%2:;zW,|2iR"!"&2;C &28;~/!/2#%M"#%5/@R7O;S6. 4:|WVz;269#$R &d7r|2 4Hgh"01,2&2!sG4&_^J[&22./832M4.W(69.tB!&2!"[69M71sG #$ .U698^LO"!"&2; ^)2Pb6!/&:,2#TWV69
M = 0.13G2 ",. 69 R"/'&O"; &c24, c ^_[C^ `2^
ZbZn
ξ / π
η/π
0 0.5 10
0.5
1
0.1
0.8
0.70.6
0.50.4
0.3
0.1
ξ / πη
/π
0 0.5 10
0.5
1
0.8
0.8
0.70.6
0.5
0.40.3
]o 4, c ^_[C^ · NQ#$R. ¥c(!869&ug>6!z#B69R<9g2pP\69¥eO]2"Sj[ "!Q!/2#%T69Pb6!/M = 0.5
69&2;CFL = 106 ^ n 'g¨p`J[R"!"&2;C &&2^z 4:p ,1.DR"!"&2;C &2 &2^
2%&2&C>R"!"&2;C &2";)/!/2#%%S 4. sb &2B!&:71T6v.U69rBR(69T9g|2$g¨"0:,2&2!sk;#B69 &2^| &(69 &2I^^| .[T,2[,2!/H6S#%2:;G@!"#$R,[.^5Pb6!/v&:,#TWVI\(^z8^ 0 &2S2(69&2;D32MR"!"&2;C &(";<!/2#%MR27:;"Q671'sdr::;u;269#$R &G69&2;D3(6|gh"!86/;~W:s2[4.W(69.¬69&(69. sO3O |B!&2!sGWVz(69&Ggm5 42zPb6!/v&:,#TWV8^KXx&v/;5$#$R(6 ´8|RRVxsr36^[R.|2S69#$R. ¥c(!869&~gm6!z9goWVH#%2:;69.&$([. &2
ξ = η69|2M/69#%
Pb6!/u&:,#TWVM69&2; ^T!"#$R(69T2# ^| <"RV"!QB2pW.:!/~!/2#%4=?[;(69#$R¥&22^ 2p",. 69M'eC W ";H &vc24, c ^_[C^ c ^ AMW271i(69|(i169RHWV ^"&~2SP\69¥eO]2"i#%2:;u69&2;H2.:!/!/2#%dp";C,2!"";ghT2G#B69. .5P\6!/)&:,#TWV8^\N¯RV&:U69.|'eOR.U69&(69&9g,2!/",. !869&HWVQgm,&2;G &H2iD"!|9g¬2iR"!"&2;C &2^Xx&2;"";D3C2[rr69._9g¬R"!"&2;C &2 &B"!/&01,2"MH#B69r$2%!"&2;C &&:,#TWVT9g52$sO#!.4$H,& xsr^%XxM#%869&2S(69S2$169RAWV ^"&69. .K2% r&:7469. ,2"SS";C,2!"";;C/69#B69!869. . sbghS2$.^Pb6!/k. #$ 8^dP<"7183t2%B!&2!sk9g2 ,2r.5R"!"&2;C &283KgmT &2/69&2!"43`@WV ^|2&Pb6!/)&:,#TWV#B69. .(69&
0.1^bN[
6A!"&2"0:,2&2!"4352~ #$R(6!G9gM#B6eC #$ ´" &2~&:,#%!869.i;C R(69&_3z6B B%2<!86HghB2Pb69¥e:]2"p!/2#%43(z."z #$RV/69&:gmz.^5Pb6!/u&:,#TWVghi^|!/<2p r&:769. ,2"i69M71s!.4M(69&Hghz 42Pb6!/H7469. ,2"8^ 6!/71¦269&(69. sO9g\2 #$R(6!¢9g<2§RV"!/69.H/6;C ,2+ #$R. ¥c(!869&_3:^§;C/6^ 269#$R. ¥c(!869&#B69R9g|.:!/k69&2;APb69¥eO]"%/!/2#%"T69SPb6!/
M = 10−3 ^<&2!"%(69S2(69RV"u9g269#$R. ¥c(!869&g>6!~#B69R2\69A #$ .U69~gmHWV#%2:;8^ P\69¥eO]2"A/!/2#%R7O;"M6Br::;b;269#$R &~7r[2@7rs~.^|mg¨"0:,2&2!s<;#B69 &_3DW,[ iB!&2!s<[7rs~.^gh2"9gt2zgh"01,2&2!s%#B69R_^`J[7r2."312Q169RdWV ^"&vWVd#%2:;&2/S #$RV/69&:8369&2;D3_69. 2,4A ["0:, "i#%@ /69&kv;269#$Rk2i#%:;"83_P\69¥eO]2"@#%2:;k!869&bWV!"#$RV 71S¥go z!"4|RVz /69&u|71'sG!/2869R_^
ZbZd
ξ / π
η/π
0 0.5 10
0.5
1
0.80.7
0.9
ξ / π
η/π
0 0.5 10
0.5
1
0.8
0.1
0.7
0.7
0.7
0.70.6
0.50.4
0.30.2
]o 4, c ^_[C^ `2NQ#$R. ¥c(!869&ug>6!z#B69R<9g2pP\69¥eO]2"Sj[ "!Q!/2#%T69Pb6!/M = 0.1
69&2;CFL = 106 ^ n 'g¨p`J[R"!"&2;C &&2^z 4:p ,1.DR"!"&2;C &2 &2^
ξ / π = η / π
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
Matrix Free without preconditioningMatrix Free with preconditioningBlock Scheme without preconditioning
M = 0.5 CFL = 106
ξ / π = η / π
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
Matrix Free without preconditioningMatrix Free with preconditioningBlock Scheme with preconditioning
M = 0.1 CFL = 106
]o 4, c ^_[C^ c iNQ#$R. ¥c(!869&bg>6!S69.&H2%6eOξ = η
69CFL = 106 ^ n 'g¨$[P\6!/
M = 0.5^
z 4:p`Pb6!/M = 0.1
^
Z [be
0.4
0.5
0.6
0.50.4
0.4
0.30.2
0.1
ξ / π
η/π
0 0.5 10
0.5
1
ξ / π
η/π
0 0.5 10
0.5
1
0.8
0.1
0.7
0.70.7
0.7
0.60.5
0.40.3
0.2
]o 4, c ^_[C^_aCtNQ#$R. ¥c(!869&Tgm6!¬#B69RSgmt2j[ "!o!/2#%69¬Pb6!/M = 10−3 69&(;
CFL = 106 ^ R`.:!/v!/2#%4^# `Pb69¥e:]2"S!/2#%4^
Z [ Z
! " #1 #% " (1( ! 6( Xx&)!/(69R · 3^5G,2;C";2~LSL!/2#%v69RR. ";\69& #$R. ! %.:!//69rv69&2; ^G,. &2";(69831,&."KRV-gm#$ &T/7r/69.2 &&2 /69&283:2|#%2:;Q^56K,&2/69W.|gmK 4G] n &:,#TWV8^[^7r83 &+2HR,RV4u9gM!"#$R, &L\(^zB&+,&2,2!,";¢#%"2"83|LSL)R:!"";C,u%7rs86s #$R.#%&:8^X&¢G"!&_3 ^ ^569&:dAO&2^¥gp,2!/6kR:!"";C,<d . .Q &:" &!",R. &'^| 6B#B69¥eOmgh"@ #$R. ! S/69r4^E@."6;b4.W(69.69&2;\.:!869.K69&a69.¥sC"i &k;iv!"#% &:%!"&2!. ,2&_^)*% )%A %& 5- - "!$#%! 2#B6eO #T,#¯69&2;B2Q6871/69rz7469. ,2"9gD2[69#$R. ¥c(!869&dg>6!71,2] n 69&2;BPb6!/B&:,#TWV69BR"/&:"; &c24, c ^ · ^ Z gh@&2B &&('@ /69&+69&2; &c24, c ^ · ^_[~gm [be~ &&2@ /69&28^0 &. s%&2 &&2¥/'69&G&2, B!&:pR7O;Q/69W . xsgm69. .a2QP\6!/d&:,#TWV8^KQ^7183,2 &'[be% &&2[ /69&2M;:"[&2Q869. . su &2!86T2@!"4QRV[ /69&k9gK2@P\]tYLpLH#%2:;W, &(69W."<"!"71@2dRV-gm#B69&2!""$9gQ2Gj[ "!$!/2#%4^L:,2!/+6u#%2:;@2'ghR#$ &B &H#%9g`!"469&2;~B!&2!s1^
0
0.25
0.5
0.75
1
Ave
rage
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL1
1.005
1.01
1.015
1.02
Max
imum
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL
]o 4, c ^ · ^ Z NQ#$R. ¥c(!869&Bg>6!9gD2P<]¬YLSL Z K!/2#%4^ n 'g¨QoN¬7r/69r4^ 4:oPb6eC #T,#~^
)*% )@ i&?./ " "!$#%! 2d.:!869.|69&(69. sCTT#B6;Bgm6u. ^5P\6!/&:,#TWV!86b
M = 10−3 p69&2;69CFL = 106 ^<Xx&
,2!/¢!"&2;C &2832~P\]¬YLSL)/!/2#%H%/69W.HW,2%¥/%B!&2!s%RV:83RV"!U69. . s &+2H.^^56871&:,#TWV|4&2S c24, c ^ · ^ · '^KZ[ & · eT &&2-> /69&(z&(69W."|T/!"7152iB!&2!s%71s!.4HA2~j[ "!vLO!/2#%4=?G&24^ (69&C%k d #$R. ! xs+69&2;+ %r::;¢!"&:71r&2!"RRV"83DLSLH.6e269&<R/1!"";C,/p0^. .¥, ";<%WVp!",R."; ^| <2pPb69¥e:]2"p #$R. ! /69r4^
Z [b[
0
0.25
0.5
0.75
1
Ave
rage
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL0
0.25
0.5
0.75
1
Max
imum
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL
]o 4, c ^ · ^_[CtNQ#$R. ¥c(!869&Tg>6!¬9g22P\]¬YLSL [ber_!/2#%4^ n 'gh_Nt71/69r4^¬ 4:_Pb6eC #T,#~^
ξ / π
η/π
0 0.5 10
0.5
1
0.8
0.7
0.7
0.7
0.9
ξ / π
η/π
0 0.5 10
0.5
1
0.8
0.7
0.7 0.7
0.70.6
0.50.4
0.3
]o 4, c ^ · ^ · @N#$R. ¥c(!869&gm6!69M = 10−3 69&2;
CFL = 106 ^ n 'ghdpP\]¬YLSL Z M!/2#%4^z 4:p`P\]¬YLSL · er'^
Z [ ·
! " #1 & # " 6( Xx&$!/(69R · 3rqo &: 6!"W(.:!/!/2#% ^56K2^|&$iWV71sT,&2/69W.z69&2; &2D"! 71^`Q^7183R/6!!869.a!"#$R,/69&(,2 &pP<]¬q #%2:;%(6KWV"&B!869";B,Kgh71'69.s1869K¥&GLOXYJ[ZQP\IK].U69WV/69s1^)X&,c2$;71.R#%&:u _[ba '3"^G,2/"; ^| +,2!"!""2v69#%G"!/&0:,2v!"#$R,Q.^5Pb6!/d&:,#TWV5\(^z8^ 2RRV"59g_#%2:; ^QW:7O,2. s$&25^. .¥>O&2^|&_^ :,283 69RRV869";~&2"!""//69sG &:71" 169igh,2z2MRV-gh#B69&(!"Q9g`,2!/<6@#%2:;D^
))%A %& 5- - "!$#%!P\]tq !/2#% ^| <&. sv&2p,WC> /69&<z/&4. sG,&2/69W.igm69. .] n &:,#TWV[69&2;HghQ69. .Pb6!/B&:,2#TWVi g-^oc24, c ^ `2^ Z '^`[^71831,&. 1Qq .1!/B!/2#%43 c eS &&2 /69&25&(69W.QR7O;p/69W . xsH7r&~ghz.U69r@] n &:,#TWV@ g^Kc24, c ^ `2^_[r'^5";"832SR.Q9g2T687r/69r69#$R. ¥c(!869&gm6!p2^zS(69S2$B!&2!sbM71s\ #$ .U69SH2%P<]¬YLSL\#%2:;D=?p&24^%Xx&2%&2'eCpR(69/694/69R_3j^l^| . .,2;Csb#%%R"!/. sb2d'B!&2!s9g|2BP\]¬q #%'2:;)69S71s4869[] n &:,#TWV8^
0
0.25
0.5
0.75
1
Ave
rage
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
Max
imum
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL
]¬ 4, c ^ `2^ Z KNQ#$R. ¥c(!869&~gm6!9g2SP<]¬q Z 5/!/2#%4^ n 'g¨pKNt71/69r4^Kz 4:SKP\6eC #T,#~^
))@ i&?./ " "!$#%! 2Hc2%.:!869.i69&(69. sC%B#B6;<69dPb6!/
M = 10−3 69&2; 69v] n106 ^ 2~",. B &+c24,
c ^ `2^ · #$R(6 ´8$2B69#$R. ¥c(!869&Ag>6!igmM&2B69&2; c ev &&2M /69&2^$P\]¬q Z [S&4. s,&2/69W.~W,B%,2~!869&+WVHR(69U69. . s+! !,#T7r&:";¢,2 &P<]¬q c er^ Xx&2;"";D3 &2/69W . "69SR &RV &:/;< &<2S71sv.^5mg¨"0:,2&2!sH4&_3a &<2 ^;83(2p#%2:; ^,.;uWVS,&2/69W.gmc2&2T#%"2"i&. s1^ ^G/694S69T2&\ #B694 &(69W.43D&(69#%. su";C,2! &v2$] n &:,#TWVS692'eCRV&29g56d4869i.4M9gB!&2!sugm[2@.^5mgh"0:,2&2!s<#%:;"83Di &2!86 &H2@&:,#TWV9gi,WC> /69&2%\R"/7rv2v &: &2!HB!&2!s69$2H'eORV&2H9g[2G4.W(69.!"48^)]¬ 4,c ^ `2^ `uR"&:p2B69#$R. ¥c(!869&gm6!p#B69R2p9g|P\]¬q c erp69
CFL = 102 69&2;A9gzP<]¬q c eber
Z [2`
0
0.25
0.5
0.75
1
Ave
rage
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL0
0.25
0.5
0.75
1
Max
imum
(ρ(
G)
)
0
1
2 Mach
0
50
100
150
200
CFL
]o 4, c ^ `2^_[C`NQ#$R. ¥c(!869&vgm6!59gt2[P\]¬q c er!/2#%4^ n 'ghiN¬7r/69r4^`z 4:iPb6eC #T,#~^
69CFL = 106 ^M5b#%2:;S69/69W.69&2;\2 M;269#$R &HRRV"p69T71su #$ .U69M'e!"R
gmd2<71s+.^5mgh"01,2&2!s¦6986C^ 2<c2&(69.i!/2!"\. "v&§,WC*-"! 71\!"&2;/69&283Q &¦,!86432^`g>687rt 4$] n &:,#TWV69&2;D3!"&2"01,2&:. s1346|#% #$RV/69&:`69#%,&:o9g2 &&2¬ /69&28^LSL~69&2;uq .U6e269&uR/1!"";C,/i"#$WVS7469. ,(69W.p!/2!""QB. 7rT69&u¥#$R2.¥! [Pb69¥eO]2"/69r4^ 2s<WVAR7:;% #$R. ! xsk69&2;kB!&2!sb!"#$R(69/$G2$P\69¥eO]2"$j[ "!S!/2#%4^J[&22."836^M&2"";vg¨,2Q69&(69. sC"zO&2^§¥g`,2!/H#%(1;2Q69S!"#$RV 7rS &<69. .t!86"8^
Z [ c
ξ / π
η/π
0 0.5 10
0.5
1
0.8
0.7
0.7
0.7
0.8
0.6
0.50.4
1.3
0.9
1
1.1
1.2
ξ / π
η/π
0 0.5 10
0.5
1
0.7
0.7
0.70.7
0.8
0.6
0.50.4
0.7
1
]o 4, c ^ `2^ · GNQ#$R. ¥c(!869&g>6!'/%69M = 10−3 69&2;
CFL = 106 ^ n 'ghHGP\]tq Z @!/2#%4^z 4:p`P\]¬q c er'^
ξ / π
η/π
0 0.5 10
0.5
1
0.7
0.7
0.70.7
0.8
0.6
0.5
0.4
0.7
0.9
ξ / π
η/π
0 0.5 10
0.5
1
0.7
0.7
0.70.7
0.8
0.6
0.5
0.4
0.7
0.3
]o 4, c ^ `2^ `2~N#$R. ¥c(!869&+g>6!d69M = 10−3 ^ n 'g¨uHP<]¬q c er$!/('#%u69
CFL = 102 ^z 4:p`P\]¬q c eber|!/('#%p69
CFL = 106 ^
Z [ba
! # & " ( +# "$#%&5&2^¢,&B,69&:&BM2z.U69 7rzRV-gm#B69&2!"9g 2P\69¥eO]2"| #$R. ! #%(1;B &%2R"&2!"S9g 4<6RV"!z/69$4;8^LO71/69.¬69,2z(6871M,2;C";H2SD!9go 4<6RV"!'z/69&).:!/A!/2#%G. 71";AW:sAN[jiX[#%2:; ZbZ TW:sk.U6e269&)R:!"";C," Z [ ^uN& &:" &;'c2& &~9g¬2[ #%i/'Rv4 71&vW:sB,2.^ 3O&(69#%. sB2 '#$ &CY] n M;'c2& &vgmI^|!/^M(687r%
∆t = max
[CFL · δx
λ+1
,CFL · δy
λ+2
] c ^ c ^ Z
[;'c2& &b&(69W."[%#B69 &:69¥&A69&<R¥#@,#] n 769. ,2p &<2@6eCU69.¬;C "!&b¥&<2T!86T9g 4)6RV"!p/69^HQ^7183¬2d69,2@2^®69.~(69T2GLSLk.:!/k/!/2#%%.:4"@6H.T9gB!&2!s<69 4b6RV"!Q/69G69&2;uQ;'c2& &b;:"Q&2Q #$R7rp2T!/2#%4^zX&<,[!8643 ^^569&:SO&2^ ^|222Q/69#%zWV(687O,W271";l^| d2P\]¬YLSL$!/2#%[69&2;%2QP<]¬q !/2#%4^
) )%A 9 !$.f="; 14;
ξ / π
η/π
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0.7
0.7
0.7
0.60.5
0.40.30.2
ξ / π
η/π
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0.7
0.7
0.6
0.5
0.40.3
0.2
0.4
]o 4, c ^ c ^ Z ¬NQ#$R. ¥c(!869&@g>6!o#B69R9g22P\]¬YLSL F`fert/!/2#%569M = 10−3 69&2;
CFL = 103 ^n 'ghp`N[RV"!|/69%9g
10−1 ^` 4:SKNQRV"!z/69%9g10−2 ^
Xx&pc24, c ^ c ^ Z 3b^R.";@2|69#$R. ¥c(!869&g>6!o#B69R$9g(25P<]¬YLSL F`fer¬!/2#%|69CFL = 10369&2;
M = 10−3 69&2;pgm¬ ^[;C D&:o7469. ,2"9g(2|6RV"!o/6934&(69#%. s10−1 69&2;
10−2 ^W,22 4 &(69.¬] n ;'c2& &Hgh52i#%#%&:8^KN[.&B652M6RV"!|/69$;"!86"83C2S69#$R. ¥c(!869&g>6!G9gS2<P\]¬YLSL#%2:;;/69"GgmB2<R,\6eCU69.Q#%:;43z(69dBgh
η = 0^ A
!869&G7O,(69. ´8Q#%[!.869. sd,2!/H6pWV(687:,5 &G2c24, c ^ c ^_[D^|!/vR"&:2QR9c2.i9gt269#$R. ¥c(!869&@g>6!69.&[2
xx6eC8^WA!"#$R(692P\]¬YLSLS!/2#%/2P<]~j[ "!KLO!/2#%43
69&2;D^5&2!"(692;269#$R &M9gVP<]¬YLSLS#%2:;¬";C,2!"";$;C/69#B69!869. . s6:&$6o256RV"!/69Gi.^i(69&
10−1 ^MP\]¬YLSLu#%2:;\ ^. .`!"&2;C &2";bgh[2T#$;O ^|687r&:,#TWV/MW,i R"&:z6p.69rQ.U6!/d9g¬B!&2!s%gm2QR,. sB.&4 ,2;C &(69.D#%:;"3: &2; !869 &$RV:|!"&:71r&2!"
Z [g
ξ / π (η / π = 0)
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
MF-SGS(40)MF - Direct Scheme
M = 10-3 CFL = 103 AR = 10-1
ξ / π (η / π = 0)ρ(
G)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
MF-SGS(40)MF - Direct Scheme
M = 10-3 CFL = 103 AR = 10-2
]o 4, c ^ c ^_[CdNQ#$R. ¥c(!869&g>6!@R9c2.~69.&x
x6eC$69M = 10−3 69&2;
CFL = 103 ^ n 'ghvN[RV"!|/69%9g
10−1 ^Kz 4:SKNQRV"!'z/69%9g10−2 ^
/694^Xx&B;p! !,#T7r&:,243^#$R.s$2 '#$ &CY] n ;'c2& &H69&2;l^z;C/6^ 2
x>R9c2.
9gt2i69#$R. ¥c(!869&Hg>6!z69AR = 10−4 ^`",. |69[R"&:";v &dc24, c ^ c ^ · ^ 2 '#$ &CY] n
;'c2& &b&(69W."[2TjQ "!SLO!/2#%T%"!"71Q2TB!&2!su9gAR = 1
3(,&Cgm,&(69. s13a [;:"&2 #$R71z2;269#$R &T9gD2P<]¬YLSL#%2:;Q^|!/B#B69 &2 Ggh6eOU69.(#%:;"8^Nz!"!";C &%2 [,2;Cs13H^T!869&\!"&2!. ,2;p(69Q2p#B69¥eOmg¨"T #$R. ca!"69 &b&2(p!869,2T9g`2p.4Q9gB!&2!s\69Q 4b6RV"![/69^ 2T &2"Q9gK2T#%2:;ughQR,. s<6eCU69.¬#%:;"Q. "i&<2LSL<69. r # ^|!/bQ,&2, ";<G 4k6RV"!i/69d4;^ AD^| . .`,2;Cs~ &\2T&2'eCi!/(69R^|22M269#%@&2;&2!s<i . .o7469. ;<gmiJi687:-YL:1"M"0:,(69&28^p[^7r83^!869&A69. 86;Cs#%&:&u(69z|Q6 ^869O&2"z9g2M#%2:;D^
) )@ 9 !$.f="; 1 ; ,5O&2^|.";Cr43D"!|9gt 4u6RV"!5/69$&Hq R:!";,i(6871Q&(5WV"'&~,2;C";D^K[&2!"3C | &:" &B%,2;Csd2MWV(687O,9g|#%2:;Hgh,2!/~!"&2;C &28^Q 4u6RV"!'z/69%(687r,R &H "!i&\2P<]¬q #%2:;D^ p"!869. .(69iq .U6e69&bR:!"";C,$!",.;\.86;<4! . .U69&k &\2;269#$R &23_69&2;b ´8"i9g,2!/b4! . .U69&2M!",.;b!869,2@ &2/69W . "8^S[^71'3^2^";v(69832!",R."; ^| <69&v #$R. ! z#B69¥eOmg¨"S/69r43C2S69. r ##B69 &2";~/69W.4^LO &2@6RV"!Q/69G69
10−1 !869"[4! . .U69&2[ &u(S.^5mgh"01,2&2!s~4&k69&2;u2S 4Cmgh"0:,2&2!s6986 g^$c24, c ^ c ^ `u.ghSR(69'^ 2B4! . .U69 &uR2&2#%&(6uR86;S~2+^|2.l^56871&:,#TWV;#B69 &~gh| 42[6RV"!z/69u c24, c ^ c ^ `% 4:zRa69'3.86;C &B$&% &2/69W . xs1^]($6RV"!@/69k69,&2;
10−4 3 @@&2"!""/69s<";C,2!"G2H] n &:,#TWV &;<R"71/69W . xs1^o](K,2!/%6RV"!/69 3r2|;269#$R &SK69. #%4`,&;C "!&(6.>31:,2^55!869&%86 . sT7O,(69. ´82SB!&2!sH9g2M#%2:;~W:sGR. &B2p69#$R. ¥c(!869&ug>6!Q69.&%2
y;C "!&_^Xx&vc24,
c ^ c ^ c 3^|W271|!.869. s@(69`(|4! . .U69 &SR2&2#%&(6S!869,25 &2/69W . xs@gh`48695] n &:,#TWV8369&2;D^&2!"(69o,2 &M6z4869`69#%,&:`9g2 &&2o /69&$;:"o&2o! !,#T71&:2RW.#¨ 4:R(699go2ic24,'^JQ&22."3CzR2&2#%&(6%!869&~WVi#$ 169";~W:sG/ &B6#%M86&(69W.] n &:,#TWV83CP\]¬q #%2:;d2&v/69W2. i69&2;G01, [B!&:M¨.'ghR(6959gt2zc24,KghR,. s
Z [ n
ξ / π (η / π = 0)
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
MF-SGS(40)MF - Direct Scheme
M = 10-3 CFL = 103 AR = 10-4
ξ / π (η / π = 0)ρ(
G)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
MF-SGS(40)MF - Direct Scheme
M = 10-3 CFL = 103 AR = 10-4
]o 4, c ^ c ^ · NQ#$R. ¥c(!869&)gm6!69M = 10−3 3
CFL = 106 69&2;AR = 10−4 ^ n 'ghG,2.
^56871&:,#TWVQ;#B69 &_^z 4:S.4'>,Ru9g4&~&2869x
x6eO8^
ξ / π
η/π
0 0.5 10
0.5
1
0.7
0.7
0.7
0.7
0.6
1
0.5
]o 4, c ^ c ^ `2`NQ#$R. ca!"69 &Hgm6!5#B69Rv9g¬2[P\]¬q c er!/2#%M69M = 10−3 69&(;
CFL = 106 ^n 'ghp
AR = δy/δx = 10−1 ^` 4:SAR = 10−2 ^
!4-869##%:;"8^KZQ&Cgh,&(69. s132P<]¬q #%2:;HR"&:|2M69#%i;C/6^|Wa6!/Oz(69&H2MP\]¬LSLv"!/&0:,243((69QQ6$71svRV:Q;269#$R &v9g`.&4 ,2;C &(69.;C,W(69&2!""¨&2[2^|&u2T &2!"2u",. v69u;&:!869.i2<&2"vW/69 &2"; ^| ¦P\]tYL2SL#%2:;('^q 69. r #~3 & 2#B69&&2M9gK2$LSL<69. r #~3_69RRV869[dWVp,&2, ";<d 4k6RV"
Z [ d
η / π (ξ / π = 0)
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
Point Jacobi (100)Direct Scheme
M = 10-3 CFL = 102 AR = 10-4
η / π (ξ / π = 0)ρ(
G)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
1.25
1.5
Point Jacobi (1000)Direct Scheme
M = 10-3 CFL = 106 AR = 10-4
]o 4, c ^ c ^ c zq`9c2.@9g2@69#$R. ¥c(!869&bgm6!M69.&G2yx6eOi69
M = 10−3 69&2;AR = 10−4 ^
n 'ghp`q Z eberz!/2#%p69CFL = 100
^Kz 4:SKq Z ebeberz!/2#%p69CFL = 106 ^
! ' & (1& (Xx&kp!/(69R83V^$6^(69p2%RV"!/69.K/6;¥,(p #$R. ¥c(!869&";C,2!""S2B;269#$R &uRRV"9gi2H #$R. ! GP\69¥eO]2"~/!/2#%4^¢[^7183,2!/ 6k #$R. ¥c(!869& %."%RV&(69. ´"¥&2)69B. ^5Pb6!/G&:,#@WV'^ :,25.4/59gt &: &2!iB!&2!sG;:"5&2z6 D"!52[4.W(69.D!"459g¬2[#%2:;R7O;C &p(69K !"4KRVK /69&BK7rs@.^S^Xx&%!/(69R`23^5z/6^a69LSL$69&2;%q 69. r #&2,p2p #$R. ! xs<69&2;~2S.^!"4Q9g`2TP\69¥eO]2"S#%2:;D^z5";"322T,2;C"(69 ^.86;% &B!/(69R2^¢(69,2!/d69. r #%569i^. .¥, ";BS/. 7rzI`,."0:,(69&2 &2!"z2sR7O;S6@r::;~;269#$R &$gm6@86&(69W.p69#%,&:9go &&('>¥/'69&28^Q^7183C2"p69. r #%#B691H(~#%2:; gm%R2,. s+6eCU69.#%:;"Q^|2& 46RV"!B/694;G69~,2";D^ 2"!/2#%"|69Q2'gm[."569/6! 71M6:&H6I^Q(687rQ@;869.^| G71sB!/2";G4;8^`X&G2&2'eCz!/(69R83H^ ^| . .t,2;Cs,^|22z zz . ._2M!86/[gm|2SJ[687O-YL:r""0:,(69&28^
`t k` 0
#1 +#% Xx&k2$R7O,2p!/(69R83¬2% #$R(6!T9g52$#B69¥eOmg¨"B #$R. ¥c(!869&)&A2$ &: &2!%B!&2!s(6KWV"'&d,2;C";$ &%2z!86z9gV2zI`,."0:,(69&28^H5&2^ ^569&:KS7469. ,(69|2|.4K9g &: &2!B!&2!s\;C,2@d2@7O!",2iRV"!/69./6;C ,2M #$R. ¥c(!869&A &\2@ #$R. ! S/69r4^EAT"!869. .`(69gm2[Ji687O'YL:r"|"01,(69&23O2Qgh#9gt2iP\69¥eO]2"i!/2#%[gh#B69. . sB,&2!/(69&r"; ^z¥"RV"!z$2M &:7:!;~!86%
C0∆wni,j + C−
1 ∆wni−1,j + C+
1 ∆wni+1,j + C−
2 ∆wni,j−1 + C+
2 ∆wni,j+1
= −∆t · Rni,j − σ1δ1µ1∆(fE)n
i,j − σ2δ2µ2∆(gE)ni,j
^| R 2z'eCR. ! K";C,(69.2 &:71. 7: &p2|7O!",2!"&: W,&v69&2;$2!869.U69K #$R. ! !:B!&:&2^4 71&HW:sb
C±
1 = −1
2σ1
(ρE1 + ρV
1
)ni± 1
2,j, C±
2 = −1
2σ2
(ρE
2 + ρV2
)ni,j± 1
2
, C0 = 1−C−
1 −C+1 −C−
2 −C+2 .
2&H2MPb69¥e:]2"M/!/2#%i69RR. ";H$. 71[7:/!",25.^5Pb6!/~&:,#TWV]\(^z83C2iR"!"&2;C¥& &v#B69¥e
P69RRV869[ &\2TR7O,2i!":B!&:B 'eOR"&F`2^_[C^_[ · '^SX&\i!/(69R83
2B #$R(6!@9gz2B7:/!",2TRV"!/69./6;C ,2@ #$R. ¥c(!869& ^| . .WV$c2T,2;C";D3o2& ^Q^| . .569.69&(69. sC@2B!&2!s\9g2@7O!",2[R"!"&2;C &2 &Hgm[2Pb69¥e:]2"%69RRr6!/_^TI`7r&:,(69. . s132pRV-gh#B69&2!"M9g`2TP\]¬YLSL~69&2;<P\]¬qK]¢!/2#%" ^| . .¬WVT,2;C";D3V"RV"!U69. . s~ &u2T!86T9g 4~4;u6RV"!z/69^
" " #%( # (.#%
Xx&~2S#B69&&2[9g2MR27:,2Q!/(69R83 ^S&2";uB;'c2&2p!869'gh,. . sv2pR(69/69#%[9g2p,2;Csr^Ng¨[&2&C;C #%&2&(69. ´6&_3_ &:7O!; 6!"WU69&2S!869&bWV'eCR"";<a69&OiG2P\6!/b&:,#TWVM
322 \(^69&4.v/u
69&2;~2S69B9gKRV"!¥c(!S2869γ
3H^| .S2S7O!",2 6!"WU69&2["01, p2
Z8·/Z
Z8· [
sO&2.;%69&2;)2Hq`/69&2;C.z&:,#TWV$ &+6;;C &_^&:71! 71~69&2;;C ,2 71dR('&(#%'&a6\RV4/"2 z"RV"! 71S!/(69/6!!p #%'R:^|!/<!869&~WVM;'c2&2";<65gm. .^zS
∆tE = min(δx
λ+1
,δy
λ+2
)
∆tV = min(δx2
ρV,δy2
ρV)
^|2λ+
1
69&2;λ+
2
69B2BRV"!/69./6;C ,2T9g|2B r&2sC# &86!/;C "!& ^z . ρV p2
7O!",2zRV"!/69._/6;C ,2i^|!/_3ghz2S69 8386;p
ρV =γµ
PrReρT!869&b!/2:4@2T #%'RA6Q2T#$ & #T,#WV ^"&b2@!"&:71"! 7r$69&2;<2@;C ,2 7r@!/(69/6!'!M #%M#T,. R. ";uW:sv6$/6gm'YsBgm6!
χ
∆t = χ · min(∆tE ,∆tV )
2&b!"&:7r"! 71 "!M69;#$ &(69&:83 xsOR!869. . s~gm69Qg¨# 2 ^569. .83^T(6871∆tE << ∆tV
3V2 #%'Ru|2'ghS4 7r&~W:sb
∆t = χ · ∆tE69&2;%2!":B!&:
χ^,.;B!""RV&2;BS2
CFL&:,#TWV8^ 0 &B22(69&2;D3f^zr
∆tV <<∆tE
^|2& 7O!",2dD!v69uRRV&2;/69&:83z(69dB & 2u2869B.U68s18352u #%'R B2&;'c2&2";~W:s\
∆t = χ · ∆tV69&2;v2M!":B!&:
χ^,.;H!""RV&2;H2
V NN&:,#TWV8^KXx&H;|.86;H/69W . xsv69&(69. sC"
gmz2pJi687:-YL:1"Q"0:,(69&283H^S&2"";\6$ 4&¥c(!869&:QR(69/69#%Q &<;B;#$ &2D^|2220\(^;#$ &(69/;vW:s%2[!"&:7r"! 71[D"!I^|225 ;#$ &(69";vW:s$2Q7O!",2D"!8^Xx&~86!/u;C "!&_3(&2S!869&u;'c2&2p6!". .tQs:&2.;|&:,2#TWVQ65gm. .^zp
Rex =u · δxρV
, Rey =v · δyρV
2S!". .tsO&2.;|&:,#TWV|2&u;'c2&2";<6|2M#$ & #T,#WV ^"&u2"i ^$7469. ,2"S
Rem = min(Rex, Rey) =u · δxρV
· min(1,AR
v/u)
Xg)^i,RRV4Q(6952 \(^69&4.["0:,(69.V45
;4""83O2&v2[/69v/u
5"0:,(69.VT,& xsG69&2;2S!". .tsO&2.;|&:,#TWV86;S
Rem =u · δxρV
· min(1, AR) aC^_[C^ Z
S!". .sO&2.;M&:,#TWVp&(69W."S,2MH7469. ,(69$2/69HWV ^"&A2$7:/!",2i #%'R)69&2;2S!"&:7r"! 71M #%'R_^Xx&2;"";D36^i(6871%
∆tV
∆tE=λ+
1 · δxρV
· min(1, AR2) ·[min(1,
AR
λ+2 /λ
+1
)
]−1
Z8·4·
^|!/_3 &~2S!86S9g`645
;4" \(^69&4.43";C,(!"H
∆tV
∆tE=λ+
1 · δxρV
· min(1, AR) =λ+
1
u·Rem = Θ
aC^_[C^_[r 2S/69%WV ^"&u2S ^% #%'R2. "&u2S!". .¬Qs:&2.;z&:,#TWVi69&2;H2S/69%WV ^"&2pW 4r"[ r&:7469.¥,(@69&2;u2pR(69!.T7r.:! xs1^|Xx&u2T!86p9gK2S &:7O!;uR"!"&2;C & &(3V2;'c2& &<9g2MR"!"&2;C &2";< r&:769. ,2SsO.;S
λ+1 ≈ u
^|!/~.86;z~∆tV
∆tE= Rem
[^7r835^ ^| . ."v(69%2~;'c2& & 9g[2HR"!"&2;C & &)2,.;WVv#%:;C¥ghs)¥&¢;%#B69 &§ "! 71ughG2<7:!",2v;#$ &(69"; \(^z8^LO,2!/§6A#%:;C¥c(!869&§#B691"d2\ r&:7469. ,2λ+
1
;C D&:Sg¨#2%R(69!.%71.:! xsr^d](SS86&L^l^| . .;&2%2%/69~9g52%7:/!,( #%'R ^| ~"RV"!z$2S!&O7r"! 71M #%'RuW:s
Θ
∆t = χ · ∆tE · min(1,Θ) = χ · ∆tV · min(1, 1/Θ)
J[^S^S!869&u86 . sv;#$ &2T69. .D2 6!W269&(Q9go2MRW.# 65gm. .^zS
AE = ∆tδxA
E = χ · min( 1ρ(AE)
, ARρ(BE)
) · min(1,Θ) ·AE(v/u,M, γ)
BE = ∆tδyB
E = χ · min( 1/ARρ(AE)
, 1ρ(BE)
) · min(1,Θ) ·BE(v/u,M, γ)
AV = ∆tδx2A
V = χ · Prγ · min(1, AR2) · min(1, 1/Θ) · AV (v/u,M, γ, Pr)
BV = ∆tδy2B
V = χ · Prγ · min(1, 1/AR2) · min(1, 1/Θ) · BV (v/u,M, γ, Pr)
SV = ∆tδxδyS
V = 13 · χ · Pr
γ · min(AR, 1/AR) · min(1, 1/Θ) · SV (v/u,M)
aC^_[C^ ·
^|2AV 3
BV 69&2;SV 69i4 71&~ &u69RRV&2;C¥eD^
]o &(69. . sr3<2"01, ";R(69/69#% ^|!/;'c2&22S^|2.7:!",2¢RW.# 692§gm. . ^5 &u
v/u3M
3γ
3χ
3AR
3Pr
69&2;Rem
^
)@ )%A i& h9<.f= 3,#1 ;H7b; #NQ&6;;C &(69.R(69/69#%Tp&2"!""/69sku!869sk,@,2;C"@69T#B69. .Pb6!/&:,#TWV83o&(69#%. sk2.^5Pb6!/AR(69/69#%
β(69 ^$ &:/1;,2!"; &!/(69R Z ^B](S &:7:!; \(^z3_ Sp!,2#B69sb
/691i 0:,(69._2SPb6!/~&:,#TWVp
β2 = min[M2, 1]
[^7r83(,2!/\6$;'c2& & ^56Q2^|&u$WVS,&(6;269R";u$7O!",2]\(^z Z ` 3(RV"!U69. . s'^|2&u2sO&2.;M&:,#TWVSi71s\#B69. .>^X&A;M/v#$R(6 ´8$2,2$;C,2v2,2$9g2 &:7O!;
Z8· `
;'c2& &_3_.M,2M!"&2;p6G.^5P\6!/k7O!",2M;#$ &(69"; \(^69M << 1
69&2;Rem << 1
69&2;.z,2z,2;CsG2T] n 69&2;H2M±[J[J&:,#TWVp
CFLx =λ+
1 · ∆tδx
V NNx =ρV · ∆tδx2
Z[ &$2M &:7:/! ;HR"!"&2;C &&ds:.;λ+
1 ≈ u69&2;u6
Rem << 1 |gh. .^zz(69p
CFLx =λ+
1 · ∆tδx
≈ u · ∆tδx
<<ρV · ∆tδx2
= V NNx
]([7:!",2i;#$ &(69"; \(^z83V2TR(69/69#%χ
&(69W."idR #$ ´82T±[J[J&:,#TWVM&. sr^ 2 &:7O!;TR"!"&2;C & &2S;:"o&2`&2,(692|] n &:,#TWV^,.;@WV5`69o R #T,#7469. ,24^Xx&u;[B! !,#T71&:QQ,243(±K&469" ^569/69& R2RV4";u & n/Z 6$&( ^;'c2& &b9gK2.^5Pb6!/~R(69/69#%T
β2v = max
(α1 (α1 − 1)
α1 − 1 + c2/u2,
α2 (α2 − 1)
α2 − 1 + c2/v2
) aC^_[C^ `:^|
α1 =1
Rex=
ρV
uδx, α2 =
1
Rey=
ρV
vδy.
aC^_[C^ c ](|2S!86M(69i^S!"&2;";D3( #$R.S!869.!,.U69&2sO. ;2S
βv ≈ ρV /δx
c(69z2S r&:7469. ,2Mz&2^"0:,(69._
λ+1 ≈ ρV /δx
^|!/u.86;|~
CFLx =λ+
1 · ∆tδx
≈ ρV /δx · ∆tδx
= V NNx
&2"0:,2&:. sr32v7O!",2R"!"&2;C &2 &)69. .^z%kR #$ ´8H2v±iJ[J&:,#TWVd69&2;2<] n&:,#TWVG69%2~/69#%H #%4^ 2~;'c2& &9g[2H.^|P\6!/¢R(69/69#%%gm$2~7:!",2B!86v2'gmM2igh. .^| &<
β2 = min[max[M2, β2v ], 1]
!869&$&2!"5(69`25&2[&2"";$9g 6;;C &(69.(;269/6[ &$;`M;'c2&252.^5Pb6!/$R(69/69#%8^
" +# &( ( ( #1" " ( (1 & "$#%&' (% (1 " (% (
Xx&;$k'e69#$ &2HR"!. s)2vWV(687:,B9g[2HP\69¥eO]2"H!/2#%Ggm2HJi687:-YL:1""0:,(69&283^z;C &4,%gh,Kxs:RV"9g #$R. ! /69r"8^ 25c2K&2|`2z!.U6!869.2.:!//!/2#%gm]^z!/u &:7:!;<69&2;H7O!",2z#%Q69p769. ,(69/;u,2 &%W.:!/ 6!"WU69&28^ 2p"!"&2;u&2MQ69&
Z8·bc
ξ / π = η / π
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
Inviscid Block / Viscous BlockInviscid Matrix-Free / Viscous BlockInviscid Block / Viscous Spectral RadiusInviscid Matrix-Free / Viscous Spectral Radius
M = 0.5 , χ = 106 , Rem = 10 , AR = 1
ξ / π = η / π
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
Inviscid Block / Viscous BlockInviscid Matrix-Free / Viscous BlockInviscid Block / Viscous Spectral RadiusInviscid Matrix-Free / Viscous Spectral Radius
M = 0.5 , χ = 106 , Rem = 10-1 , AR = 1
ξ / π = η / πρ(
G)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
Inviscid Block / Viscous BlockInviscid Matrix-Free / Viscous BlockInviscid Block / Viscous Spectral RadiusInviscid Matrix-Free / Viscous Spectral Radius
M = 0.5 , χ = 106 , Rem = 1 , AR = 1
ξ / π = η / π
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
Inviscid Block / Viscous BlockInviscid Matrix-Free / Viscous BlockInviscid Block / Viscous Spectral RadiusInviscid Matrix-Free / Viscous Spectral Radius
M = 5.10-2 , χ = 106 , Rem = 10-1 , AR = 1
]o 4,5aC^ · ^ Z tNQ#$R. ¥c(!869&Tgm6!tR9c(. "69.&[;CU69r&(69.C6eO_ght2`gm,t;C D&:¬ #$R. ! o/69r"8^ RC n 'ghv
M = 0.53Rem = 10
^ RC 4:vM = 0.5
3Rem = 1
^u5# n 'ghvM = 0.5
3Rem = 0.1
^`5#z 4:TM = 0.05
3Rem = 0.1
:sOW;b/69r@gh ^|!/k2@ &:7O!;\#%p69 #$R. ¥c(";b,2 &HRV"!/69.`/6;C ,2 ^z . 27:/!,(#%p69869";k. 1@2$.:!/b!/2#%4^ 2 ;k/69riW, . M,2 &v &O7:!; 6!"WU69&2@69&2;7O!",2pRV"!/69./6;C ,28^ 2&_3¬2B.U6T&2%T,@P\69¥eO]2"B/!/2#%$gm ^|!/WV &:7O!;69&2;G7O!",25#%z69i7469. ,(69";v(69&C/RV"!/69. /6;C ,2^`Xx&GR/6!!"432i"!"&2;~69&2;G2[ ;!/2#%"T69/$&2S869. . sb &:" &uWV"!869,2%2sA69%'eCRV!";AHR/7:;%."SB!&2!s^| 2,";C,2! &B2M#%#%sH/69r4^KJ[&22."83,2 &%2""|gh,z #$R. ! Q/69r"z&(69W.",2|O&2^^|!/ #$R. ¥c(!869&_3t &:7O!;ki7O!",2S&243DM#B69 &. s<RV&(69. ´" &uHW/69 &Ar::;kB!&2!sr^$Xx&i"!&_3-^@!"&2;M,W2&! \(^zS69&2; ^T7469sH2!". .sO&2.;[&:,#TWV[g¨# !"&:71"! 7r;#$ &(69"; \(^z%
Rem > 1zG7O!",2M;#$ &(69"; \(^z%
Rem < 1'^ 2@",. i &<c(4,+aC^ · ^ Z
R"&:52M69#$R. ¥c(!869&Hg>6!z69.&2i;CU69r&(69.t6eO59g¬2 ^5687r&O,#TWV|;2#B69¥&t^ AQc252@Pb6!/\&:,#TWVM69
M = 0.569&2; ^T:u,2!"!"" 7r. s~2@!". .sO&2.;Q&:,#TWVS69
1031
69&2;0.1
^ TW271T(69QghRem = 10
3V2T.4[9gB!&2!su[#B69 &. su;C,2Td2T &:7O!;\RV"!/69.
Z8· a
/6;C ,2S #$R. ¥c(!869& &2!"@2$ #$R. ¥c(!869& &b27:/!",2iR(69S;:"i&2S;/69%B!&2!s1^0 &k2$('S(69&2;D3¬69
Rem = 0.127O!",2MRV"!/69.`/6;C ,2Si2%!"69,($9g2W(6;k;269#$R &
RRV"8^5](Rem = 1
3(690^|2&<!"&:7r"! 71T69&2;~7O!",2 "![69p9g2p/69#%p;83(2.4@9gQB!&2!s@"01,(69. . s;C W,";WV ^5"'&2d ^\ #$R. ¥c(!869&_^kN|!"!";C &buTc269&(69. sC83`&2G!869& &A(692GD"!9g2d7:/!",2@RV"!/69.| #$R. ¥c(!869&+&)2GB!&2!s69S#B69 &. sH;C 71&uW:sG2S7469. ,2S9g`2p!". .¬s:&2.;&:,#TWV83V6zz.U69[;"!86"2S7:/!,( #$R. ¥c(!869&bWV"!"#%"[#%pRV&(69. ´" &2^[Xx&~g>6!83(Qi6$W [#%T!"#$R.'eD3a &2;"";D3VW:sHr"R &2T!". .osO&2.;&:,#TWVi69
0.169&2;< &d2TP\6!/<&:,#TWVi69
M = 0.05^T!869&<&2!"p(69
2v "!9gQWV #$R. ¥c(!869&WV"!"#%v9g2v/69#%G;8^&2"0:,2&:. sr3K2d #$R(6!$9gQ27O!",2zRV"!/69._/6;C ,2Q #$R. ¥c(!869&\"#%z%. sv&~2S/69
Rem/M/692(69&
Rem^
/69$|O&2^|&<6|2 "6!",2!SsO&2.;|&:,#TWV[69&2;H¥zz;&2";<65gm. .^zS
Rec =c
ν/δx
aC^ · ^ Z 2&
Rec > 13:6!",2!D"!;#$ &(69Q2]\(^S31O,22z7O!",2RV!69.(/6;C ,2 #$R. ¥c(!869&
;:"@&2%6 D"!@2vB!&2!s)9g2G#%2:;D^ 0 &)(G2(69&2;D3"^|2&Rec < 1
3`2G7:!",2!/(69/6!!T #%Sz2p#B69. .'i69&2;~2p #$R. ¥c(!869&b&~2S7:!",2#%z2'gmS 4. sRV&(69. ´" &2^B[^7183_ &,MR/6!!869.!"#$R,/69&(83¬2 "6!",2! %Qs:&2.;M&:,2#TWVSM/69. s#B69. .5(69&v,& Ysr^V[!869&v86&(69W. sd &B(6952[7O!",25RV"!/69. /6;C ,2|;:"5&2|!869,2[:W k6~.4@9gB!&2!sr^\Xx&2B"@9g2d!/(69R83K,,2;C"D^| . .|69 #69@.^5RV"";D3`7:!",2;#$ &(69";'\(^z3C(69z5gm
Rem << 13M << 1
69&2;Rec > 1
^
" !( ( "$# Xx&$"!&_3f^]^|69&:KS,2;Cs@R"!. s2zB!&2!s%9g 2zPb69¥eO]2"|#%2:;% &B2z!86z9g.^5Pb6!/~7O!",2z;#$ &(69";:\(^z8^K5'gmM/6!/:. &Bz,WC*-"!83.z,25c2z,2;CsG2M #$R(6!Q9g2|7:/!",2`R"!"&2;C & &@&%25 #$R. ! W.:!//!/2#%^VA|/`2zP\6!/%&:,#TWV69
M = 10−569&2;v2M!". ._sO&2.;5&:,#TWVQ69Rem = 10−1 69&2; ^i;C/6^§2M69#$R2.¥¥c(!8694&Hg>6!|#B69Rvgh|2
&:7O!;69&2;k2%7:!",2SR"!"&2;C & &2 c24, aC^ `2^ Z '^ 2$7O!",2SR"!"&2;C & &k69.¥.^ &2!86S;C/69#B69!869. . sv2M &: &2!SB!&2!sH9go2M.:!/G/!/2#%M71|2M&: ^56871&:,#TWV;#B69 &_^NQ%#%&:&2"; 869. 83 d69RR8eC #B69. s+"0:,(69. ´""B2<6!",2!'Y] n &:,#TWVG69&2;+2±iJQJ&:,#TWV0^|!/~#%71"z2M V&2"5gh#2SJi687:-YL:1"sO#~^&2^ #B6912+/69#%+,2;CsghA2¢P\69¥eO]2"+!/('#%4^ (",. 69+R"&:";l &c24, aC^ `2^_[C^dZQ&. 1$2%.:!/k!8643t2$7O!",2SR"!"&2;C & &b;:"S&2p #$R7rB 4&¥c(!869&:.¥s2 &: &2!5B!&2!sT9g(2Pb69¥e:]2"#%'2:;D^J'71&@&2(69o( &:7:!;TR"!"&2;C &&R7O;"@6HWV@;269#$R &<ghp2%.^5mgh"01,2&2!sA#%:;"8^ 2%86&2@9gz,2!/)6HWV(687O,69&2|!'.869. sd &2;&:¥c(";D36^[,RRV4[(692[7:/!",2RV"!/69. /6;C ,25 #$R. ¥c(!869&~56952[ 4 &9g[2vRW.#~^J[7r2."3"^v";\c2&2;+6\#%869&+A6b #$R71v2HB!&2!s9g[2#%2:;D^ z"869!/~.";H,2|$#%:;C¥g¨sG2M7O!",2|R"!"&2;C & &2d;'c2& &_^
Z8· g
ξ / π
η/π
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0.3
0.1
0.4
0.4
0.5
0.6
0.4 0.4
0.5
0.2
ξ / π
η/π
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0.3
0.1
0.4
0.3
0.3
0.3
0.3
0.4
0.3
0.2
]o 4, aC^ `2^ Z NQ#$R. ¥c(!869&g>6!~#B69R9g$2A.:!/§!/2#%. 71";;C "!. s69M = 10−5 3
Rem = 10−1 69&2;χ = 106 ^ n 'g¨HGXx&:7:/!;R"!"&2;C &2 &2^z 4:Hd±[!",2R"!"&2;C &C
&2^
ξ / π
η/π
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0.3
0.6
0.6
0.7
0.6
0.4
0.6
0.5
ξ / π
η/π
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0.6
0.6
0.5
]o 4,DaC^ `2^_[CNQ#$R. ¥c(!869&ugm6!#B69R<9g2pPb69¥e:]2"p!/2#%p. 71";~;C "!. s~69M = 10−5 3
Rem = 10−1 69&2;χ = 106 ^ n 'g¨pKX&:7O!;vR"!"&2;C & &22^z 4:S`±Q/!,(|R"!"&2;C & &(^
Z8·bn
&( ' ( # & "' #]o @"869!/2"< &2!86/d2G;269#$R &A9g2GPb69¥eO]"v!/2#%dgmT2G.^5Pb6!/7:/!,(;#$ &(69";$4 #%|7r869.";$(69`25R #T,#l7469. ,2|9ga25.^5P\6!/%R(69/69#%
β/.";WV ^"&
βv69&2;
M^](2Q!86"(695^,2;C";D3M^,2,(69. . s$(687r
βv > M^|!/d#%869&2(6927:!",2
R"!"&2;C & &2T!869"K#%5&:,#%!869.a;C R(69&%(69&%25 &:7O!;$&24^NQ`257:/!",2`RV"!/69./6;C ,2M #$R. ¥c(!869&kR7O;"M7r-;C/ R(69&_3-^5T";\G";C,2!"@[&:,#%!869.K;C/ R(69&bW:s &u
β2MF = βv ×M
2T",. iW/69 &2"; ^| b[!/2!"$69TR"&:";<gmQ2@.:!/<69&2;<2@Pb69¥eO]"!/2#%" &\c24,+aC^ c ^ Z ^AA@&2!"@(69ii&2h^71&b9g2@7O /!",2[R"!"&2;C & &2~;:"i&2M&(69W.2u.:!/!/2#%uAWVH#%<B!&:G!"#$R(69uA2u!.U6!869.71&_^ 0 &¢2u2B(69&2;D32%Pb69¥e:]2"%/!/2#%$869R2T6G4/"69pWV&2'c2T9g5M&2h^!/( !"4^dX&2;"";D3t2%p6H 4&¥c(!869&: #$R71#%&:S9g2;269#$R &u7ri2&: ^56871&:,#TWVS;#B69 &69&2;b2@RV-gh#B69&2!""S &b271sM.^5mg¨"0:,2&2!sS4&$69&2^71sS #$ .U69o2 &:7O!;pR"!"&2;C &&2^ :,28392!.U6/!869.71&T9g227:!",2tR"!"&2;C &2 &i¬!.869. s ^. .¥, ";Sgmt2.:!/S!/2#%^| .2P\69 eO]2"69. r 2#"01, "S69&k69. &(69¥717r&_^MXx&<(T"M9g2@R(69/694/69R^ ^| . . &:7r" 169gh,2z2i #$R(6!Q9gz&(69&2!"";~R"!"&2;C &2 &G&H2SPb69¥e:]2"M#%2:;D^
ξ / π
η/π
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0.3
0.1
0.2
0.4
0.4
0.3
0.3
ξ / π
η/π
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0.4
0.6
0.5
0.5
]o 4,+aC^ c ^ Z QNQ#$R. ¥c(!869&bg>6!i#B69Rk9g2j[ "!pLO. 71' ^| b2&(69&2!"";bR"!"&2;C &&69M = 10−5 3
Rem = 10−1 69&2;χ = 106 ^ n 'ghp`.:!/v!/2#%4^Kz 4OS`Pb69¥eO]"S/!/2#%4^
2Tr::;uR/RV'M9gK2T&2h^;'c2& &k9gK2T7O /!",2R"!"&2;C &2 &v/'#B69 &b769. ;<R7:;";(69H2A6!",2!ksO&2.;v&:,#TWVHv4869~(69&§,& xsr^LO6v!"&Cc2# H &:" &WV(687O,83 ^bR"&:~ &¦c24," aC^ c ^_[69&2; aC^ c ^ · #%k!"#$R(69/&2HWV ^"&69. .M2bxsORV~9gR"!"&2;C & &2T &%2z!86z9g 2zPb69¥e:]2"!/2#%z. 71";%;C "!. sr^VAz;C/6^+269#$R. ¥c(!869&g>6!%69.&k2H;CU69r&(69.z. &2
ξ = η^]($71s!8643K2G &:7O!;)R"!"&2;C &2 &R7:;"B6
WVT;269#$R &<9g52$.^5mgh"01,2&2!sk#%:;" ^| .%2$7:/!",2MR"!"&2;C & &2<M#%BB!&: &)2d"@9g2 ^56871&:,#TWV$;#B69 &_^ 2d&2h^¯;'c2& &&(69W."uR"71dWV)xsORV"@9g
Z8·bd
B!&2!sr^MX&2;"";D3^T&2T(69[2@&(69&2!"";\R2"!&(;C & &HR7O;"M6B #$ .U69i;269#$R2¥&2Ga69&2% &:7:!;&2gmS2%.^5mg¨"0:,2&2!sb#%:;" ^| .% pS#%B'B!&:T(69&2%7:!",2p&2% &2M"z9g2 ^5687r&:,2#TWV;#B69 &_^
ξ / π = η / π
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
Inviscid PreconditioningViscous PreconditioningEnhanced Preconditioning
M = 10-3 , χ = 106 , Rem = 10-1 , AR = 1
ξ / π = η / π
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
Inviscid PreconditioningViscous PreconditioningEnhanced Preconditioning
M = 10-1 , χ = 106 , Rem = 10-1 , AR = 1
]o 4, aC^ c ^_[C|NQ#$R. ¥c(!869&<g>6![9gK2TP\69¥e:]2"T!/2#%@. 71";u;C "!. s<69.&G2T;CU69r&(69.. &2
ξ = η^ n 'ghT5#$R(69&k69
M = 10−3 3Rem = 10−1 69&2;
χ = 106 ^5z 4:T5#$R(69/&b69M = 10−1 3
Rem = 10−1 69&2;χ = 106 ^
(69&C[`&(69&2!"";$;'c2& &B9ga27O!",2oR/!"&2;¥ &2 &2312.4`9gVB!&2!s@;C,25[2Pb69¥e:]2" #$R. ¥c(!869&2MQ";C,(!";D^ 2@#%2:;b!869&bWVT2&b7rsu!"#$RV 71 ^| \"RV"!$(S.:!/H!/2#%4^J['eOzR(69/694/69R<;71";~%2S,2;Csv9g`2SPb69¥eO]"p/!/2#%p. 7r";W:sLSL)Bq .U6e69&+R2:!";C,2"8^¢[ 71&b2vr::;",. %W/69 &2"; ^| 2~'&2(69&2!"";R"!"&2;C & &223 ^ ^| . .t&. sv!"&2; 5gm|2SP\])!/2#%"^
( " ( P\]tYLpL\69&2;bP\]¬q 69. r #%T69RRV869";b/GWV!"#$RV 71#%2:;Sv. 71@IK,.'p"01,(69&2^| 2,T 4)6RV"!p/69 !/(69R c '^GXx&2;"";D3¬WV#%2:;pR7O;%r::;;269#$R &ugmT686&(69W.p&:,#TWV[9g &&2-> /69&28^;Ap'eORV"!BW/69 &<2S69#%MRRV"gmz2pJi687:-L:1"B"0:,(69&28^ n B,2d!"&2;B2u!86u;/69 .";¢ &¢2~R7:,2d"!&
M = 10−5 69&2;Rem = 10−1 '^SX&<c24,laC^_aC^ Z 3-^@R.";b2$69#$R. ¥c(!869&kgm6!M9g2P<]¬YLpL F`fer#%2:;h&~2M.'g¨|69&2;~9go2SP\]¬q c erz69. r # h&~2M 4:'^N['eORV"!";D3P<]¬YLSLB!/2#%i&(69W.",2@W/69 &H;269#$R &RRV"|!.4/QT2[&2"5; 71";gh#¯2ij[ "!|!/2#%4^K[^7r83/^[&2i(69gm2i7rsB.^ ^|687r&:,#TWV832P<]¬YLpLB/!/2#%6M. .W (69&d2Qj[ "!#%2:;D3: &2;C!869 &$6M.4/9gtB!&2!sr^Kj["R z83OP<]¬YLSL!/2#%S!869&~WVM169;";<66@7469. ,(69W.S!/2!"S$. 71i2SJi687:-YL:1"0:,(69&28^L: #$ .U69. s13tP<]¬q c eri#%2:;bR7O;"p6dr::;A;269#$R &<69WV7r%6G.U69rR(69S9g2 ^56871&:,#TWV;#B69 &_34&2&22."834 oWV"!"#%"869. . sT %gm¬2.^5mg¨/0:,2&2!s@69/"6C^ o,2!869&@WVR(69U69. . s! !,#T71&:";bW:su,2 &v#% &&2-> /69&2T69i2'eORV&29g569&b &2!86$9g2!"4[RVi&(&C. &2869o /69 &t^Q^71834P<]¬q !/2#%#B69 &2K69.[!"#$RV 7r|69&2; ^!869&@&2!"(6983,&. 1
Z `fe
ξ / π = η / π
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
Inviscid PreconditioningViscous PreconditioningEnhanced Preconditioning
M = 10-5 , χ = 106 , Rem = 10-2 , AR = 1
ξ / π = η / πρ(
G)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
Inviscid PreconditioningViscous PreconditioningEnhanced Preconditioning
M = 10-3 , χ = 106 , Rem = 10-2 , AR = 1
]o 4, aC^ c ^ · |NQ#$R. ¥c(!869&<g>6![9gK2TP\69¥e:]2"T!/2#%@. 71";u;C "!. s<69.&G2T;CU69r&(69.. &2
ξ = η^ n 'ghT5#$R(69&k69
M = 10−5 3Rem = 10−2 69&2;
χ = 106 ^5z 4:T5#$R(69/&b69M = 10−3 3
Rem = 10−2 69&2;χ = 106 ^
gm|2SI`,."0:,(69&2832M#%2:;~#B69 &2/69W.igm| 4~7469. ,2M9g2S/6ghxsdgm6!χ
^Xx&H2MR7:,2!/(69R283H^MW2/7r";H(69zWV\69. r # ^M&2i^. .¥, ";Hgmz/. 7: &BI`,."0:,(69&2z &H2MR"&2!"S9go 4<6RV"!z/69$4;8^KZ[ &%!/2";H4;zz!,2#B69sG%. 712[J[67:-YL:1"50:,(69&25 &2!" ^Q&2"";d#%QRV &:@;"! WVQ20\(^ &G2[2869.U68sr8^`Xx|2'gmM&2"!""/69sGB69&(69. s:´8M(iRV-gm#B69&2!"p9gKLpLv69&2;~q 69. r9¥#%z &u,2!/u!"&2;C &28^ z|2SWC*-"! 71S9go2M&2'eCz"!&_^
ξ / π
η/π
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0.7
0.90.6
0.5
0.6
ξ / π
η/π
0 0.25 0.5 0.75 10
0.25
0.5
0.75
10.7
0.90.8
0.50.6
0.5
0.6
0.6
0.7
0.7
]o 4, aC^_aC^ Z @N#$R. ¥c(!869&g>6!p#B69R)69M = 10−5 3
Rem = 10−1 69&2;χ = 106 ^ n 'ghBTP\]¬
LSL F`ferp!/2#%Q^| &(69&2!"";7:/!",2pR"!"&2;C & &22^< 4:dP\]tq c erp!/2#%Q^z¥)&C(69&2!"";~7:!",2|R"!"&2;C &2 &2^
Z ` Z
# & " ( +# "$#%&Q 4k4;6RV"!S/69u69%!,2#B69sk&2!",&:"; &kR/6!!869.!"#$R,/69&2T &2!"$2869S.U68sr"0:, \6A.U69r\69#%,&:v9gSRV &:d & 2<;C "!&¦&2#B69.[A( ^569. .Q &;d". 71u2 44/6;C&:@4&28^~Xx&!/(69R c 3V^d;869. D^| ),2!/ 46RV"!T/69u,2"Sgmp2dI`,."0:,(69&2@69&2;L^%2^";A(69p2%Pb69¥eO]"B #$R. ¥c(!869&);C;A&2pr&2/69G69&:sb;CXB!,. "^0 &k2$2M(69&2;D3^&2!"";k(69TLSLb69&2;kq 69. r #% ^&2 ^. .¥, ";bgmiRW.#% &:71. 7: &[71sS!/2";T#%"2"^ 2""o#%2:;K69'eORV"!";@QWV(687r #$ .69.¥sSgm¬2Ji687:-L:1"["0:,(69&28^iX&\;id2^2@;CXB!,. xs<9g;869. & ^| \7O!",20\(^zi71Q 4k6RV"!/69B4;83a.Q,2Q!"&2;M69&\'e69#$R.4^ n [,2Q,RRV4p(69
Rem < 13M << 1
69&2;AR << 1
^v,RRV4H69.b(69+^G,2G2H!.U6!869.|7:/!,(@R"!"&2;C &2 &)69&2; ^H!"#$R,v(
CFL&:,#TWVQ69&(;v(V NN
&:,#TWV &~86!/u;C "!&\6|gh. .^zp
CFLx =λ+
1 ∆t
δx≈ ρV ∆t
δxδy= V NNy ×AR
CFLy =λ+
2 ∆t
δy≈ ρV ∆t
δy2= V NNy
V NNx =ρV ∆t
δx2= V NNy ×AR2
V NNy =ρV ∆t
δy2
Xx&v2i!86M9g¬4;z!/2";H &v2x
;C "!&_322i7:!",25R"!"&2;C &2 &d&(69W."|$R #$ ´82±[J[J¢69&2;T2z] n &:,#TWV &25!'/4-869#;C "!&_^N[`6",. 832
CFLx&:,#TWV69&2;
2V NNx
&O,#TWVM69T#B69 &:/69 &2";k69i#B69. .¬7469. ,2p¥g26RV"!Q69dQ7rs~#B69. .>3a &2;C!869 &~6RV:;269#$R &T9g 25.&4 ,2;C &(69.VK#%:;"8^KL:,2!/G6i/694s@`/gm";B6`2 '#B6eOY] n #B6e:±iJQJ ~!"#TW &(69 & &2!"u vR #$ ´8"v2< 42"G7469. ,2\9g@] n 69&2; ±[J[J R(69/69#%8^ 2#%:;C¥c(!869&¦9gS2~7O!",2BR"!"&2;C & &2gmB2uPb69¥e:]2"<!/2#%<;:"B&2G!/(69&ru ,(69&_^ d! !,#T71&:iQ,243a±'&26h^569/69&k69&2;\P\:.pRRV4/";< & n/Z 6%#%:;C¥c(!869&9g¬2[7O!",2R"!"&2;C &&2^ 2 |;86T5@ #T,. /69&2",2. sGR #$ ´8M2[7:!",25R:!"""5 &2@!4-869# ;C "!& ^| b26!",2! ^5687rTRR(69169&k &\2T.&4 ,2;C &(69.K;C "!&_^MXx&,[!8643(2p&2h^;'c2& &b9gK2S7O!",2R"!"&2;C &2 &v"B#B69 &:/69 &\2
CFLx&:,#TWV
69&2;2V NNy
&:,#TWV692 KR #T,#l769. ,2"8^ K!869&WV|6!/71";W:sT!/(69&4 &S25;'c2& & aC^_[C^ c |9g
α169&2;
α265gh. .^zp
α1 =1
Rex· 1
AR2, α2 =
1
Rey·AR2 .
aC^ gO^ Z
t/694si_'gh";p6_2 '#$ &CY] n #d6eO>±[J[J K!"#TW &(69&@ &2!"` tR #$"_2K#$¥&2¥#@,#] n &:,#TWV569&2;B2#B6eO #T,#®±[J[J &O,#TWV8^`[^7r83O7r& ^| B&2h^ R"!"&2;C & &2V NNx
&:,#TWV[#B69 &2Q71sH#B69. .V^|!/\ . .¬!869"[ &2"zghQ2pR,p.&4 ,2;C &(69.o#%:;"^P\"7r83K$;'c2& &+9gQ2GR"!"&2;C & &AR7O;"#%G&:,#%!869.;C R(69&_^Xx&2;"";D3 #$R.S!869.!,.U69&2sO.;S
βv ≈ ρV /δy
c· 1
AR>>
ρV /δy
c
Z `f[
2&<26RV"![/69dQ71s~#B69. .>3a2TR"!"&2;C &28;b!/('#%@!869&b71&<,2&\W(6!/~B Q&(&CR"!"&2;C &2";§7r&_^[&2!"43,2!/§6/694s "#%v!.869. s¢,&(6;269R";),vPb69¥eO]2"!/2#% ^|!/"01, "$."&:,#%!869.;C R(69&_36@#%&:&2";+869. 8^ 2~69&(69. sO(69+^RV-gm#%";\ &2;C!869"S!."69.¥s<(6983Vgm[i!/2!"43D2@Pb69¥e:]2"!/2#%@i )7ri2@&: ^56871&:,#TWVz;#B69 &_^KN.65/694s ^56|69.;" 4&2";GWOsBP\:."8=O869# ZbZ 3 n/Z 69&2;G'gh";H62 '#$ &CY] n #% &C>±iJQJ !"#TW &(69& &2!"@ i"iG#B69 &:/69 &A2@#$ & #T,# ] n 69&2;b±[J[J&:,#TWVi69z2 [R #B69.¬769. ,24^ !/2!"T!869&uWVp&Cgh!"";uW:sv/69: &B2S#$ & #T,# &286;u9g2M#B6eO #T,# &u'eOR"& aC^_[C^ `:M
β2v = min
(α1 (α1 − 1)
α1 − 1 + c2/u2,
α2 (α2 − 1)
α2 − 1 + c2/v2
)
^| T2 4 &(69.C;'c2& &@9g22!":B!&:α1
69&2;α2
^ ¬!/( !"'&a69W.¬QW/69 &6|7rsir::;;269#$R &< &2$.&4 ,2;C &(69.5;C "!&_3o2^7183¬ S #$R. "pa69p2$7469. ,2"p9g|2G] n &:,#TWV69&2;@25±[J[J&:,#TWVK &25!4-869#;C "!&d6971spW ^|!/%!869&r&2/69z#%5 &2"712gh"01,2&2!sd;#B69 &_^KP<"7183O2[!/2!"[9gt2Q#$ & #T,#¯ &G2Q7O!",2R"!"&2;C && #$R. "K(692|.^5P\6!/BR(69/69#%
β!869&%WV"!"#%"0:,(69.2i2zPb6!/%&:,#TWV6gm`2| &:7O!;
R"!"&2;C & &2B69&2;G",. &$ &2"5!869&G2&~69RRV8698^KJ['eOR(69/694/69RH5;71";GT2i,2;Cs9g2"M;C D&:/694"z &H2S!86S9go2Sj[ "![LO!/2#%4^ ) )%A L# #; .7:!$.f=; 14;n S,2S!"&2;T6 \(^69
M = 10−5 3Rem = 10−1 69&2;b.S,2M/69r
AR = 10−2 ^ R.";k &c24, aC^ gO^ Z ([",2.¥/|W/69 &2";vgm52M.:!/G!/('#%S69
χ = 106 69&2; ^| H2 '#B6eOY] n #B6e:±iJQJ BR"!"&2;C &2 &2^ 2H!/2#%HR7:;"B69&+'eC!". .&:B;269#$R &712H&: Ggh"0:,2&2!s;#B69 &<69&2;H71&vgm52iR2,'>.&4 ,2;C &(6.t#%:;"6| z!869&HWVM"&~&H2S!. 4/'>,R_^ 2"ir::;RRV"[69.&%2
xx6eO69M#B69 &. sH;C,2M2M,2/M9go2¥4u769. ,2"9go #%'RF^M"!869. ._a69
χ = 106 '^K[&2!"43 &~(69!86436^M!869&u" #B69M(69CFLx ≈ 104 ^| .
V NNx ≈ 102 ^|!/\69 . ._ 4~769. ,2"|gh|2"iR(69/69#%^Xx&dc24, aC^ gO^_[C3/^[R"&:|2[",. |W/69 &2"; ^| H2iPb69¥e:]2"M!/2#%Qgm52i/69#%i!"&2;C¥&2Q'e!"R(69 ^S,2";~2p#%:;C¥c(";u7O!",2R"!"&2;C & &(^AS&2!"p(69Q2p;269#$R &dr::;b71M6B.U69r@R(69M9g2Tg¨"0:,2&2!s<;#B69 &_^SJ[&22."83-^$69.d&2@(69i2 &2"Q#B69 &. su!"&2!"&:/69";k69.&d2
xx6eC83V6Q&2T!869&\"S &<2T!.4'>,R_^Z&. 1S2T.:!/~!/2#%43
2Q;269#$R &9gD2QP\69¥eO]2"#%2:;d;/69"gm2R,.&4 ,2;C &(69.V#%:;"569&2;BR(69!',.U69. sGgmz2S 4~gh"0:,2&2!"8^[&2!"432 Q!"69&<WVM &:" &BB,2;Csv2 '#$ &CY] n #$ &>±iJQJ !"#TW &(69&b69&2;H zD!&~2MRV-gh#B69&2!"S/69igm|2MR,M.&4 ,2;C &(69.¬#%:;"8^Xx&$c24, aC^ gO^ · 3M^R.";B2[69#$R. ¥c(!869&dg>6!#B69RGW/69 &2"; ^| d2 '#$ &CY] n #$ &>±iJQJ /694s1^N[z'eCRV"!";D322p;269#$R &d71z2 ^567r&:,#TWV[;#B69 &ui^M(69& ^| u2MR7O¥,257r&_^`Q 4v769. ,2"|9g¬2[/69&271'/M] n 69&2;G±iJ[J§&:,#TWVz69&2;G2Qg>6!5(6952[7:/!,(R"!"&2;C & &2\"01,(69. ´8"p2$ &:7O!;A!86B69%2$#B69 &86&2p9g52% V&2"^ 0 &A2%2(69&2;D3_6i&2!869&k"T &\2!.4'>,R_3 (@;269#$R &~9g2@R,@.&4 ,2;C &(69.K#%:;"[[ #$R71";D3R(69!,.U69. sGgm|2M 4Hg¨"0:,2&2!"69&2;H2M7rsv#B69. .^56871&:,#TWV8^[^7r83±K&469" ^569/69&69&2;bP\O.@#$R(6 ´8(69 ^| ki: &2;b9g694su2/. ,&2i#B68s\;C 7rr%69&2;\2s<R9RV4%u,2%2%#%d!"&27469 7rB7O!",2pR"!"&2;C & &b &2%WV4 & &b9g|2d!"#$R,/69&69&2;~2&< ^| !/u/$2 '#$ &CY] n #% &C>±iJQJ p!"#TW &(69&_^|N|Q69&:sH!864322pB!&2!sHghzR2,6eCU69.a#%:;"9g_(QPb69¥e:]2"[!/2#%0^| . .V. sB69.T&G2". ,&G#%2:;D^KJ['eO"!&,^| . .R"!. su;869.j^| \2pRV-gm#B69&2!""i9g2LSLu69&2;\q .U6e269&\R:!"";C, ^| \ 4b6RV"!/69$4;8^
Z ` ·
ξ / π
η/π
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0.4
0.5
0.2
0.3
0.3
0.2
0.3
0.1
0.4
0.30.4
ξ / πη
/π0 0.25 0.5 0.75 10
0.05
0.1
0.5
0.2
0.6
0.1
0.33
0.4
0.3
]o 4, aC^ gO^ Z |NQ#$R. ¥c(!869&\gm6![#B69R\9gK(Tj[ "![.:!/u!/2#%69M = 10−5 3
Rem = 10−1 3χ = 106 69&2;
AR = 10−2 ^ n 'g¨@I`&: ^56871&:,#TWVi;#B69 &_^zz 4:@5.4/'>,R<&2869Q2T69W2!/6eC8^
ξ / π
η/π
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0.4
0.50.60.5
ξ / π
η/π
0 0.25 0.5 0.75 10
0.05
0.1
0.5
0.2
0.8 0.82
0.4
0.3
0.50.5
0.6
0.7
]o 4, aC^ gO^_[C|N#$R. ¥c(!869&\gm6!Q#B69Rb9g`2Tj[ "![Pb69¥eO]2"T!/2#%@69M = 10−5 3
Rem =10−1 3
χ = 106 69&2;AR = 10−2 ^ n 'g¨$QIK&: ^5687r&O,#TWVS;#B69 &_^Tz 4:$i.4'>,Rk&2869M2
69W2!6eC8^
Z `b`
ξ / π
η/π
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0.6
0.8
0.7
0.8
0.5
0.6
0.7
ξ / π
η/π
0 0.25 0.5 0.75 10
0.05
0.1
0.2
0.60.62
0.7
0.5
0.4
0.3
0.5
]o 4, aC^ gO^ · |N#$R. ¥c(!869&\gm6!Q#B69Rb9g`2Tj[ "![Pb69¥eO]2"T!/2#%@69M = 10−5 3
Rem =10−1 3
χ = 106 69&2;AR = 10−2 ^ 2 '#$ &CY] n #$ &C±[J[J u/694s)%,2";D^ n 'g¨<~I`&:
^56871&:,#TWVQ;#B69 &_^z 4:S.4'>,R~&2869z2p69W2!/6eO8^
) )@ 9 -5> 9 !$.f=; 14; !Xx&vzR(69/694/69R_3H^S69 #69,2;Cs: &%2MRV'gh#B69&2!""Q9gP\]¬YLSLv69&2;~P<]¬q !/2#%"| &~2R"&2!"H9gQ 4¢6RV"!$/69b4;8^N[@#%&:&2";+869. 832s69v'eCRV"!";\R7O;GRV:;269#$R &<gmp2BR,v6eOU69.#%:;"8^uQ",. Sg¨/#c24, aC^ gO^ `\2^(69Tgh
AR = 10−2 3¬WV#%2:;69iB!&:|/;269#$Rv2[#$;O ^567r&:,#TWV#%1;2|W,52sBgm69 . . #$ &(69MR,[.&4¥,2;C &(69.`^iZ[ & '#$ &CY] n #$ &>±[J[J @;'c2& &A;:"[&2i&(69W2. d #$R71@ 4&¥c(!869&:.¥s2dRV-gh#B69&2!""8^bXx&Ac24,/+aC^ gO^ c 69&2; aC^ gO^_aC3?^BR"&:@2v69#$R. ¥c(!869&gm6!$69.&\2
x
6eC`gh2P<]¬YLSL F`fer69&2;BP<]¬q c er!/2#%"8^](WVd!/2#%"8312 '#$ &CY] n #$ &>±iJQJ /694sAg>69¥.@< #$R71d2GR,v6eCU69.5;269#$R &2^k]2,2 &:7r" 169&2@71869.";)(69@/69O &6d7rsu.U69r$69#%,&:S9g &&2- ^"R2S&(69W."iWVk#%2:;\G"!"71S6B. . &: &2!B!&2!sW,|,2!/~6@/694sd5&2z6 D;269W.i &v#%z9g`qKZ> #%MB!&2!s1^KN[|6T",. 83C2i,2i9g¬2'#$ &CY] n #$&C>±iJQJ |!"#TW &(69&d!869&&2KWV586&(69W. s$!"&2;";gh`2|Pb69¥eO]/|#%2:;!",R.";:^| \LSLvzq 69. r #~^
' & (1& (Xx&bS!/(69R283)^%,2;C";k2 #$R(6!T9g27O!",2SRV"!/69.K/6;C ,2T69&2; ^%#$R(6 ´8";k(69836T.&b6
Rec > 13`,2!/6u #$R. ¥c(!869&;:"T&26 D"!T2dB!&2!s9gz2d#%2:;D^ 2&
Rec < 13C2S #$R. ¥c(!869&<| 4. sGRV&(69. ´" &dW,]^5M!869&~86&(69W. sH!"&2;(69,2!/u6$!86
/69gm.^5P\6!/v&:,#TWV \(^z8^`5;"83M^i2^";d(6952[ 4 &(69.D7:!",2R"!"&2;C &&v&2v869. . s ^. . , ";gmG2bPb69¥eO]"b"!/&0:,2b &2!"\ vR7:;"v:#T,2!/¦&:,#%!869.;C R(69&_^+[&2!"35^H,. &2"; 6\&2h^;'c2& & 9g[2v7O!",2$R(69/69#%Q^|!/¢&(69W."%2#%2:;u%R7O;T6$r::;<;269#$R &Bgm71sGgh"01,2&2!sH#%:;"8^5Xx&~2p!86p9g` 4b6RV"!/69
Z ` c
ξ / π
η/π
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0.60.8
0.7
0.9
ξ / π
η/π
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1 0.60.80.7
0.9
0.6
0.6
0.70.8
]o 4,QaC^ gO^ `2SNQ#$R. ¥c(!869&Ag>6!S#B69R69M = 10−5 3
Rem = 10−1 3χ = 106 69&2;
AR = 10−2 ^n 'gh@P\]tYL2SL F`ferz!/2#% ^| \&(69&2!"";\7:!",2R/!"&2;¥ &2 &2^Q 4:TP\]¬q c er/!/2#%^| u&(69&2!"";~7O!",2|R"!"&2;C & &22^
ξ / π (η / π = 0)
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
1/AR = 51/AR = 101/AR = 201/AR = 100
M = 10-5 , Rem = 10-1 , χ = 106
ξ / π (η / π = 0)
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
1/AR = 51/AR = 101/AR = 201/AR = 100
M = 10-5 , Rem = 10-1 , χ = 106
]o 4, aC^ gO^ c `N#$R. ¥c(!869&vg>6!59gt2iP<]¬YLSL F`fer!/2#%M69.&2x
x6eO8^ n 'g¨MI`&(69&2!"";7O!",2|R"!"&2;C & &22^ 4:S '#$ &CY] n #$ &C±[J[J S!"#TW &(69&_^
4;83^T2^";<(69[2'#$ &CY] n #$ &C±[J[J T!"#TW &(69&A!869&< &2!86/T2TB!&2!s<9g2#%2:;\gm[2@R2,$6eCU69.#%:;"8^pZ&gh,&(69. s13D2/69#%/694sugm69 .iG#B69r%LpL\69&2;bq 69. r #%T6MB!&:T6i2$j[ "!S!/2#%4^@[^7183D/#%/69/'4"T69@ #B694 &(69W.% &k;v6871;\ 4A6RV"!i/69G,2"83 ,2!/k6[,&2!/ &H4;Qg>69Qg¨# 2 ^569. .oWV,&2;(69S/G6R"/'7rMB!&2!s1^X]^56z2^|&~ & n/Z (696eCU69.D#%:;"!869&HWVS. ¥#$ &(69/;uW:sd2iWV,&(;269s!"&2;C &<RV"!¥c(!869&28^Xx&~!/(69R n 36^ ^| . .Dgh:!,2&~,2!/HR/W.'#%[69&2;'^ ^| . ._sG$,. &2R/6!!869.¬. ,&28^
Z `fa
ξ / π (η / π = 0)
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
1/AR = 51/AR = 101/AR = 201/AR = 100
M = 10-5 , Rem = 10-1 , χ = 106
ξ / π (η / π = 0)
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
1/AR = 51/AR = 101/AR = 201/AR = 100
M = 10-5 , Rem = 10-1 , χ = 106
]o 4, aC^ gO^_aCzN#$R2.¥¥c(!869&bgm6!i9gK2@P\]¬q c erQ!/2#%69.&G2x
x6eC8^ n 'g¨|I`&(69&2!"";7O!",2|R"!"&2;C & &22^ 4:S '#$ &CY] n #$ &C±[J[J S!"#TW &(69&_^
`t k` i u ¯
#1 +#% ZQ&286;Cs \2,;<;CsO&(69#$!pR:!"""i69Sg¨"0:,2&:. s~&2!",&:";\ &\&4 &2"'¥&2GRW.#%8^N[2s69<r7r&2";W:s¢6W^|;</69&r\9gp #%\!869."b¨xsOR!869. . sgh# 6!",2!< #%<R(69 !.\ #%'32sb69$!/(69. .&4 &uH!"#$R,4^](M.^5P\6!/A&:,#TWV \(^z83D,&286;Cs\RW.#M71&b#%!"#$R.'e &2!"u2<G6&);CR(69 xs+WV ^"&¦2<!/(69/6!!\ #%<.U69"; )6!",2!^56871"B69&2;+2~!/(69/6!!u #%~.U69";+kR(69!.~7r.:! "^¢Xx&B!/(69R83 ^vgh:!,2B&2SI`,.,&286;CsH0:,(69&2/. 7r";HW:sv6$;C,(69.¥> #%pRR &d!/2#%436^|2 &u2MR:sC!869.¥> #%; 7469 7r"Q69p#$R.sr";~gm. .^2MR:sC!869._/69&2&:[69&2;H2SR2,2;9> #%p; 7469 7r"Q716p69&A /69 7r%;7O!"4^$Xx&A!/(69R · 3)^%2^";k2^lHW/69 &A2%'eCR"&A9g52$;C,a69. >¥#%RR &d!/2#%M &H2S](,R(6!"4^`](z2SI`,.!8643 z|4 7r&~W:sG2[gm. .^| &dsO#
H · ∆wn,mi,j = −K · wn,m
i,j ,
^|
K =3
2
∆τ
∆tId + 2
∆τ
δx(1 − ε)D1 z1 + 2
∆τ
δy(1 − ε)D2 z2
+∆τ
δxε (1 − κ)D1 z
21 +
∆τ
δyε (1 − κ)D2 z
22
+7(
∆τ
δxAE s1
(1 +
ε
2(1 − κ) z1
)+
∆τ
δyBE s2
(1 +
ε
2(1 − κ) z2
))
H = P−1 +3
2
∆τ
∆tId + 2
∆τ
δxD1 z1 + 2
∆τ
δyD2 z2 +
7(
∆τ
δxAEs1 +
∆τ
δyBEs2
)
^|2:^H"!869. .(69∆t
%2HR:sC!869.z #%'R69&2;∆τ
%2HR2,2;9> #%<R_^¢Xx&+;~RV-gh#6v±&J[,#B69&&)69&(69. sC83V^$&2"";~!,2#$ ´8B2%,&286;CsAsC#~^ (69T p2,WC*"!z9g2M&2'eOzR(69/694/69R_^
Z `Mg
Z ` n
" " #%( # (.#%
Xx&2d#B69&&29g2BR7O,2!/(69R83^d&2"";R(69/69#% ^|!/&(69W.G,2T<!"#$R,G2 6!"WU69&G#B69!""569&2;%2;C/ R(69&d#B69!""8^`NQ2^|&d869. 83:2Pb6!/B&:,#TWV
M3r2i\(^
;C "!&v/u
69&2;d2Q/69@9gtRV"!¥c(![2869γ
69[, B!&:5T;#$ &2i,2!/d#B69!""8^K5";"83^M(687rM$!"#$R,S2M/694
∆τ/δx3∆τ/δy
69&2;∆τ/∆t
^ 2MR2,2;9> #%pR~86;p
∆τ = CFLτ · min(δx
λ+1
,δy
λ+2
)
^|2λ+
1
69&2;λ+
2
69i2SRV"!/69.t/6;C ,29go2MR"!"&2;C &28;usO#~^`[&2!"436^SW/69 &
∆τ
δx= CFLτ · min(
1
λ+1
,AR
λ+2
)
∆τ
δy= CFLτ · min(
1/AR
λ+1
,1
λ+2
)
gO^_[C^ Z
&2"0:,2&:. sr3O ^@2R(69/69#%z69Q"0:, ";D3O&(69#%. sB2i6RV"!/69AR = δy/δx
69&2;d2;C,(69.t] n &:,#TWV
CFLτ^`]2,2#%43/^[(687r[@;'c2&2[2[R:sO!869.¥> #%iR_^ 2M69[ ^
RV4 W.T;'c2& &283a2Sc2[&2SW(6/";u&<2T6!",2!p #%T!869.43(2T"!"&2;<&2SW(6/";u&<2R(69!.S #%S!869.%
∆t = CFLt · δx · min(1
u+ c,AR
v + c)
∆t = CFLt · δx · min(
1
u,AR
v)
N|69&:sv!"6/43/^M(6871$∆τ
∆t=
∆τ
δx· δx∆t :,283M^!/2:42R:sC!869. ] n &:,#TWV
CFLt62Q6;;C &(69. R(69/69#%p2R:sC!869.
#%vR_^k'gmdRV-gm#$ &A69&(69. sC"83?^ ^569&:\;'c2&2G ^5<R/6!!869.|&:,#TWV83`&(69#%. s2] n &:,#TWVW(6";~&~2SR(69!.S7r.:! xs~69&2;u;&2";
CFLu3(69&2;H2T] n &:,#TWVWa6";~&
2p6!",2! ^56871S69&2;~;&2";CFLu+c
CFLu = u · ∆t
δx
69&2;CFLu = (u+ c) · ∆t
δx2&\2Pb6!/\&:,#TWVM[7rs<#B69. .>3V2!/2#%@(6iG;869.^| A6d&v;CR(69 xs<WV ^"&2" ^Q&:,#TWV^o]( &2/69&2!"434¥g(25P\6!/&:,#TWV`69
M = 10−3 69&2;@¥g ^|69 &:" & &¢2u6!",2!~R2&2#%&(6C3|2& ^~(687r
CFLu+c ≈ 169&2;
CFLu ≈ 10−3 ^ [&2!"43I^u!869&&2!",&:T#%$RW.#%pWV"!869,2%9g52"$71s\;C D&:S #%'!869.4^ n S,2M&2^;869.^| A2.^5Pb6!/~R(69/69#%0^|!/<&(69W."z,2|%! !,#T7r&:Q,(!/H,2"^
( #% " #% N[5#%&:&2";u869. 8322M.U69rS;CR(69 xsGWV ^"&\6!",2!T] n &:,#TWV[69&2;HR(69!.T] n &:,#WVM(69 ^&2!",&:Mgmi.^5Pb6!/k&:,#TWV \(^W2.¥ r"S,2iG";'c2&2$RRV. s\2@.^|P\6!/R(69/69#%8^LO71/69.V,(;C569WV,RW.#69[6769 .U69W.Q &d2. /69,M69&2;dP<O.4=?869#%
Z ` d
R7O;";d69&d&(69&2!"#%&:59g_86;CsR"!"&2;C & &2T & n/Z ^ZQ &T/69&2;269;B86;CsR"!"&2;C &C &S!869&$WV5RV"!'U69. . s@ &2 "! 71gmK6!",2!5RW2. #%^X&@c(4,0gO^ · ^ Z 3^R.";$2z69#$R. ¥c(!869&g>6!W/69 &2";l^| B2zP\]¬YLSL · er`!/2#%]^| B2z86;CsR"!"&2;C & &H¨.'g¨`69&2;l^| 2,69&:sR"!"&2;C & &§¨ 4:'^ AH&2!"~a69%2~R"!"&2;C &2";!/2#%~WV"!"#%"BR(69!,.U69. s gmM2$.^5mg¨"0:,2&2!sA6986H69&(;D3¬6p6G",. 83t¥SM &(69RRRU69%ghS,2!/kRW.#%8^ 0 &A22(69&2;D312z/69&(;269;%/!/2#%69RRV869MWVi^. .¥, ";$gmK!8669&2;%R7:;"6i7rs@r::;;269#$R &d71|2S&: ^56871&:,#TWVQ;#B69 &_^
ξ / π
η/π
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0.8
0.9
0.6
0.6
0.7
0.6
ξ / π
η/π
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0.5
0.4
0.3
0.1
0.7
0.6
0.2
]o 4,lgO^ · ^ Z iN#$R. ¥c(!869&kg>6!M9g2P\]¬YLSL · er[!/2#%$69M = 10−3 3
CFLu+c = 169&2;
CFLτ = 106 ^ n 'g¨pL:86;CsvR"!"&2;C & &(^ 4:S`J[R"!"&2;C &2 &2^](GR(69!.bRW.#%83Q(69v,^|2&
CFLu = 132</,. H69b;C D&:8^lX&(;";_3 & c24,
gO^ · ^_[C3^@!869&b&2(69i2%L:86;Cs<R"!"&2;C & &u!/2#%R7:;"S6dr::;b;269#$R &'^| .2,&R"!"&2;C 4&2";¢!/2#%v 4. s) ¢gm$,2!/+6b!864^L:86;CsR"!"&2;C &2 &2&#%69RRRU69pghR(69!.pRW.#%83(2^710^S#B69H(692p;269#$R &G71z(S.^5mg¨"0:,2&2!s4&u . .D &2D"! 7r432 &2;C!869 &d.^!"&:71'r'&(!4^5L:/69&2;269;~R"!"&2;C & &2Bz2'ghS&271/69 .v&2,4+b;2"69.]^| (G. #$ %9gi#B69. .z69&2;.U69rvR:sC!869.| #%'R28^)P\"7r83 &2S!86M9g &:#%";CU69T!/2!""9go #%'Ru ´843&2 286;CsGR"!"&2;C &2";<!/2#%M&2z,&CR"!"&2;C &2";\&2pR7:;Sr::;<!"&:71'r'&(!TRRV"8^5Q"869!/2"Q2^";~(69[69&\6;;C &(69.;'c2& &u9g¬2[.^5Pb6!/HR(69/69#%i^5652&v&2"!""69sr^ 2i,&286;CsdR(69/69#%z5;'c2&2";~6gm. .^zS
βu = max(lx
π ∆t c,
lyπ ∆t c
)
^|2lx
69&2;ly
69!/(69/6!!.&42S9g2@R:sO!869.oRW2. #~^TXx&\R/6!!"43_¥iM!,2#B69sT/69r2"/.&425"0:,(69.aT2[ ´8Q9gt2Q!"#$R,/69&(6.D;#B69 &_^ 2Q#B69 &G86&G(69I^^569&:oQ;269#$RB!&:. sp2.^5mgh"01,2&2!sTo#%:;"o¥&(!2sM,2,(69. . sS!"&:.O2!"&:71r&2!"/694^ 2"Q#%:;"|69Q.U69";dT2Q. &2rI^5687r.&425(695!869&GWVQ!869R,";v712[;#B69 &_3 pp2'ghB86/&(69W.Bu!"&2;T(69@,2!/W^56871.&4269B9gz2B;@9g|2B ´8B9g|2!"#$R,/69&(69.t;#B69 &_^KN[|6p",. 83C2[/69
lx/∆t!869&vWV[169;";~62iRV"";v9g¬2" ^|687r"
69&2;H2M,&286;CsGR(69/69#%[!869&~WVM!"&2;";<6|2SPb6!/H&:,#TWV0^|!/<;'c2&2"z2#~^
Zc e
ξ / π
η/π
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0.4
0.4
0.5
0.60.6
0.7
0.5
0.6
ξ / πη
/π0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0.95
0.990.980.97
]o 4,:gO^ · ^_[CdN#$R. ¥c(!869&gm6!$9g2vP\]tYLpL · erT!/2#%H69M = 10−3 3
CFLu = 169&2;
CFLτ = 106 ^ n 'g¨pL:86;CsvR"!"&2;C & &(^ 4:S`J[R"!"&2;C &2 &2^
[^7r83D2$'eCM69&22 ^568s<v;#$ &2$2".'&2428^TXx&\gm6!83D2!"&:71r&2!"$/69@!"&:. .";W:s2G.^5mgh"01,2&2!s#%:;"$W,2 $R"&:/69&+ &2v](,$R(6!"G. "$&2d#%" ´84^bP<dR"!/. sr3?^|2&2d4;WV"!"#%"@c2&283K2dR"&:/69&+9g2d.U69r"^56871.&4b!""RV&2;iB.^Qgh"01,2&2!"^QQ&2!"43V269p ^G!/(69/6!!@.&42 ^|!/(6871|SWVz/691&B &:T6!"!",&:83:&(69#%. s$2 ´8z9gD2i^56871Q69&2;%2 ´89gD2z#%"_^K&2; &2@gm6" ^56871@RR(69169 &\69i2$RV"";
(u + c);C, &H2R:sO!869. #%'R
∆t3-^$W/69 &
2igh. .^| &B.&4
Lacous = (u+ c) · ∆t = CFLu+c · δxNz!"!";C &T2[7469. ,2[9g
CFLu+c3O.&4~!869&GWVi"&H62[&:,#TWV|9goR(6!"'Rv(6952
g>6"i^56871p!"7rQ;C, &G&2S #%'R_^|P<"710^p!869&<!"&2;(69Q R"&:Q2p!869.9g|2$.U69r" ^5687r.&4_^vN[M2gh"01,2&2!sbR"/&:/69&);RV&2;69.~&2%#%" ´843^!869&~/691S6|'gm&2!"M.&4u2ir"#%su687r/69rS9g
(Lacous × δx)
lx =√Lacous · δx = δx ·
√CFLu+c
:,2832M,&286;CsGR(69/69#%Q &H2x
;C "!&\!"69&~WVM";'c2&2";<65gm. .^zp
βu =δx/∆t
c·√CFLu+c
πn ~,2H!"&2;<69&6!",2!kRW.#~3CFLu+c
HH,& xs¦69&2;§2bR:sO!"69.M #%'R§2'gmS"0:,(69.D
∆t = δx/(u+ c)^ 2M,&286;CsGR(69/69#%z2&~4 7r&~W:sb
βu =u+ c
c· 1
π≈ 1
πZ[ &BQR(69/69#%M&(69W."Q,2QBWV"!"#%T7rs~!.4pB2p,&R"!"&2;C U&2";b!/('#%D^|!/<^. .¥, ";gh@6!",2!BRW.#%8^ 0 &2d2p(69&(;D3¬.T,2p/
CFLu~,& xs^|!/#%869&2
Zc/Z
(69∆t = δx/u
69&2;CFLu+c ≈ 1/M
3/^SW/69 &
βu =u
c·√
1/M
π≈
√M
πJ[^2$,&286;Cs\R(69/69#%SS!.4v2$86;Csb&2%69&2; ^$!869&A'eCRV"!p6Gr::;A;269#$R &u9g2%S#%:;"^%](S &:#%";CU69d!86"83_2%0:,(69:T9g(B6!,2!d] n &:,2#TWVp&(69W.",2[v!869.@RRV. s<2@,&286;Cs<R(69/69#%Sv6[/dR"71@r::;b!"&:71r&2!"RRV"8^pXx&c24, gO^ · ^ · 3I^~2^2~",. BW/69 &2"; ^| ¢B;'c2& &9gi2H,&286;CsR(69/69#%BghCFLu+c = 1
&~2i.'gh69&2;CFLu = 1
&~2i 4:8^ i&2!"M(695gm|WV~!86/"83C2M!/2#%,2!"!""";H &u;269#$R &BB!&:. sH2i.^5mgh"01,2&2!sHz#%:;"83 &2;C!869 &%g>6z!"&:7rr&2!"S/694^
ξ / π
η/π
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0.3
0.4
0.3
0.1
0.2
ξ / π
η/π
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0.30.2 0.4
0.50.6
0.8
0.7
0.1
0.9
]o 4, gO^ · ^ · NQ#$R. ¥c(!869&Bg>6!9gD2P\]¬YLSL · er`/!/2#%i^| B2z,&286;CsR"!"&2;C &&69M = 10−3 69&2;
CFLτ = 106 ^ n 'g¨pCFLu+c = 1
^` 4:SCFLu = 1
^Xx&c24, gO^ · ^ `23I^~;Ch^2~;CU69r&(69.R9c2.u9gi2u69#$R. ¥c(!869&¢g>6!d69
CFLu+c = 169&2;
CFLu = 1^ 2v,&2/"6;sR"!"&2;C &2"G69RRV869$bWVv2v#%v71/69 .v#%2:;k;269#$R
!""!. sB2.^4^56871&:,#TWV &GWVG!86/"8^P\"7r83:gh &:#%";CU69M!"&Cc24,/69&283C&(69#%. sgm
CFLu = 10−1 69&2;ghCFLu = 10−2 3425,&286;CsTR/!"&2;¥ &(R7O ;2Kr::;$!"&:71r&2!"
RRV"[6z2^|&~ &Hc24,DgO^ · ^ c ^
,UMJLE65^|69&:i"&$2g>6!(69`2,&286;CsTR"!"&2;C &2 &S(66Q&2&C>&24. 4 W.zD"!`&$26!"!,/6!s<9gK2@. ,&_^QXx&2;"";D3a [ #$R. [a69Q2T.^5P\6!/bR(69/69#%i Q.U69r[(69&<gmQ286;CsR"!"&2;C &2"5M(692z&:,#%!869.V;C/ R(69&BWV"!"#%"#%z #$RV/69&:8^KXxK "RV"!U69. . s,2H &2~!86H9gi2u6!,(!HRW.#%$gm+^|!/+2H,&286;Cs)R"!"&2;C &2 &69RRr6!/2"2B&2~R"!"&2;C &&<gm#T,.U69&_^ 2%&:,#%!869.5;C/ R(69&p2&RV:. sA!869.B."6;¥&2< &(6!"!,/6!s1^
Zc [
ξ / π = η / π
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
No preconditioningSteady preconditioningUnsteady preconditioning
M = 10-3 , CFLdual = 106 , CFLu+c = 1
ξ / π = η / π
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
No preconditioningSteady preconditioningUnsteady preconditioning
M = 10-3 , CFLdual = 106 , CFLu = 1
]o 4,]gO^ · ^ `2¬j[U69r&(69.CR9c2.59g(2569#$R. ¥c(!869&g>6!9g(2P\]¬YLSL · ert/!/2#%569M = 10−369&2;
CFLτ = 106 ^ n 'ghpCFLu+c = 1
^`z 4OSCFLu = 1
^
ξ / π
η/π
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0.30.2
0.4
0.5
0.6
0.8
0.7
0.1
ξ / π
η/π
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0.3
0.2
0.4
0.5
0.6
0.7
0.1
]o 4,DgO^ · ^ c NQ#$R. ¥c(!869&ugm6!Q9g2SP<]¬YLpL · er5!/2#% ^| ~,&286;CsvR"!"&2;C &&v69M = 10−3 69&2;
CFLτ = 106 ^ n 'ghSCFLu = 10−1 ^`¥4OS
CFLu = 10−2 ^
Zc4·
" &(% & "$#1 ))%A ; 3 25 ,#43&#f# >!
ξ / π = η / π
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
Block-ALGS(1) schemeMF-SGS(30) scheme
M = 10-3 , CFLdual = 106 , CFLu+c = 1
ξ / π = η / π
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
Block-ALGS(1) schemeMF-PJ(51) scheme
M = 10-3 , CFLdual = 106 , CFLu+c = 1
ξ / π = η / π
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
Block-ALGS(1) schemeMF-SGS(30) scheme
M = 10-3 , CFLdual = 106 , CFLu = 1
ξ / π = η / π
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
Block-ALGS(1) schemeMF-PJ(51) scheme
M = 10-3 , CFLdual = 106 , CFLu = 1
]o 4,]gO^ `2^ Z ¬j[U69r&(69.CR9c2."`9g(2569#$R. ¥c(!869&g>6!`69M = 10−3 69&2;
CFLτ = 106 ^| 2,&286;CsBR"!"&2;C &&2^ Rb#$R(69&vWV ^"&~.:!/1N n SL Z /!/2#%i69&2;vP<]¬YLSL · er!/2#%4^ n 'ghz
CFLu+c = 1^o 4:|
CFLu = 1^o5##$R(69/&$WV ^"&%.:!/:N n pL Z
!/2#%p69&2;~P\]¬q c er|!/2#%4^ n 'g¨pCFLu+c = 1
^` 4:SCFLu = 1
^ 2$",. MR"&:";A &bc24, gO^ `2^ Z 6971s< &:" &< &2!"2sb2^(69S2$Pb69¥eO]2"#%2:;\!",R."; ^| ALSLu69. r #[q 69. r # i6[B!&:S6Q2T.:!/u!/2#%@. 7r";W:sN n SLR:!"";C,4^)Xx&2;"";D3`gm%6!",2!~69&2;R(69!.vRW.#%83(69$@gh
CFLu+c = 1
CFLu = 13¬2BPb69¥e:]2"d!/2#%BR7:;"6H7rsbr::;;269#$R &b9g|2%.^|mg¨"0:,2&2!sA
#%:;" ^|!/ &2;C!869"T6Ggm6S!"&:71r&2!"%/694^ :,23tP\69¥eO]2"$#%2:;69RRV869SHWV71s69RRRU69Sgh|2M,&286;Cs,\(^z8^
Zc `
))@ V7#;67.M="; > 3&#f# >!Sc2i;869.^| \2$6!",2!@RW.#%S69&2; ^@[2$6RV"!i/69v69
10−4 ^i](i,2!/k6d!86432P\]¬YLSL<!/2#%@i71su!"#$RV 71$ &2!"@ iR7:;"p69&k'eC!". .&:S;269#$R &~7ri2&: ^56871&:,#TWVz;#B69 &<69&2;G z.¥ #$ &(69"!"#$R.. sG2iR,S6eOU69. #%:;"p g-^oc(4, gO^ `2^ [r^ 2"",. H69\,R &)WV"!869,2 ^\'eORV"!";(69G2<#%2:; R27:;";§6AW(6;;269#$R &9gp2.&4 ,2;C &(69.¬#%:;"Q6| |z2M!86/[gmz86;Cs \(^z8^
0.1
0.2
0.3
0.3
0.4
0.4
0.4
0.4
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.6
0.6
0.7
1
ξ / π
η/π
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
ξ / π
η/π
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0.2
0.1
η / π (ξ / π = 0)
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
Block-ALGS(1) schemeMF-SGS(30) scheme
M = 10-3 , CFLdual = 106 , CFLu+c = 1 , AR = 10-4
]o 4, gO^ `2^_[C n 'ghM`NQ#$R. ¥c(!869&Hg>6!|9go2MP\]tYLpL · er!/2#%S69M = 10−3 3
CFLu+c = 169&2;CFLτ = 106 ^| <2S,&286;CsvR"!"&2;C &2 &2^zz 4:pq`9c2.T9g`2T69#$R. ¥c(!869&<g>6!
69.&v2yx6eC$
ξ = 0|gmQ2/69#%@!"&2;C &28^T#$R(69&bWV ^"&A.:!/:N n SL Z Q!/2#%
69&2;~P\]tYLpL · er5!/2#%4^Xx&Bgm6!836!"!";C &T2[!/(69R · 3M^[!869&v86 . sB'eOR.U69 &vWV'a687:,8^KZQ&2;5,2!/v!"&2;C &2832~R:sC!869. #%'; 769 71u# B 4. s;#$ &(69&:dghB2
x>#%:;"G69&2;D3|!"&2"01,2&:.¥s1352
RV/69#B69!""H1
69&2;H2
^|!/u;'c2&2S2pLpLG!/2#%B 'eOR//& · ^ Z [C^ Z 586;A
H1 ≈ ∆τ
∆tId
69&2;H2 ≈ ∆τ
∆tId
P\"7r832M #$R. ! QRV/69H
69&2;H2S'eCR. ! RV/69K
86;u69./H
H ≈ ∆τ
∆tId
69&2;K ≈ ∆τ
∆tId
N[|6p",. VSGS
5"0:,(69. T´8%69&2;G2M69#$R. ¥c(!869&~#B69¥ev9g¬2SLSLd/!/2#%$GSGS
K52/69#%[(69&v2ijQ "!z!/2#%4=?|&2$
G∗
'^`5";"83O2M69#$R. ¥c(!869&~#B69¥eG9g¬2[j[ "!|/!/2#%|4 71&HW:sb
G∗ = H−1 · (H −K) ≈ H−1 · (∆τ∆t
Id− ∆τ
∆tId) ≈ 0
[&2!"432M;269#$R &B|#B6eC #T,#gh|2MR,p6eCU69.D#%:;"^
Zcbc
n u,2~&2^ ,2;Cs¦2kR(69!.RW.# ^| 46RV"!u/69+4;8^N R>3 ^A!869&'e:RV"!i(6983_6[2@R:sO!869. #%'; 769 71# [."i;#$ &(69&:83D2TWV(687O,M9g2P<]¬YLSL!/2#%WV"!"#%"k #$ .U69\ 2)86;Cs!864^ ]o¥4,2 gO^ `2^ · R"&:\2",. bW/69 &2";ghAR = 10−1 69&2;
AR = 10−3 ^VA&2!"Q(6983: &d2#B69&&2|9g_2Q86;Cs%!86/4312i69#$R. ¥c(!869&g>6!|WV"!"#%"Q \gh|R,M.&4 ,2;C &(69.¬#%:;"8^
ξ / π
η/π
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0.30.2 0.40.5
0.6
0.8
0.7
0.1
ξ / π
η/π
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0.30.2
0.40.5
0.6
0.8
0.7
0.10.1
]o 4,:gO^ `2^ · dN#$R. ¥c(!869&gm6!$9g2vP\]tYLpL · erT!/2#%H69M = 10−3 3
CFLu = 169&2;
CFLτ = 106 ^| ~2M,&286;CsvR"!"&2;C & &(^ n 'g¨pAR = 10−1 ^Kz 4:p
AR = 10−3 ^[^7r83\2769. ,29gk26RV"! /69 #$R. " (69¢gm¢.&4 ,2;C &(69.\#%:;" 2R:sC!869. #%'; 7469 7rM#WV!"#%"|.U69r83(69&2;D326|6T",. 83O2i.459goB!&2!sG|'eORV"!";v@WV[."/ #$RV/69&:z(69&Hgh|2i86;CsG!864^`Xx&Gc24,DgO^ `2^ `236^i&2i(69|,&:¥.
AR = 10−3 3O2MP<]¬YLSL#%2:; G!"#$RV 7rb!"#$R(69"; 2\.:!/:N n SL¢!/2#% ^|!/ ^. .¥>:&2^|&¦ARV4/"r::;RRV"B &¢,2!/¢!"&2;C &28^ JQ712."83gm
AR = 10−4 3P\]¬YLSL!/2#%HWV"!"#%":u ¢69&2;D3¬. 1$ghp2B86;Csk!8643¬ pS,2&2, ";kgmp!"#$R,2 &\,2!/RW.#%8^HXx&R B9g ^869:&2"3)^$!869&!"&2!. ,2;%(69S2%P\69¥eO]2"$#%2:;A T6G7469. ,(69W.%!/2!"%H!869sk,,&286;CsH!"#$R,/69&20^| ~86&a69W.M4;~!/2¥&22^
' & (1& Xx&bM!/(69R83)^$,2;C";b2RV-gm#B69&2!""S9g2$P\69¥eO]2" #$R. ! S#%2:;\gmi2$I`,.,&286;Cs~"01,(69&2Q/.¥71";H,2¥&2d6$;,(69.¥> #%TRR &d/694s1^2AS;"/! WV";~2p;CXB!,. "Q'.U69";$M2|.^5Pb6!/%&O,#TWV!"#$R,/69&v69&2;+^z,. &2";d6MRV"!¥c(!z;'c2& &G9g 25.^|P\6!/R"!"&2;C &2^Z[ &b$,&286;CsR"!"&2;C &2"83I^H2^";(69$2HPb69¥eO]2"v#%2:;!",R.";'^| <LpLG|q 69. r #5 4. svB!&:z$;269#$R~2i.^5mg¨"0:,2&2!sG#%:;"5gm5WV6!",2!B69&2;bR(69!.$RW.#%8^$]2,2#%43^$#$R(6 ´8";k(69M2$#%'(1;AS69.v71/69 .&2,4H@;869.-^| H2Q &:/'#%;69/M!86"5(69|!869&~69 ^|2&H&2[,2" 4~6RV!5/694;8^ 22"!869. ±&dJ[,#B69&&v69&(69. sC&(69W.";d,2T,2;Cs$2#B69 &dgh869,/59gD2QPb69¥eO]2" #$R. ! B#%2:;¢,2!/+(69Q^~!869&¢!"&2!. ,2;~(69%2 %#%2:;+R//&:d6b7469.¥,a69W.~!/2!"~
Zc a
ξ / π (η / π = 0)
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
Block-ALGS(1) schemeMF-SGS(30) scheme
M = 10-3 , CFLdual = 106 , CFLu = 1 , AR = 10-1
ξ / π (η / π = 0)
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
Block-ALGS(1) schemeMF-SGS(30) scheme
M = 10-3 , CFLdual = 106 , CFLu = 1 , AR = 10-3
ξ / π (η / π = 0)ρ(
G)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
Block-ALGS(1) schemeMF-SGS(30) scheme
M = 10-3 , CFLdual = 106 , CFLu = 1 , AR = 10-2
ξ / π (η / π = 0)
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
Block-ALGS(1) schemeMF-SGS(30) scheme
M = 10-3 , CFLdual = 106 , CFLu = 1 , AR = 10-4
]o 4,:gO^ `2^ `2dN#$R. ¥c(!869&gm6!$9g2vP\]tYLpL · erT!/2#%H69M = 10−3 3
CFLu = 169&2;
CFLτ = 106 ^| ~2M,&286;CsvR"!"&2;C & &(^ 4:SAR = 10−1 ^ n 'ghp
AR = 10−3 ^
!"#$R,z,&286;Cs \(^z8^Q^7183:6`#%&:&2";d869. '3^|(68715M/691|!86969WV,25gm6!K(692z,&286;CsR"!"&2;C &25!869&B(687r6M&2169 7rD"!&B26!"!,/6!s%9gD2z!"#$R,/69&v &2!"43"RV"!U69. . s%gh56!",2!QRW.#%83O |69RRr6!/2"52&(pR"!"&2;C &&gh#T,.U69& ^|!/G #$R. "6HRV:@/!869.B9gz2%&O,#%!"69.|;C R(69&_^l^| . .,2;Cs69. .(/%RRV"@ &2%&:,#%!869.'eCRV #%&:8^
k` )` `
#1 +#%
i!/(69Ri;869. ^| \&:,#%!869.`'eORV #%&:i(69[(6871pWV"&b!869";\,[,2 &H6%#%:;.!":;43^| <6@ ^9mgh.;";uWC*-"! 71%
• G!"&Cc2# W:s~R/6!!869.K!"#$R,/69&2M2 69&(69. sO[9g¥&O &2!B!'&(!'s<W(6/";\&2<,2;Cs+9gS2\69#$R. ¥c(!869&g>6!Bghd2<86;Cs¢69&2;¢,&286;Cs. &2869 ´8";¦I`,.H69&2;J[687O-YL:1"z"0:,(69&283
• )!"#$R.\2<RV-gh#B69&2!"k69&(69. sCv9gp2\RRV4";#B69¥e:mgh"\869#%&:HW:s+/69O & &:\6!"!",&O@2d!"#$R,69&(69.|!"49gz2B#%2:;6D^. .|6p @#%'#%s"0:, #%&:83:,2@169 & &6!"!""@b6<R. #$¥&a69s6"/#%&:@9gQ2d4.W(69.zB!&2!sR7:;";)W:s2#B69¥e:mgh"p69RRr6!/_^
2T,2/T9g6B#%:;.`!"1;2D^56[/69 &2";<gm0\('eC W . xsu86&|i!":;.. su;71.RV";<ghi,^|&bR,RV4/@R71";\GWV$6B71su,2'g¨,.&:,#%!869.`R.U69-gm# ghM!869s: &~,S6d.U69r@&:,#TWVS9g&:,#%!869.t'eCRV #%&:]^| ~/7r/69.D769U69&:z9go2[ #$R. ! !/2#%"|,&2;z,2;CsBWV'gmS6!,(69. . s #$R.#%&: &+ &IKN@=?uNiL · P r&2/69.¥>R,RV4/bR.U69-gm# 2\WV"H!/2!"b;&:¥c(";69v2!"&2!. ,2&B9gaR. #$ &(69s,2;Csr^ 25#%:;.2!":;5j^| &%,2 &S6c2& ';C D&2!"Q69RRr6!/&H569"U69&G4;8^Xx69RR. ";dp""-!86"83O;" 4&2";GT6@69. .^ 2 &:71" 169&v9gt6^|;M/69&rS9g!"&Cc24,/69&2T&(69#%. sH69&~ &:7:!;~86;CsdRW.#~32WVH &~2M!"#$R" W.T69&2;.^5Pb6!/\4 #%v\(^71i6d &2'>W,#$Ra'3D6B7:!",2Q.^5Pb6!/b86;CsHRW.# hqo/',2¥. .D\(^[69&2;6v7O!",2S.^5Pb6!/,&286;CsbRW.# \(^®7r@69&4/! .¥.U69 & \a69pR.U69'^v](T86!/!8643^D^569&O[v,. &2@2;C &:iRRV"S9g2P\69¥eO]2"@#%'2:;D3 #B69 &. su2!"&:71r&2!"/69u69&2;2v#%#%s/69rH"0:, ";D3569&2; ,(69v2"vRRV"Q^| +"RV"!%b2~&2"R7O;";+W:s/69&2;269;+W.:!/1> #$R. ! d869#%&O\¨R"&:/69 71u9gM2u687469 .U69W.~W.:!/1> #$R. ! 869#%&: &\NML · Pk'^ 2M",. z(69i^p69p69WV,%!"#$#%&:0^| . .t&(69W.S,2z[*-,2¥ghsbh&28µ S2769. ,(69W.&2"/o9g(,t#%(1;T &@;¬ #$R.#%&:o ¬ &:25NML · P!":;9g225IKN@^
Zc g
Zcbn
&"$# " n p,2MR2"&:S2$r7r& &<"0:,(69&2T69&2;A2$&2&C;C #%&2&(69. ´694&(69 ^d69RR. ";A &A2#%:;.R4/69#~^dZ&. 1$2$R:!""S(69 ^%;"/! WV";k &!/(69R Z 3^%!/2:4$2$g¨"'869#71.:!¥YsH6|2M'gh&2!"pRV"";D^KXxgm. .^zz(69z&2&C;C #%&2&(69.o01,a69&: "Q69S;'c2&2";HW:sb
t~r =~r
L, ρ =
ρ
ρ∞, ~u =
~u
u∞, T =
T
T∞, p =
p
ρ∞u2∞
,
µ =µ
µ∞, λ =
λ
λ∞, t =
t
L/u∞, E =
E
u2∞
, H =H
u2∞
,
^|2S,W2! R ∞ ;&2"z2i'gm&2!"S0:,(69&: "8^ i!869&<69.$;'c2&2p6@'gm&2!"S,&2;~RV""; &u;z$;#$ &2p2M'gh&2!"pP\6!/~&:,#TWVp
c2∞ = γRT∞ ⇒ M2∞ =
u2∞
γRT∞L:,2!/u!/2!""z.86;H$2igh. .^| &d;C #%&2&."Q"01,(69&<9g/69%
p =1
γM2∞
ρT
2d;C #%&2&."Ji687:-YL:1"@"01,(69&2T(69 ^% #$R.#%&:";) &),TR4/69#³69B2&2gm. .^| &u
∂ρ
∂t+ ~∇ · (ρ~u) = 0
∂ρ~u
∂t+ ~∇ · (ρ~u ~u) + ~∇p =
1
Re(~∇ · τ7 )
∂ρE
∂t+ ~∇ · (ρ~uH) =
1
Ec Pr Re~∇ · (µ~∇T ) +
1
Re~∇ · (τ7 · ~u)
^|2M2MR #B69sH;C #%&2&."R(69/69#%[69M4 7r&~W:sb
Re =ρ∞u∞L
µ∞, P r =
µ∞Cp
λ∞, Ec =
u2∞
CpT∞= (γ − 1) M2
∞ .
Xx&);/<R71&:@(d,&2;O¦ ^|!/!869&:!"!,%69@71sA.^5Pb6!/&:,#TWV83"^vR. 83 &k2$#B69&&2T9g Z ` 3D2R", &:H ^HR(6983_&(69#%. sk6v2#%:;Cs:&(69#$!BR",%9g5;O(1/M2)
^|!/\#B69 &2M!"&2/69&: ^| \"RV"
Zcbd
Xx5gh. .^zz(69p
p′ = p− 1
γM2∞ 2S;&2 xsGzW/69 &2";~,4u2S"0:,(69&u9g/69$
ρ = (1 + p′/p) / T
^| .M2M/69.t&24sv86;p
ρE =p0 + p′
γ − 1+
1
2ρ(u2 + v2)
[&2!"436^M!869&<69.%;'c2&2p6";C,2!"";~&24sk
e′ =p′
γ − 1+
1
2ρ(u2 + v2)
N[z6",. z2i&2h^/z9g!"&2/769 71M769U69W."z|2igm. .^| &u
w = (ρ, ρu, ρv, e′)T
Xx& `fe 3V^$R27r";A(69T,2!/)6~!""!&)&(69W."p2B!":;BH86!/2%#B6!/ &2B´"<7r&AghPb6!/H&:,#TWVQ!.4M
10−8 ^KP<"71832S01,(69. xsv9g2S. ,&u!869&~WVp69.R"/7r";D^n ¢,2+&2^ ;"! WV#%¦gm869,"+(6969!"#$#%&69. .u"-!86"8^ ]o ¢9gA69. .>3 ^&. s;869.5^| )2G:' ,1.!/2#%G'eC&2;";u 42$6!"!,/6!s(69&OT<2B&2&C>. #$ ";P\ZiL n 'eO/69RV.U69&_^LO"!"&2;D3^769. ,(69/z27:/!",2?\2,Ce"(69&CT6S"!"&2;O;5!"&:";gm#T,.U6C^SIeOR2.¥! MWV,&2;269s\!"&2;C &2p69@,2";<W:s<;'gm69,. iW,83 &k#%@!86/43^ ^| . .869i2WV,&2;269sH!"&2;C &2 #$R. ! . sr^
& ' " (1& )*% )%A ; !$. #f#%3"7# &NQ&% &:7:/!; \(^¢ &G6S!/(69&&2.a !"#$R,";D^ (zr"#%'s%9g 2!/(69&&2.V!"&29gt6i"!/69&4,.U69;#B69 &
(x, y) ∈ [0; 4] × [0; 1]3i^| 6/69 4:v,RRV ^569. .S69&2;§6!,71"; .^5 ^569. .>30^|24
!":;C &(69"[69M4 71&HW:sb
x < 1 y = 0
1 ≥ x ≥ 3 y = 0.1 (1 − cos[(x− 1)π])
x > 3 y = 0
Xx&d;@6;269Rr"#%sBT,||69"U69&G4;D3:2Q.^I^569. .a#B69 &25/69 4:W,83M^|2&x ∈ [1; 3]
3C2M&2#B69._$2 ^569. ._z!"#$R,";<65gm. .^zSnx = −0.1 · π · sin[(x− 1)π]
ny = 1
Z abe
"[4;|69[#$R.s1";d@!869sd,2[,2;C"83:&(69#%. sG6T!"r69[&281× 21
'3C6p#%";C ,#&2121×31
69&2;d6ic(&2&2@161×41
'^`]286!/d4;D3f^Q. 7rz2QI`,."0:,(69&2gh69&d;869.a1683^|
γ = 1.4^](
M∞ ≤ 0.52 \(^l#B69 &2p,W2&!43¬69&2;k2&2!"$&:R!B69&2;k /69&(69.
¨WV"!869,2|gm` 42Pb6!/%&:,#TWV&2|r6i/69&2&!]\(^ ^| G6M2:!/3169&2;%2zPb6!/%. &2"69%'eCRV"!";kHWV%!"#$R.. sAsO#$#%!869."^| "RV"!SH2$r"#%s+ RV&:U69."\(^Q'^J[&C;C #%&2&(69.V769. ,2"69z/691&G6Kgh. .^zQ
ρ0 = 13u0 = 1
3v0 = 0
3P0 = 1/γM2
∞
^](&2&C>.^Pb6!/H&:,#TWVi\(^z83(69z]^|2&
M∞ > 0.136^M!"&2 ;2'Q,W2&!MWV,&2;269sH!"&2;C &2T
• 6952i &.'p¨.'gh^[ #$RV4i2i'eC&(69. /69._&:(69. R:s13C2i'eC&(69._&:R:sv69&2;G2 \(^69&4. ^| . ^S'eO/69RV.U69M2iR"/,[gh# &2;4
• 69p2B,.G¨ 4: ^%¥#$RV4/%2B/69!%R",l^| .l^B'eO/69RV.U69B2B;&2 xs69&2;2M7r.:! xsdgh# &2;4^
0 &G2[2(69&2;D3^|2&v2[P\6!/G&:,#TWV5#B69. .`M∞ ≤ 0.1
'3^#%:;C¥ghs%2Q &.5!"&2;C &2W:s\!"&2; & ' &2!"#$R"/ W.$ &.S!"&2;C &28^<@ #$RV4$2$'eO&(69.;&2 xsk69&2;b7r.:! xs^| . ^\'eC/69RV.U69\2\R",ugh# &2;4^X& g>6!83i^ ^569&:v,2<2\/69#%<WV,&(;269s!"&2;C &2Sgh
M∞ = 0.169&2;
M∞ = 10−5 v(69S2%!"#$R(69&2SWV ^5"'&!"&:71r&2!"%/69"69S&2z &/\2,2&2!"";uW:sv2MWV,&2;269s~D"!8^ 2p!/2!"p9g2S &2!"#$R" W.T!"&2;C &2Q|,2.¥s*-,2¥c(";Hgh
M∞ = 10−5 W,z9gK!",p;W(69/69W.SgmM∞ = 0.1
^N|Q69&:sH!8643(,QWC*"! 7r &H2 !/(69R!"&2!"&2#%M2iRV-gm#B69&2!"SRRV"Q9g,!/2#%M/692z(69&~2S0:,(69. xs9g2M/. ,&_^
)*% )@ ; .7b# -; !'&)( 7=; !7 25>?[43^c2 ^569&:p~,2;Csb2$ #$R(6!@9g|,TP\69¥eO]2"gh/69#%h^k &A#%T9gqKZ® #%4^Xx&2;"";D3 2T±&bJ[,#B69&&A69&(69. sCi!869&b. .,2[2^2RV"!/69./6;C ,2M #$R. ¥c(!869&69. i2!"&:71r&2!"%/69W2,M Mgm69 .MH &Cgh#,2p69WV,p2169 &W6 &2";AW:s\2% #$R. ¥c(";kR:!"";C,(69i^M;"! WV";~ &~!/(69R0`2^`](|z86&_36^M!869";~ &H2S!":;p69&H #$R. ! Q/69ri&(69#%";[LhiL~/69&2;C &Ggh [68s1&kL:RV"!/69. @SL:RV"!/69.`Q6;C ,2'^ i[LH #$R. ! M/69rTQ &<g>6!i6/69&2;269;W.:!/ #$R. ! %69rd & ^|!/ ^GR.U6!"";2G&:,#%!869.z;C R(69&#B69!""$W:s2;&: xs)#B69¥e#T,. R. ";+W:s2 BRV"!/69.z/6;C ,28^ 0 gM!",43,2!/¢69& #$R. ! d/69rv(6$&2R/6!!869. &:" &2!"B @!"#TW &2"@2B.4T9gB!&2!s;C,2d~2diLk #$R. ¥c(!869&R. ,2T2 4G!"4RV /69&v9gD2zW.:!/%/!/2#%4^[^71831¥g_&2Q!"#$R(69" #$R. ! /69rQ!",R.";^| b6Bqo &: 6!"Wo.U6e269&<R:!"";C, ^| <"RV"!Q/%2TP\]¬q /!/2#%43a&2T!869&b!869. !,.U692p69#%,&:Q9gKqKZ #%S/6871";vW:sd2SPb69¥e:]2"igh/69#%h^ ^5";"83 ^S^569&:¢l#$R(6 ´"2§ #$R(6!9gb2§.^5Pb6!/R"!"&2;C && &#% 9gB!&2!sb69&2;k6!"!,/6!s1^AA"RV"!U69. . s<¥&(i&b2@ &2;RV&2;&2!"%9g2!"&:7rr&2!"/69 ^z¥"RV"!u2GPb6!/)&:,#TWV ^|2&)(d &:7:!;R"!"&2;C &&k@#$R.s1";D^AL:,2!/",. $69^. .¥"/69W. 2";&2^lghS2%W.:!/k!/2#%"SW,p2sb(687r$~WV%!"&Cc2#%";AghS2%Pb69¥eO]2"#%2:;D^ 0 ,<'gm&2!"#%2:; ^| . .TWVA2W.:!/:N n SL Z u!/2#%W^|!/ ^. . >O&2^|&RV4"/@71sr::;!"&:71r&2!"HRRV"8^ ,^| . .|.86;sC#B69!v!"#$R(69/&2$WV ^"&+!/2#%Q69&(;B,P\69¥eO]2"zR:!"";C,gh6M,W2&!!86Q69
M∞ = 0.569&2;$gm6M7rs.^5Pb6!/
!86p69M∞ = 10−5 ^ 2SP<]!/2#% ^| . ._WVS!",R.";:^| <6%LSLvQ6q .U6e69&uR:!"";C,4^
Z a Z
)*% )*% " "!$# ! &)( 7="; 9<7#f# #;H; ( #14; ,& # /n %,2%!"&2;G6b,W2&! \(^69
M∞ = 0.5^NQ$#%&:&2";+ &2HR7O,2$R(69/694/69R_3 ^
;869.^| 6G[L~ #$R. ! p/69r$69&2;\2Pb69¥eO]/ #$R. ! S/69r4^p5b#%2:;p69/. 7r";\W:s2qo &: 6!"W`.U6e69&kR:!"";C,%69&2;\2s\69#$R.s1";\v!"#$R,2,W2&!D\(^&2i!"r69i4;D^ 2M;C &2!"iWV ^"&~2"[ ^/69r"|/69&2;5 &v2Qg>6!5(6952iPb69¥eO]2"!/2#%K#$ & #$ ´8"¬2`&:,#TWVo9gRV/69&2¬(69o69`"0:, ";S| &:71t2KsC#~^ 2K",. o69R"&:";% &$c24, n ^ · ^ Z ^`%#%2:;"01,2¥/|2z69#%|&:,#TWV9g /69&BM86!/B2z86;Cs/6943 &2;C!869 &(69%2HPb69¥eO]2"vg¨/69#%h^(6$&2k&2169 7rH #$R(6!d&2~!"&:71r&2!"/694^Xx& ` Z 3H^S2^";H2p/69#%S",. |gm;C &:4;u ´8"Q69&2;<69.gmz2Sq, . . \(^S^5";"83H^S&2!"S &u2p,#$#B69 ´" &d/69W.S(692pPb69¥e:]2"Mgh/69#%h^~&(69W.",2zB/6871qKZ #%4^Xx&2;"";D3K2G #%Gb!"&:71rv;C/69#B69!869. . s)";C,2!""; ^| 2vP\]!/2#%k£gh#g$[G"!"&2;i ·bn "!"&2;'^ 2"@c2[/,. M'#$R2(6 ´8(69i2Pb69¥e:]2"#%'2:;b(69 ^;71.RV";RV4/""T6~71'sb.^®!"4TRVT /69&69&2;(69T2B &:7r. 71"; #$R. ¥c(!869&2;u&26 D"!|2S!"&:71r&2!"S/694^
Iterations
Log(
Resid
ual)
0 200 400 600 800 1000-12
-10
-8
-6
-4
-2
0RS implicit stageMF implicit stage
M = 0.5CFL = 106
Mesh : 81 x 21
Point Jacobi relaxation procedure (20 inner-iterations)
P\2:; Xx/69&2 qKZ
[L1q [ber d ebe g$[/!4^P<]¬q [ber d ebe ·bn /!4^
]o 4, n ^ · ^ Z oXx#$R(6!`9ga2|P\69¥eO]2"gh/69#%h^V^ n 'gh|&:7rr&2!"|s@9ga2";C,(69.2&2p!"r69S4;zgmiL1q [berQ69&2;uP<]¬q [ber|#%2:;i69
M∞ = 0.569&2;
CFL = 106 ^ 4:p 69W.z,#$#B69 ´" &p2z!"&:7rr&2!"zRRV"9gVWV%#%2:;83r&(69#%. s25&:,#TWV9gV /69&2$86!/~2S86;Csv/69p69&2;H2S6/:!U69";<qKZ #%4^
)*% ) * &143,#f# !2&'%&?. / !$.f=; 14; 9 !$.f=; 14; ( &&# ! 2 5!2&V#%.5I&Xx&GR(69/694/69R_36^ ^569&:5!"WV/69i2Q",. 5W/69 &2"; ^| v2[±&vJ[,#B69&&~69&(69. sC8^5'ghi.86;C &%!"#$R(69&23/^i(687r[$;'c2&2i2i!"4|9g6T&:,#%!869._#%2:;D^ |!"4|!869&HWV'eCR"";~6|gh. .^zp
C = N · Ic^|2
C24.W(69.V!"483
N2&:,#TWV59g_ /69&2|69&2;
Ic2Q!"4RV /69&_^KX&%g>6!83
2@&:,2#TWVS9g /69&2S!869&[email protected]";bG2%69#$R. ¥c(!869&kg>6!M9g(@#%2:;A6!"!";C &~
Z ab[
2igh. .^| &d'eOR"/&A(ρ(G))N = ε
^|2ρ(G)
o2|69#$R. ¥c(!869&$gm6!K69&2;ε
o2.7r.9gV!"&:7rr&2!"|86!/2";$6ghN
/69&2^ :,283a&2p!869&<R";C!Q2S169RuWV ^"&\ ^%#%2:; ^|!/<86!/<2p/69#%S.71.¬9gK!"&:71r&2!"65gm. .^zS
log(ρ(G1))
log(ρ(G2))=N2
N1 igm#T,.U6GM,2@gmM&2%M#%:;$ &k2$](,pR(6!"4^X&A;MvrT69&A" #B69&9g2M169RHgh69. ._2M#%:;"836^S!869&~#B69rp69&<687r/69rS9g2p69#$R. ¥c(!869&ugm6!p
ρ(G)ave =1
π2
∫ π
0
∫ π
0ρ(G(ξ, η)) dξ dη
Xx&H/69W. n ^ · ^ Z 3H^MR"&:2p6871/69rT69#$R. ca!"69 4&ugm6![69M∞ = 0.5
9g2M#%2:;(690^^569&:QB!"#$R(6943a&(69#%. sH2T n N n pL Z z!/2#% ^|!/<,2";<62S'gm&2!"T&243(2pP\]¬LSL Z er|!/2#%T69&2;H2pP<]¬q · erz!/2#%4^X&~2S/69#%M/69W2. 43H^T69.$R7:;S2M2"!869./69uWV ^"&69. .2"%#%2:;^ %&2!"B(6983o &2"s13¬2dP<]¬YLSL Z erS!/2#%B"0:, "2.5
#%"z6#B69&:sd /69&26([ n N n pL Z !/2#% ^| .i2iP\]tq · er!/2#%i&2"";2.8 #%"6#B69&:s /69&28^L:,2!/B",. #$R(6 ´8(6983:¥g^i^569&:2P\]¬YLSL Z er`/!/2#%zSWV
!"#$RV 71S &~#%9gqKZ #%43 !"4zRVz /69&u(6z$WVp69z."62.5
#%"z."/z'eORV&C 71@(69&A2$ n N n SL Z i/!/2#%4=?S&24^ M#%2:;k5*Y,2p6v01,(69. /69 71$:.K &2!"2$±K&
P\2:; n N n SL Z P\]¬YLSL Z er P<]¬q · er
ρ(G)aveeC^ c ` eC^ g n eC^ n e
Ni/NrefZ [C^ c [C^ n
69W. n ^ · ^ Z ¬N¬7r/69r569#$R2.¥¥c(!8694&g>6!9ga71/69.O#%2:;K69M∞ = 0.5
t n N n SL Z o!/2#%43P\]tYLpL Z er`!/2#%Q69&2;%P\]¬q · erK!/2#%4^ 2|2"!869.V/69p9g /69&dWV ^"&G69. .(2"#%2:;zQ69.R7O;";D32 n N n SL Z |!/2#%S !/24&<6|2M'gh&2!"4^J[,#B69&&<69&(69. sOz;:"|&2z/69rM &:%6!"!",&:2S "!z9g2iWV,&2;(69sv!"&2;C &2^KP\"7183^[:&2^¦(6952[!"&:71r&2!"i/69[#B69 &. sd;C 71&GW:sB2[RV-gh#B69&2!"S695.^4^|687r&:,#TWV83/^2,.;% &2!86z2i^ 4:9gD2|.^5mgh"0:,2&2!s &d;MWVz#%zR"!4^`JQ712."8312" #B69&B4 7r&%W:sT69&(69. sO KRV &:K,K2z K(69K#T,2`WV|#B6;zi(69K2|Pb69¥eO]2"#%2:;~z!"#$RV 7r4^Xx&+c24, n ^ · ^_[C3]^5~R"/&:d2<;CU69r&(69.[R9c2.\9gS2\69#$R. ¥c(!869&gm6!G9gS2"u#%2:;69&2;2H!"&:71r&2!"~s9g[2H";C,(69.%&+2Gc2&(v4;8^ v&2!"H(69%2v2 /69/!/:sR7O;";uW:sv2p±&<J[,#B69&&b69&(69. sO#B69 &2Q2T/69#%Mgm2p&:,#%!869.o"-!86d2T n N n SL Z !/2#%[2[#%45B!&:83C 5"01, "
N1 = 720 /69&25@86!/v2Q´8@#B6!/ &24
/69r"N2 = 1320
/69&2gmK2P\]¬YLSL Z er`!/2#%M86!/d2/69#%z.7r.V69&2;N3 = 1400 /69&2zghz2MP\]¬q · er|!/2#%4^CAi(6871M2&~2[gm. .^| &B.69¥71SB!&2!sk
N2
N1= 1.83 ,
N3
N1= 1.94 .
Z a ·
ξ / π = η / π
ρ(G
)
0 0.25 0.5 0.75 10.25
0.5
0.75
1
BL-ALGS(1) schemeMF-SGS(10) schemeMF-PJ(30) scheme
M = 0.5 CFLBL = 20 CFLMF = 106
IterationsLo
g(Re
sidua
l)0 500 1000 1500-12
-10
-8
-6
-4
-2
0
BL-ALGS(1) schemeMF-SGS(10) schemeMF-PJ(30) scheme
M = 0.5 Mesh : 161 x 41
]o 4, n ^ · ^_[C@#$R(69&WV ^"& n N n SL Z M/!/2#%43¬P\]¬q · erM/!/2#%B69&2;P<]¬YLSL Z er!/2#%@69
M∞ = 0.5^ n 'gh@5j[U69r&(69.oR9c2.T9gK2@69#$R. ¥c(!869&\g>6!8^z 4:@|5&:7rr&2!"
sv9g2M";C,(69.&H2[c2&2M4;D^
:,283z2<±K&¦JQ,#B69&&69&(69. sCG71" #B69"G2u.4/G9gS &: &2!\B!&2!s¢.U69";2Pb69¥e:]2"[#%2:;D^KN[5'eOR.U69 &2";H869. 83C2i!"&:71r&2!"i/69[. "5#B69 &. sd&v2i;269#$R &%9g2T.^ ^56871&:,#TWV8^[Xx&2;"";D3a2@!"#$R(69&k9gK269#$R. ¥c(!869&\g>6!i'#$R2(6 ´8"[(69[2169RBWV ^"&d2;C D&:#%2:;."K #$RV/69&:57r2z.^5mgh"01,2&2!s$/'4&_^ 2"z",. 2&4 71H!";&2!"Hk,2!/¢69&¢6,#$R&_^J[7r2."/8369& #$RV/69&OB!"4$";C,2!&¢ % . .&2"!""/69s@ &B;i#B691|2zP\69¥eO]2"|#%2:;B!"#$RV 7r4^ 25g¨/69#%h^(69"^55R"/&: &H!/(69R `$&2,"|,2!/H6T";C,2!&_^ 2[",. 5W/69 &2";~69
M∞ = 0.569[R"&:";G & 69W.
n ^ · ^_[C^ [email protected]!". ,#$&b[.U69";\d2!"4[RV[RV &:iRVi /69&_3-^;&2@ SqKqKqKX'^ 2qKqKqX#B69 &2M01, @!"&2/69&: ^| \"RV"!id2@4;\ ´"43V iQ2'gm@71s~,2'g¨,.oG;C 71!"#$R(69&28^T](i2Tc2&24;D3 [email protected] 71$!"4i9g2#%2:;i(69 ^ ^569&:Mv!#$Ra69%692igh. .^| &<
CPPPI1CPPPI2
= 2.52 ,CPPPI1CPPPI3
= 2.07 .
2SPb69¥e:]2"igh/69#%h^H&(69W."z$";C,2!"p;C/69#B69!869. . sH2p5qKqKqKX'^(Q&2!"4322MPb69¥eO]2"#%2:;B!",R.";l^| %2QLSL$69. r #K2zq R:!"";C,zWV"!"#%"4.W(69. . s#%z!"#$RV 71(69&)2G.:!//!/2#%4^ n @,2@&2^gh:!,2&86!/)"!/&0:,24^bN[T2d4;WV"!"#%"@c2&283`2P\]tq !/2#%G"0:, "%6u.U69r%69#%,2&:$9gQ &&2-> /69&2% &;$<#B69 &+B!&:8^ k &lr::;694"#%&: ^| l2±K&JQ,#B69&&®69&(69. sO832+P\]tq 69. r # &2"";A#B69&:s &&2-> /69&($/\6T\;269#$R!""!. s2d.^5mgh"0:,2&2!s@#%:;"^bQ^7183^GW271#%T &2/69W . "[gmQ71s~.^ ^5687r&:,#@WV'/ ^| .@2@!/2#%@[/69W.T &\R/6!!"4^ 2T4;<RV(69R2&2`c2&2z&2,4BS!869R,z2"|.^5mg¨"0:,2&2!s#%:;"83O6969&:s!86/5K69&22K/"6/& &2!86/M2i&O,#TWV9g,WC> /69&28^ 0 &~2M2z(69&2;D322MP\]¬YLSLG!/2#%M#B69 &2z71sB!&:Qgh[69. .¬4;0^| \&. s
10 &&2-> /69&28^ Q [69.d!"&Cc2#%";<W:sv2p±K&<J[,#B69&&
",. 8^pXx&b2 69W. n ^ · ^_[C3^$69.GR7O;@2",. MW/69 &2"; ^| k2iL~ #$R. ! S/69rTgh2%!"r69$4;D^dXMS &:" &uH&2$(69 ^|2&A SS!",R.";L^| 2%N n SL Z [.U6e269&R:!"";C,432z#%2:;~|2M#%4zB!&:[69#%&B2SP<]¬>xsORVi"!/&0:,2"8^
Z a2`
[&2!"43B2¢.4/9g~ &: &2!B! &2!s &2&:) &2 RV"!/69.G/6;C ,2)¥#$R2.¥¥c(!869&¯W(69.¥69&2!"";W:sk2GP\69¥eO]2"B #$R. ! @g¨/69#%h^(69D^G,. &2"; &)!/(69R `2^uXx&2;"";D3o2d!"4RV /69&&4. s";C,2!"";<(692G4.W(69.z!"4@WV"!"#%"$ 4. s!"#$RV 71H!"#$R(69d2@!'.U6/!869.oW.:!/u!/2#%4^[P\"7183 ^@!869&b'eCRV"!'S6B;C/69#B69!869. . s\;"!86@9g2T#%#%s"0:, #%&:83:,(832~P\69¥eO]2"H#%2:;$,. s) &:" &k!"#$R,2~,W2/&!v &:7O!;\(^z8^ n z,2z&2^!869sv,,2;C"z7rz2i.^5Pb6!/~/'4 #%4^
Z a c
JQ5I n P\2:; ] n Xx/69&2 qKZ qKX qKqqKX
Z g$e Z n N n SL Z 20
' 53
8^0.13 7.8 · 10−5
iL1N n SL Z 106 710 83
8^0.12 6.9 · 10−5
[L1q [ber106 900 72
8^0.08 4.7 · 10−5
P<]¬q Z er106 1950 53
^0.03
·10−5
P<]¬q [ber106 900
Y G0.04 2.5 · 10−5
P<]¬q · er106 830 48
8^0.06 3.4 · 10−5
P<]¬YLSL Z er106 800 40
8^0.05 3 · 10−5
P<]¬YLSL [ber106 750 66
8^0.09 5.1 · 10−5
P<]¬YLSL · er106 730 92
8^0.17 7.4 · 10−5
· g c/Z n N n SL Z 20
m mH169
8^0.31 8.2 · 10−5
P<]¬q Z er106 JM
P<]¬q [ber106 1250 133
8^0.11
X ·10−5
P<]¬q · er106 1120 165
8^0.15 3.9 · 10−5
P<]¬YLSL Z er106 1070
/Y LG0.12 3.2 · 10−5
P<]¬YLSL [ber106 1000 215
8^0.21 5.7 · 10−5
P<]¬YLSL · er106 970 300
8^0.31 8.2 · 10−5
ababe Z n N n SL Z 20
HXH396
8^0.55 8.3 · 10−5
P<]¬q [ber106 3000 581
8^0.19
X / ·10−5
P<]¬q · er106 1400 370
8^0.26 4 · 10−5
P<]¬YLSL Z er106 1320
X/ LG0.22 3.3 · 10−5
P<]¬YLSL [ber106 1250 482
8^0.39
8^5.8 · 10−5
P<]¬YLSL · er106 1200 667
8^0.568 8.4 · 10−5
69W. n ^ · ^_[CiXx&:7O /!;b86;Cs:\(^71S6v &2W,#$R69M∞ = 0.5
^T4,.U69T569"U69&A4; ^| AR = 1
^ 2S!"#$R,/69&20^S;&2 ^| <6qo&: ,# `232.4 GHz
31GB
N[P#%#%s1^KJ[I np2%&:,#TWV@9g|RV &:8^uqKZ¯S(% #%%"0:, ";u!"&:7rr4^uqX[p2B!"4pRVT /69&^| .TqKqqKX|2S!"4zRVzRV &:zRVz /69&_^
Z aba
)*% ) * &143,#f# !2&!:# 7=; %&"h9<.f= #; 3# 14;5'gh[RV-gm#$ &B!"#$R(69&283/^ ^569&:5,.¥ &2M2[&2"!"" xsG9g¬2[R"!"&2;C &&% &H;\R"71GB!&2!s69&2;6!"!,/6!sr^],2#%43^,^| . .|#$R(6 ´8v2G &2;RV&2;&2!"~9gQ2!"&:71r&2!"S/69 ^| ~"RV"!z$2MPb6!/~&:,#TWVi^|2&u2M/!/2#%M|R"!"&2;C &2";_^
Iterations
Log(
Resid
ual)
0 200 400 600 800 1000-12
-10
-8
-6
-4
-2
0 BL-ALGS(1) without preconditioningBL-ALGS(1) with preconditioning
M = 0.1 CFL = 50Mesh : 121 x 31
Iterations
Log(
Resid
ual)
0 200 400 600 800 1000-12
-10
-8
-6
-4
-2
0 MF-SGS(30) without preconditioningMF-SGS(10) with preconditioning
M = 0.1 CFL = 106
Mesh : 121 x 31
qKQI P<2:; ] n X/69&2 qKZ qKX qKqKqKXJ[ n N n pL Z
50 850 2228^
0.26 6 · 10−5
P\]¬q · er106 Ji
P\]tYLpL Z er106 Ji
¶" n N n pL Z 50
' 6144
^0.31 8.2 · 10−5
P\]¬q · er106 990 144
^0.15 3.9 · 10−5
P\]tYLpL Z er106 790
/ G 0.12
Y NX ·10−5
]o 4, n ^ · ^ · ~Xx&:7:!;86;Cs \(^71B6b &2W,#$R 69M∞ = 0.1
gm$2v#%";C ,# 4;3751RV &:'^`4,.U69|569"U69&d4;Q^|
AR = 1^ RC n 'gh[`&:71r&2!"Qs%9gD2z";C,(69.(gh
2| n N n SL@!/2#% ^| B69&2; ^| 2,KR"!"&2;C & &(^ RCz 4:o&O7rr&2!"|s@9gV2";C,(69.Ogho2P\]¬YLSLp!/2#%I^| $69&2; ^| 2,R"!"&2;C & &22^`/#¯ 69W.5,#$#B69 ´" &2MRV-gh#B69&2!"T9go2MR"!"&2;C &2";\69&2;H,&R"!"&2;C 4&2";<!/2#%"Q69
M∞ = 0.1^
Xx& c24, n ^ · ^ · 3 ^R"&:2",. W/69 &2"; ^| 2R2"!&(;C &2"; 69&2; 2&(&CR"!"&2;C &2";v!/2#%"569
M∞ = 0.1&B2#%";C ,#4;D^`NQ'eORV"!";D3M^Q!.869. s$&2(692
R"!"&2;C &2";+!/2#%"$69Bg>6@(69&)2G,&R"!"&2;C &(";&2"8^bN[pgh/!86";W:sA2G±K&J[,#B69&&69&(69. sO 35^H#B69(69B2uP\69¥eO]2"~!/('#%"Q^| 2,BR"!"&2;C &2 &RV4/"RV:Q!"&:71r&2!"S/69/^2SW271p69.%(69|gmz2p/69#%M4;D3aP<]R"!"&2;C &2";\!/2#%"[69
Z ag
WVB69M∞ = 0.1
(69&+69M∞ = 0.5
&2!"G2G&:,#TWV%9gQ /69&2$"01, ";\86!/2#B6!/ &2´8p#B69. .8^ WV(687O,!"&Cc2#%2z",. 9gD2!/(69R2' c 3r(RV"!/69.a/6;C ,2 #$R. ¥c(!869&<|."|RV&(69. ´" &G69z.^5P\6!/u&:,#TWV &2!"i2S r&:769. ,2"Q9go2iR"!"&2;C &(";sC#69M7rsv!.44^
]o 4, n ^ · ^ `2oXx/. &2"9ga25Pb6!/&:,#TWVM∞ = 0.1
3#%"~121×31
'^ RH¬Z&R2"!&(;C &2";!/2#%4^K5# `q`"!"&2;C 4&2";<!/2#%4^Xx&c24, n ^ · ^ `23"^v;C/6^2G. &2"$9gQ2vP\6!/&:,#TWVB69
M∞ = 0.1&2G#%";C ,#³4;D^
S&2T(6983V71'&<¥gK2su!"&:71rpB#B6!/ &2@´83a2p,&R"!"&2;C 4&2";b/!/2#%"QR7:;69& &2!""!|. , &t^ I^. .¥>:&2^|&v",. ;C,2QT2[71-;C R(69&H9gt2Q&:,#%!869.D/!/2#%^|!/<69/i^|2&~2SsC#| . .¥!"&2;C &2";)m69z.^5Pb6!/~&:,#TWVzgh| &2/69&2!"'^NQ&226RV"!9g~2 .^5Pb6!/®R"!"&2;C &&2=?D"!)2¢ &2;RV&2;&2!8¦9gu2!"&C71r&2!"T/69D^| \"RV"![B2@Pb6!/<&:,#TWV8^iX&\;[G#$R(6 ´8@QRRVxs13^p#B6916$!"#$R(69&<WV ^"&\69. ._2S#%2:;Q&~2S#%";C ,#4;u69
M∞ = 0.169&2;
M∞ = 10−5 ^ 2",. i69pRV/;\ &</69W. n ^ · ^ · ^QN[Q'eORV"!";D3V2T!"&:71r&2!"T/69"[9gK2T;C &:[#%2:;;@&2|769sB#@,2!/Ggm5WVHPb6!/v&:,#TWV^ Ai!869&v2'ghM!869sG,69.¥.D2i!"#$R(69&2|ghz6,&0:,2SP\6!/~&:,#TWV8^CAM!/2:4/p60:,(6¥> &2!"#$R" W.p!8643&(69#%. s
M∞ = 10−5 ^Xx&¢2<#B69&&2v9gS2uR7O,2G/!&_3i^uR"&:G &¢c24, n ^ · ^ c 2<;CU69r&(69.iR9c2.\9gS269#$R. ¥c(!869&g>6!'/9g[;C &O#%2:;83K&(69#%. s2v n N n SL Z @!/2#%43K2vP<]¬YLSL Z er!/2#%H69&2;2GP\]tq · erT!/2#%4^ Av69.\;C/6^®2G!"&:71r&2!"G/s9g2 @";C,(69.^[\69169 &_3z2u/69!/Os¢R7:;";W:s(u±&J[,#B69&&69&(69. sOG#B69 &2G2\/69#%ughG2&:,#%!869.|'eCRV #%&:H$2G.1!/!/2#%dT2d#%4@B!&:$¥&)#%9g /69& ^| .G2P\]tYLpL<!/2#%i2Tg>6"iG86!/k2#B6!/ &2´8v &b#%S9g|qKZ #%4^ 2"",. 69b!"&Cc2#%";&69. .i4;A 69W. n ^ · ^ `:'^NQGghv2b,W2&!b!86432\.4v9gT &: &2!
Z a n
P\NQ P\2:; ] n X/69&2 qKZ qKX qKqKqKX
10−1 n N n SL Z 50
' /144
8^0.31 8.2 · 10−5
P<]¬q · er106 990 144
8^0.15 3.9 · 10−5
P<]¬YLpL Z er106 790
/LG0.12
Y NX ·10−5
10−5 n N n SL Z 50
' /144
8^0.31 8.2 · 10−5
P<]¬q · er106 970 141
8^0.15 3.9 · 10−5
P<]¬YLpL Z er106 780
/ mG0.12
Y NX ·10−5
69W. n ^ · ^ · HX&O7:!;86;CsW\(^7rB6b &2W,#$R 69M∞ = 0.1
69&2;+69M∞ = 10−5 gh$2
#%";C ,# 4;3751
RV &:''^Q4,.U69v569"U69& 4; ^| AR = 1
^J[5I n %2H&O,#TWVd9gRV &:8^|qKZz2S #%S"01, ";~B!"&:7rr4^|qKXz2p!"4zRV /69& ^| .@qKqKqX5z2!"4zRV|RV &:RVz /69&_^
B!&2!sb &2&:S &A2$RV!69.`/6;C ,2p #$R. ¥c(!869& MW(69.U69&2!"";AW:s<2%Pb69¥e:]2"$ #$R. ! gh/69#%h^V^ 2d";C,2!&9gz2G!"4@RVT /69&&(69W."@2GP\]!/2#%d<"!"7r$6~71sr::;4.W(69.BB!&2!sr^ &2/"01,2&:. s13$2Pb69¥e:]2"#%2:;67469. ,(69W.¢69. &(69 71!"#$R,S &:7O!;v.^5Pb6!/u&:,#TWV]\(^S^K5";"832MPb69¥eO]2"M #$R. !¥Q#%'21;~(6Q69&226;C7469&:/69rS!"#$R(69S$2S.:!/v/!/2#%"83&(69#%. sG2M";C,2!&<9g2M#%'#%sv/69r4^
ξ / π = η / π
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
BL-ALGS(1) schemeMF-SGS(10) schemeMF-PJ(30) scheme
M = 10-5 CFLMF = 106CFLBL = 50
Iterations
Log(
Resid
ual)
0 200 400 600 800 1000 1200 1400-12
-10
-8
-6
-4
-2
0BL-ALGS(1) schemeMF-SGS(10)MF-PJ(30) scheme
M = 10-5
Mesh : 161 x 41
]o 4, n ^ · ^ c @#$R(69&WV ^"& n N n SL Z M/!/2#%43¬P\]¬YLSL Z erM!/2#%B69&2;P<]¬q · er!/2#%i69
M∞ = 10−5 ^ n 'g¨ijQU69r&(69. R9c2.i9g_2i69#$R. ¥c(!869&vgm6!^ 4:[K&:71r'&(!sv9g2M";C,(69.&H2[c2&2M4;D^
Z a d
J[5I n P<(1; ] n Xx/69&2 qKZ qKX qKqKqKX
Z g$e Z n N n SL Z 50
Y 649
8^0.13 7.8 · 10−5
P\]¬q · er106 700 40
8^0.06 3.4 · 10−5
P\]¬YLSL Z er106 550
X G 0.05
Y ·10−5
· g c/Z n N n SL Z 50
' 6144
^0.31 8.2 · 10−5
P\]¬q · er106 970 141
8^0.15 3.9 · 10−5
P\]¬YLSL Z er106 780
/m G 0.12
Y NX ·10−5
ababe Z n N n SL Z 50
m 308
^0.55 8.3 · 10−5
P\]¬q · er106 1230 325
8^0.26 4 · 10−5
P\]¬YLSL Z er106 960
XH G 0.22
Y Y ·10−5
69W. n ^ · ^ `2Xx&:7O!;H86;Cs \(^7rQ6$ &2W,#$Rb69M∞ = 10−5 ^4,.U69i569"U69&u4;i^z¥
AR = 1^ 2S!"#$R,/69&20^S;&2 ^| <6qo&: ,# `23
2.4 GHz31GB
N[P#%#%s1^KJ[I np2%&:,#TWV@9g|RV &:8^uqKZ¯S(% #%%"0:, ";u!"&:7rr4^uqX[p2B!"4pRVT /69&^| .TqKqqKX|2S!"4zRVzRV &:zRVz /69&_^
)*% ) 9<; 14&&#b !7$& #3);
2%Pb69¥e:]2"$ #$R. ! Sgh/69#%h^b(69 ^%;71.RV";A¥&)(69R `:^56T6r";kW:s\2%]&2!/N|#$!QIK&24sG#$#$/&k>IKNMK &v;p";C,2!"i;C/69#B69!869. . sB2Q"0:, #%&: &G#%#%s/69r~9g[2v #$R. ! d!/2#%"8^ (HWC*"!'¥71H9gM,2!/¢6\";C,2!&+$k!869s,$&:,#%!869. #T,.U69&v9gt!"#$R.'eB &2;C,2U69.VRW.#% &:7r. 7O &T4;5^| v6S7rs$.U69r[69#%,&:59g_RV &:8^`Xx&R(69/694/69Rt3M^Q!"#$R(6943:gh69. .a4;8312#%#%s$"0:, #%&:59g_86!/d#%2:;Q^| d"RV"!\2v'eCR. ! $/!/2#%4^ 2G",. %69GR"&:"; & 69W. n ^ · ^ c ^Rd&2v(692HP<]¬YLSL!/2#%Q56SW #%Q4. ,&2,2I^| G"RV"!T2QP\]¬q !/2#%4^`Xx;C,2T2zg>6!(69I^&2"";~%M2S/69"i^|!/\69S7469. ,(69";u;C, &d86!/< ^"RuB6|B!"#$R,S2M&2'eC&2"8^0 &2G2T(69&2;_3o2GP<]¬q R:!"";C,v;:"T&2@&2"";<G69&:s: &23TT2B/"6/&^|:s 2H!/2869RV$ #$R. ! $#%2:; &#%%9gQ#%#%s/69r4^)NQ$'eORV"!";D32H.1!/!/2#%4=?"0:, #%&:|6924869"8^`Xx&2"";
110 #%Q",!""(69&d2Q'eCR. ! !/2#%
^| .B2dP\]!/2#%"T"0:, S*-,260
#%4^uQ&2!"43o2dPb69¥eO]2"%g¨/69#%h^&(69W."@";C,2!"p9g69WV,(69.¥g`2S#%#%sH"01, #%&:[9g`2M #$R. ! [!/2#%4^5";"3(62^z&~ &u2R7O,2K"!&2834(|P\69¥eO]2"5 #$R. ! K#%2:;$K69.i71sT!"#$RV 71| &$#%K9g_qKZ¢¥#%% #T,.U69M.^5Pb6!/~ &:7O!; \(^z8^
Z g$e
81 × 21 121 × 31 161 × 41
!"#$%!&! '$(! ) %"*$!&% )'+$(! ) !"*,$!&! )'+$(!- /.10 2) - /.10 2) - /.10 2
3546 '78$97:) 3.1 1 6.6 1 12.2 1
/<;>=@? 7:A 4.7 1.52 10.1 1.53 19 1.56
/<;,B B 7:) 4.8 1.55 10.5 1.59 19.6 1.61
.DCE;GF9C B 7:A 6.7 2.16 14.7 2.23 25.8 2.11
69W. n ^ · ^ c ~P\#%s/69rvgmB69. .z2H#%'2:;l^| +"RV"!%k2v4;+ ´84^ 2~'eOR. ! !/2#%M|/691&<6|2M'gh&2!"p%6|%769. ,(69S2i.U69 71S ´8"^
)*% ) * &5.6 25!2#%&5! 2$P\69¥eO]2"@ #$R. ! S#%2:;bi2&A6d7469. ,(69W.%69. &(69 7r$v #T,.U692 &:7:!;b86;Cs\(^z8^dX&2;"";D3o'71&A¥g5 S &: &2!BB!&2!skMRV:83¬ p!"4SRVS /69&pH!/2869R(69p2#%2:;~|4.W(69. . sG#%SB!&:(69&<6.:!/G #$R. ! Q!/2#%S!",R.";:^| <6N n SLd.U6e269&R:!"";C,4^]2,2#%43(2SP\69¥eO]2"ig¨/69#%h^H&(69W."z$";C,2!"S;C/69#B69!869. . sH2M&2"";~9g#%#%sv/69r4^L:,2!/~",. |(6871iWVS!"&Cc2#%";vgm|7O /!",2I\(^8^ (69z|2S,WC*-"!z9g2&2'eCz"!&_^
Z g Z
&(% &))%A ; !$. #f#%3"7# &N7O!",2 \(^ &§/69 4:H!/(69&&2.iv!"#$R,";D^ 2\r#%s 9gT2\!/(69&&2.S!"&2v9g@6"!/69&4,.U69[;#B69 &
(x, y) ∈ [0; 10] × [−1; 1]^
h=1
L=10
]¬ 4,/ n ^ `2^ Z #$R,/69&(6.¬;#B69 &~ghz2Sqo, . . \(^S^](69& &2!"#$R" W.v86;Cs \(^ ^| 2,@#$RV/69,H69&2;4/687O /69&(69.|.D"!83o2dR",gh,&2!&@. &2869%69&2;)2vJi687:-YL:1"$"01,(69&2";C,2!"v<2G#%#%&:,# &2b _aba gh#%S;/69 .M
1
Re· d
2u
dy2=dp
dx^|2M2SQs:&2.;|&:,#TWV|4 71&HW:sb
Re =ρ∞ u∞ l∞
µ∞^| l∞ = h
^|!/uz!/24&<6z2M.&4u!869.S9g2MRW.#~^[ 71&H$2M&29. Ru!"&2;¥ &(&u86!/:^569. .>32S'e6!. ,&~z2&
u(y) = −Re2
· dpdx
· (1 − y2)
2[71.:! xsBR9c2.i2'gm[R(6969WV.¥!p69&2;G2[#B6eC #T,#¯7469. ,2[5W/69 &2";H&v2M6eC59g¬2!/(69&&2.
umax = −Re2
· dpdx 2p6871/69rM7r.:! xsG86 . sv!"#$R,";~,4u2[gm. .^| &B'eCR"&
uave =1
2
∫ 1
−1umax · (1 − y2) dy =
2
3· umax
LO &v69W /69 . sH2T687r/69rS71.:! xsH"0:,(69._%,& xs1322M &2!"#$R"/ W.T'e6!Q. ,&uz2&4 71&HW:sb
u(y) = 1.5 · (1 − y2)$'eCRV!pHW/69 &2%/69#%$R(69/69WV. !%R9c2.d69S2%,.p9g52%!/(69&&2.>^BJ[&C;C #%&2&(69.7469. ,2"i69T/691&b6Qgh. .^z
ρ0 = 13u0 = 1
3v0 = 0
3P0 = 1/γM2
∞
3M∞ = 10−5 3
Re = 15^
!"&2;z(i &2!"#$R" W.SWV,&2;269sH!"&2;C &2T
Z g$[
• 69%2~ &.'\¨.'gh ^H #$RV4~2~;&2 xs132H71.:! xs69&(; %R(69/69WV. !~R9c2.43|69&2; ^'eO/69RV.U69S2MR",[gh# &2;4
• 69p2B,.G¨ 4: ^%¥#$RV4/%2B/69!%R",l^| .l^B'eO/69RV.U69B2B;&2 xs69&2;2M7r.:! xsdgh# &2;4^
2M,RRVQ69&2;H.^0^569. .69S!"&2;";<6;CU69W(69!T69&2;H2M&29. Ru!"&2;C &u69RR. ";D^
))@ ; .7b# -; !'&)( 7=; !7 25>?Xx&A!/(69R aC3)^%,2;C";k2$ #$R(6!T9g52$7O!",2SRV"!/69./6;C ,2p #$R. ¥c(!869& ^| 2$±K&J[,#B69&&69&(69. sC8^ \,. &2";¦(69GgmG.^5Pb6!/69&2;.^5Qs:&2.;,\(^p32\Pb69¥eO]2"!/2#%I^56o0:, B!&:R7O;C &i(69o256RV"!'/69Q#B69 &2`9ga;`&24^JA569.QR"&:";69&d&(69&2!"#%&O59gD27O /!",2KR/!"&2;¥ &2 & ^|!/G&(69W."2Pb69¥eO]"#%2:;dS#B69 &!"#$RV 71$"RV"!U69. . sugm[2@.^5mgh"01,2&2!s<#%:;"8^p](ii"-!8643^D^|69&:iv!"&Cc2#2",. i9g2T±K&bJ[,#B69&&k69&(69. sC8^[]o 83 ^D^| . .`,2;CsHgh
AR = 12TRV-gm#B69&2!""M9gK2
#$R. ! ¬#%2:;¬(69^K(67r69. 86;CsS!"#$R(69";pgmt2 &2W,#$R_3&(69#%. sM2 n N n SLS!/2#%432|P<]¬YLSL@!/2#%z69&2;@2|P\]tq !/2#%4^ 2&_3^I^| . .2#$R(6 ´8525.4`9gVB!&2!s9gV2P\]!/2#%" ^|2&k2%6RV!S/69v &2!86"8^]o &(69. . s13^ ^| . .`R7O;$#%R/6!!869./. ,&2 &u;z"!"71B!&2!s1^
))*% * &143,#f# !2&'%&?. / !$.f=; 14; 9 !$.f=; 14; ( &&#AR = 1
Xx&$R(69/694/69R_35^H!"&2;d6<.^5Pb6!/+7:!",2 \(^ M∞ = 10−5 69&2;
Re = 15@69&2; ^
!"#$R,S2S. ,&u&~"M4;i^| <69&<6RV"!z/69%"01,(69._$,& xsb
• 101 × 21h!"r69'
• 201 × 41¨#%";C ,#d'
• 301 × 61£c2&2'^
2"" ´8"9g24;@!""RV&2;@Q";C D&:!". .CsO&2.;¬&:,#TWV8^ZQ &[2;'c2& & aC^_[C^ Z '3^Q!"#$R,
Recm = 0.763Rem
m = 0.3869&2;
Refm = 0.25
^](2zc2&2Q4;D3 gh2[!86gm5^|!/G2Q7O!",2D"!|69Q2Q#%4 #$RV/69&:83C2i6!",2!is:&(. ;2&:,#TWV54 71&dW:su
Refc =RefmM∞
= 2.5 · 104 >> 1 .
Q!869&G2'gmi'eORV"!(6952[RV"!/69.V6;C ,25 #$R. ¥c(!869&'^,.;dWVQ&25:TRV&(69. ´" &2^ n ,2z&2^!"&2;2ic2&2S4;D^N[z71'sG #$R. ! Q!/2#%Sz!",R.";:^| u2S/69#%S'eCR. ! Q/69r4369. .V2Q&:,#%!869._. ,&2|69Q2[/69#%4^`Xx&Bc24, n ^ `2^_[C3^[;C/6^2[;C W,&H9gt2Q7r.:! xs69z2M,.Q9go(M!/(69&&2.o69&2;'^S!"#$R(69M i^| ~2S'e6!z &2!"#$R" W.p. ,&_^ 269#$R. ¥c(!869&dgm6!69K2|'gh&2!"6969&2;$2z!"&:71r&2!"zs$9g 2|";C,(69.9g 2" #$R. ! 5!/2#%"569zR"/&:";B &%c24, n ^ `2^ · ^VAz&2(69gm2z71s.^ ^56871&:,#TWV832GPb69¥e:]2"v!/2#%"@R7O;v69&6@r:1;;269#$R &A6@2G n N n SL Z T!/('#%43` &2;C!869 &6ur::;!"&:71r&2!"v/694^ 2G!"&:71r&2!"vs9g2G";C,(69.$!"&Cc2#%$(69Pb69¥eO]2"!/2#%"bhP<]¬YLSL Z erB69&2; P\]tq F`ferd69<6dB!&:v6B2< n N n pL Z '^§XxB%&2!"869W.(69QWV\P\]+!/2#%"QR7:;p2T69#%pB!&2!s1^Q^7183V62TP\]¬q !/2#%T"01, " `fe
Z g ·
Velocity
Y
0 0.5 1 1.5-1
-0.5
0
0.5
1Numerical solutionIncompressible solution
]o 4, n ^ `2^_[CQj[ W,&k9g2T7r.:! xs\69[2,.M9g2@!/(69&&2.K69M∞ = 10−5 3
Re = 1569&2;Bgm2c2&2[4;b301× 61
'^KLO0:,(69[sO#TWV.R"/'&O2Q&:,#%!869._. ,& ^| .Q2[. ;. &2S/69&2;|gh|2S'e26!z¥&(!#$R2" W.p. ,&_^
&&2-> /69&(83K !"4@RV@ /69&@ 42@a69&)2GP\]¬YLSL!/2#% ^| &. s Z eu &&2- ^"R28^ 2d",. T & 69W. n ^ `2^ Z #$R(6 ´8d2Br::; &: &2!GB!&2!s9g2dP<]!/2#%"gmQ69. .¬2S4;^ Q &ur::;\694"#%&: ^| <2pW27469&2[9g(69R aC32(69Q2pP\] #$R. ! #%2:;B 4. s%!"#$RV 71Qp!"#$R,Q7O!",2"\(^z714;5^| v6RV"!/69T!.4i,& xs1^X&2;"";D3r2zP\]<#%2:;$R7:;"K 4%¥&O &2!zB!&2!s%69&2;%6[.^¢!"4`RVK /69&_31 |2'gmM71sd &:" &%/$!"#$R,p,2!/'\(^z8^KJ[7r2."3!"#$R, &d7:!",2 \a^zz9gh&"0:, "K!/2";$4;69&2;+^z/6^+ &d(69RIai(69K2zP\69 eO]2"|#%2:;B!",R.";l^| GLSLQq .U6e269&<R:!"";C,TWV"!"#%"Q."B!&:QghQ,2!/<!86"^A ^| . .¬,2;Csv,2S &u2&2'eC|R(69/694/69R_^
)) * &143,#f# !2& ( &&# !7#;67.M="; > 3&#f# >!Xx&d;p!869sB,!"#$R(69&(5&G!/2";d4;83M^,2";B"Q;C D&:4;8^ 2Q ^Sc2&2"z69/[W, . |,2 &%!"&2/69&:zR(6!"MR2 ^| .i2i.U6|&2 ^565#B6;i,2 &$769U69W.MR(6!"iR2 &~2
y;C "!&
• 21 × 41^|
AR = δx/δy = 10
• 21 × 81^|
AR = δx/δy = 20
• 21 × 41^|
ARmax = δx/δymin = 5069&2;
ARmin = δx/δymax = 2^
&:71r&2!"H"%9g[2v";C,(69.zW/69 &2";+&+86!/4;¢69GR2"&:"; &c24, n ^ `2^ `2^N['eCRV"!";D352uB!&2!s+9gM2uP\]¬YLSL!/2#%u;"!86\6%2<6RV"!B/69AWV"!"#%"dW 4r8^ .4/9g_ &: &2!QB!&2!sB!869&&2WVzW(69.U69&2!"";dW:s2.^!"4RV /69&v9gD2#%2:;G6 [i2^|&\ & 69W. n ^ `2^ [C^SQ&2!"43 2@Pb69¥e:]2"@ #$R. ! M!/('#%@[&2M6Br::;k69. &(69 71@ &,2!/H!86"^ 0 &H2i2|(69&2;D3C2i n N n SLG!/2#%i#B69 &2zB!'&O5gm|69. ._6RV"!5/694;83
Z g`
ξ / π = η / π
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
BL-ALGS(1) schemeMF-SGS(10) schemeMF-PJ(40) scheme
M = 10-5 CFL = 106Rem = 0.25
Iterations
Log(
Resid
ual)
0 500 1000 1500 2000 2500 3000-12
-10
-8
-6
-4
-2
0
2BL-ALGS(1) schemeMF-SGS(10)MF-PJ(40) scheme
M = 10-5
Mesh : 301 x 61
Re = 15
]o 4, n ^ `2^ · @#$R(69&WV ^"& n N n SL Z M/!/2#%43¬P\]¬YLSL Z erM!/2#%B69&2;P<]¬q 8`fer!/2#%M69
M∞ = 10−5 69&2;Re = 15
^ n 'g¨Mj[U69r&(69.DR9c2.i9gt2M69#$R. ¥c(!869&Hgm6!8^` 4:i&:71r&2!"Ssv9g2i";C,(69.&H2ic2&2M4;D^
JQ5I n P\2:; ] n X/69&2 qKZ qKX qKqKqX[ Z [ Z n N n SL Z
106 ' m 75
8^0.17 7.9 · 10−5
P\]tq · er106 690 53
8^0.077 3.6 · 10−5
P<]¬YLpL Z er106 570
Y G0.065
Y ·10−5
n [2` Z n N n SL Z 106 Y
903^
0.7 8.4 · 10−5
P\]tq · er106 1700 578
8^0.34 4.1 · 10−5
P<]¬YLpL Z er106 1310
Y LG0.27
Y Y ·10−5
Zn4· a Z n N n SL Z 106 2780 4375
8^1.57 8.6 · 10−5
P\]tq F`fer106 X ' m
2383^
0.97 5.3 · 10−5
P<]¬YLpL Z er106 X ' m /m Y G
0.63Y ' ·10−5
69W. n ^ `2^ Z ±Q!",2T86;Cs \(^ &6~/69 4:!/(69&&2.569M∞ = 10−5 69&2;
Re = 15^H4,.U69
569"U69&4;+^| AR = 1
^JQ5I n 2G&:,#TWVB9gQRV &:8^qKZ2G #%v"0:, ";!"&:71r4^qKX|2S!"4|RVz /69& ^| .TqKqqKX|2S!"4zRVzRV &:zRVz /69&_^
QQ &<r::;b694"#%&: ^| <2T",. [9gK±&469" ^569/69& Z [ ^X&\;[d! !,#T71&:5,243^S6;;G2[ #$R. ! z869#%&:z9g¬2iWV,&2;269sG!"&2;C &2ThXxz|(69]^i;"! WV";v &
Z g c
!/(69R]`2^`ZQ&Cgh,&(69. s13C 5;21"5&2|&(69W.i2[#%2:;v!"&:71rQg>6I^|2&~!"&2/69&:zR(6!"R2$69G,2";D^kQ^7183?^|2&2v6RV!/69\!/24&,2!/6@<WVGW b!.4G<2,^569. .69&2;k#B69. .`gm69ig¨# 83D2%P<]¬YLSL\!/2#%+^| AXY|"!"71'/p6G0:, r::;b &O &2!$B!&2!s1^P\"7r83t,2!/6G869#%&:T;:"M&2M &2!86$2$!"4MRVM /69&9g2#%2:;D3_('&(!43t2Pb69¥e:]2"|!/2#%|!869&$#B69 &%7rs@!"#$RV 7r4^`Xx&@c24, n ^ `2^ `23^z69.[R"&O`25;C W,&9g(M71.:! xs~692S,.[9g2p!/(69&&2.oW/69 &2";<7r86!/\!/2";~4;D^2M&2S(6926!"!,/6!sd50:, Q([69#%gm571s%#%"H69&2;G(695 5.U69 71. sBRV:8^ 5.U6!/d9go6!"!,/6!sd&. sB;C,2Q/p2[#B69. .D69#%,&:|9g_RV &:z69.&@2
x;C "!&vW, I^56&22Q#B69 &vrr69. 9gt2
,2;CsG &~z"!&_^
Iterations
Log(
Resid
ual)
0 200 400 600 800-12
-10
-8
-6
-4
-2
0
2
BL-ALGS(1) schemeMF-SGS(10)MF-SGS(10) scheme with IBC
M = 10-5
Mesh : 21 x 41
Re = 15AR = 10
Iterations
Log(
Resid
ual)
0 500 1000 1500 2000-12
-10
-8
-6
-4
-2
0
2
BL-ALGS(1) schemeMF-SGS(10)MF-SGS(10) scheme with IBC
M = 10-5
Mesh : 21 x 81
Re = 15AR = 20
Iterations
Log(
Resid
ual)
0 500 1000 1500-12
-10
-8
-6
-4
-2
0
2
BL-ALGS(1) schemeMF-SGS(10)MF-SGS(10) scheme with IBC
M = 10-5
Mesh : 21 x 41
Re = 15ARmax = 50ARmin = 2
Velocity
Y
0 0.5 1 1.5-1
-0.5
0
0.5
1
Mesh (41 x 21) with AR = 10Mesh (81 x 21) with AR = 20Mesh (41 x 21) with ARmin= 2 and ARmax= 50Incompressible solution
]o 4, n ^ `2^ `2o5&:7rr&2!""¬9g2K";C,(69.tght2 n N n SL Z t!/2#%432P<]¬YLSL Z er!/2#%b69&2;¢2\P<]¬YLSL Z erB!/2#% ^| ¦69& #$R. !¥v869#%&:H9gS2uWV,&2;269s !"&2;C &2hXY||'^
M∞ = 10−5 69&2;Re = 15
^ RC n 'g¨%AR = 10
^ RC 4:$AR = 20
^S5# n 'gh%ARmax = 50
69&2;ARmin = 2
^T5#z 4:B[j[ W,&9g271'.:! xsk69i2$,.S9g2!/(69&&2.¬W/69 &2";u7r86!/u!/2";~4;D^
Z g$a
P<" P\2:; ] n Xx/69&2 qKZ qKX qKqKqX
21 × 41 n N n SL Z
106 Y 25
8^0.067 7.7 · 10−5
AR = 10P<]¬YLSL
(10)exp 106 720 178^
0.024X ·10−5
P<]¬YLSL(10)imp 106 590
' G0.024
X ·10−5
21 × 81 n N n SL Z
106 m/ G0.14 8.1 · 10−5
AR = 20P<]¬YLSL
(10)exp 106 1850 1018^
0.055Y X ·10−5
P<]¬YLSL(10)imp 106 1900 104
8^0.055
Y X ·10−5
21 × 41 n N n SL Z
106 1240 84^
0.068 7.9 · 10−5
ARmax = 50P<]¬YLSL
(10)exp 106 [ n [be67
8^0.024
X ·10−5
ARmin = 2P\]¬YLSL
(10)imp 106 6XH G0.024
X ·10−5
69W. n ^ `2^_[C±Q!",2T86;Cs \(^ &6~/69 4:!/(69&&2.569M∞ = 10−5 69&2;
Re = 15^H4,.U69
!/2";§4;:^| AR = 10
3AR = 20
^¯X/'4,2.69u!/2";§4; ^| ARmax = 50
69&2;ARmin = 2
^
)) 9<; 14&&#b !7$& #3); 2@Pb69¥e:]2"pg¨69#% ^5~Q. #$ ";\d!"#$R,D\(^z ^| \71s~!/2";\4;8^Q[^7183a &#%$9g#%#%s/69r43`2G#%2:;) . .|6u7469. ,(69W.v!/2!"v &2!"G @"01, "$&. s
60 9g
6;;C &(69.t#%#%s ^| ~"RV"!z$2M'eCR. ! !/2#% ^| .M2S.:!/v!/2#%M&2"";Q69WV,110
#%4^ ('/Mz2&<6$&%/;,2!&<9g2M#%#%sv"0:, #%&:0^|!/<!869&$*-,2¥ghsd2M,29g`,2!/u6#%2:;H%!"#$R,S7:/!",2 \(^z8^)) * &5.6 25!2#%&5!N[gm"!86";vW:sB2i±K&~JQ,#B69&&u69&(69. sC5 &~!/(69R0aC3C2iPb69¥eO]"M!/2#%"|R7O;[71sr::;v &: &2!MB!&2!sd/. 7r[.^5P\6!/H7O!",2I\(^z ^| H4; ^|24M6RV"!5/69469i!.4<,& xs1^ 0 &2v2(69&2;D3`WV"!869,2v9gQ2d,2/G9g2~LSL$q .U6e269&R:!"";C,432#%2:;WV"!"#%"% §62~6RV"!$/69b &2!86"B69&2;2v.^!"4RV$ /69&+ &2&:% &2$Pb69¥e:]2"g¨/69#%h^\M&2S, B!&:SvW(69.U69&2!"$2$.4M9g5B!&2!sr^ MR2W.# !869&WV[R(6969. . sv! !,#T71&:";H¥go&2M#$R.sO|WVu69&v #$R. ! z869#%&:9g¬2iWV,&2;269sv!"&2;C &269&2;b6%&2&C!"&2/69&:S6RV"![/69b¨xsOR!869. .¥suW G!.4/p2D^569. .69&2;\#B69. .Qg>69g¨# '^Q]([,2!/!"&2;C &283:2zPb69¥e:]2"z/!/2#%|"!"716i71sTr::;% &: &2!B!&2!s1^5";"3r2zP\69 eO]2"[ #$R. ! |#%2:;~&(69W."5@/;,2!"i;/69#B69!869.¥.sG2[#%#%sd"0:, #%&:]^|!/H#B691"52#%2:;~,. sG &:" &B$/. 7ri7O!",2 \(^z;"R p6RV"!z/69% /,2"8^
Z gbg
101 × 21 201 × 41 301 × 61
!"#$%!&! '$(! ) %"*$!&% )'+$(! ) !"*,$!&! )'+$(!- /.10 2) - /.10 2) - /.10 2
3546 '78$97:) 3.8 1 15.3 1 33.8 1
/<;>=@? 7:A 5.8 1.53 23.7 1.55 52.6 1.56
/<;,B B 7:) 6 1.58 24.5 1.6 54.3 1.61
.DCE;GF9C B 7:A 8.4 2.21 32.2 2.10 71.5 2.11
69W. n ^ `2^ · ~P\#%s/69rvgmB69. .z2H#%'2:;l^| +"RV"!%k2v4;+ ´84^ 2~'eOR. ! !/2#%M|/691&<6|2M'gh&2!"p%6|%769. ,(69S2i.U69 71S ´8"^
Z g n
#% ' ( (% ' ) )%A ; !$. #f#%3"7# &J[^.k,2A!"#$R,2 \(^ 71A69& &Cc2& \a69A4/! .¥.U69 &R.U69#%7O &69k27r.:! xsu(t) = u∞ cos(ωt)
^ @#%& &:71. 7r"86;Cs4/! . .U69&2@a69!869&)WVG!869.!,.U69";+69&(69. s:¥!869. . sb
u(y, t) = u∞ · exp(−y√
ω
2ν) · cos(ωt− y
√ω
2ν)
Xx&%;M&2&2;C #%&2&(69. ´82zRW.#~3O6i'gh&2!"z.&4d!869.|(6KMWVz;'c2&2";D^ p6!/7r832&2S!869&H,2/i2M7r.:! xs
u∞69&2;H2MR,./69&
ω
L∞ =u∞ω
2&~,2 &%2igh. .^| &dsO&2.;|&:,#TWVQ;'c2& &_3Re =
u∞L∞
ν=u2∞
ν ω
32&2MW/69 &2T
u(y, t) =u(y, t)
u∞= exp(−y
√Re
2) · cos(t− y
√Re
2)
z!869&G;'c2&22z!/O&2"δ
9gD2.U68sr62zRV &:5^|227r.:! xsB69#$R. ,2;Q(6;CRRV"; Z RV!"&:Q9g
u∞^ :,283Cgh|2S;C #%&2&."/zRW.#~3H^SW/69 &
exp(−δ√Re
2) = 0.01 = e−4.6
δ ≈ 3.5 ·√Re
2S2&\!/2:4@6%/01,(69";<!"#$R,/69&(69.`;#B69 &b,2!/\62TD"!Q9gK2T4! . .U69 &GR.U69@69&24. 4 W.T69z2MR~9g2S;#B69 &_^CAMxs:R!869. . sG69r%
L = H ≥ 2 · δLO &d2TsO&2.;&:,#TWVi69
10.86;Qd6$!/O&2"Q9g
δ = 23(l^T!"69&\!/2:4
L = H = 4^
2S!"#$R,/69&(69.o;#B69 &u|2'ghp!/24&u0:,(69";~69&2;H z|R"&:";~ &vc24, n ^ c ^ Z ^LO¦6\;#$ &226!"!,/6!s9gG86!/#%2:;83 ^)!869&!"#$R,)2O &gh!&!":'g¨c(!&:i^|2 !/<!/(69/6! ´8"2 \(^
Cw =2 · τwρ · u2
∞
=2 · µ
ρ · u∞ · L∞
· ∂u∂y
∣∣∣∣w
=2
Re· ∂u∂y
∣∣∣∣w 2p69&(69. sO!"69.t7469. ,2M9g!":B!'&:Q|4 71&HW:sb
Cw =2√Re
· sin(t− π
4)
Z g d
X
Y
0 1 2 3 40
1
2
3
4
]o 4, n ^ c ^ Z #$R,/69&(6.`;#B69 &<gm2L:1"8=a"!"&2;<RW.#~^ 2T#%"<R"&:";\2|2S!"r69S&24^ 2S4! . .U69 &Q^569. ._69z2MWV#~^
2%6!"!,/6!sk9g56G&:,#%!869.. ,&^5,.;bWV$7469. ,(69";k,4A2$S;7OU69&L^|!/k;'c2&2";<65gh. .^zp
I`=
√√√√ 1
2π
2π∑
0
(Cw − Cnum)2
^|2M2M&:,#%!869.¬: &vgh!&u!":B!&:z!869.!,.U69";~,2 &B6$"!"&2;O;zgm#T,.U6H
Cnum =2
Re· −u(jw + 2) + 4u(jw + 1) − 3u(jw)
2(y(jw + 1) − y(jw))
^| jw
2M769. ,2S9g2SR(6!"M!",&OQ69z2 ^569. .>^J[&C;C #%&2&(69.H & U69.H7469. ,2"69/691&6gm. . ^z¦
M∞ = 10−3 3Re = 10
3ρ0 = 1
3u0 = 0
3v0 = 0
3P0 = 1/γM2
∞
^ 2<&29. R§!"&2;C &¦v69RR. ";§69G2<#%7O & ^569. . ^| .2v &2!"#$R" W.u!"&2;C &2d69H!"&2;";gm%69. .|2H2%WV,&2;269"83(69$ ^v #$RV4/2G/69!dR",v69&2;W^5d'eC/69RV.U69G2d;&2 Ys69&(;2d71.:!¥Ysr^\X&);@\!"#$R,G2,&286;Cs \(^z83 ^~,2/<6A;C,(69.¥> #%'RR&gh/69#%h^ 3 ^|2 &2uR:sO!869.[; 769 71"v69#$R.s1";~$gh. .^2SR:sC!869.t/69&2&:[69&2;~2SR2,2;9>¥#%@; 769 71"Q71p6Q69&u /69 7r;7O!"4^@#$R,/69&2T69;&2,2 &v ^v;C D&:M4;83 &(69#%. sb6G!"r69&2H
21 × 81Q69&2;A6
c2&2S&2B41 × 161
'^) )@ ; .7b# -; !'&)( 7=; 7$; !7./!;Xx&B!/(69R gO3f^,. &2";G6i&2h^ ;'c2& &G9gD2z,&286;Cs.^|P\6!/dR(69/69#%8^Vz(^";%(69,2!/\6$;'c2& &\&(69W."Q2MR"!"&2;C &(";\!/2#%S$R7:;Sr::;u;269#$R &BgmzWV\6!",2!69&2;d#B69U69. RW.#%8^`Xx&dR(69/69469R_3^ ^569&:T!"&Cc2#®(692Q,&286;Cs%R"!"&2;C &(";!/2#%571/69 .|&2,4$ghK7rsTxsORV59gVRW.#%8^HI^| . .2,2;Cs69.i25RV'gh#B69&2!""9gV2
Zn e
#$R. ! M#%2:;M(69 ^@(6871$69. 86;Cs\#$R.s1";bv!"#$R,$86;Cs:\(^z8^ 2@±&bJ[,#B69&&69&(69. sC#$R(6 ´8";G(695Pb69¥e:]2"[!/2#%"52,.;dWVQ71s%!"#$RV 71i!"#$R(69QT2[.1!/!/2#%4^) )*% L5!7b;H>" 3 #; ./&5># 7# &5# 3Xx&k!/(69R gO3)^$,. &2";6G&2h^l;'c2& &9g2,&286;Cs\.^5Pb6!/AR(69/69#% ^z!/A&(69W.",2|B687r;H2M!/OsG!/2!"p9g`6RRVz.&4<!869.4^ Ai/!869. .t|&2h^;'c2& &
βu = min
[δx ·
√CFLu+c
∆t · c · π ,δy ·
√CFLv+c
∆t · c · π
]
2&_32M.^5Pb6!/~R(69/69#%Qz!.U6!869. . sv;'c2&2";<65gm. .^zS
β = min [max(βs, βu), 1]
^|2βs
S2B/"6;sb.^5P\6!/R(69/69#%8^HXx&ApR(69/694/69R_3,TWC*"! 7rBSu#$R(6 ´82$r::;!"&:71'r'&(!B/69$R7O;";AW:s\2$,&286;CsbR"!"&2;C &28;!/2#%gmS71s\R:sC!869. #%'R_^)LOk6\,2;CsA RRV"83^G!"&2;2vP\]¬YLSL Z erT!/2#% ^| +6uR2/,2;] n &:,#TWVSS69
106 ^A@!"#$R,2$&2R:sO!869. /69&A71i2@c2&2@4;A69&2; ^5$69. &(6 71. s%!/2:42Q&29>R"!"&2;C &28;v!/2#%43:2[86;Cs$R"!"&2;C &28;v!/2#%[69&2;d2,&286;CsR"!"&2;C &2"; !/2#%4^ 2H,WC> /69& !"&:7rr&2!"~%,2;C";gm%7r/69.z #%'R28^ 2 #%'Rb!869&<WVp!/(69/6! ´8";\W:s
CFLu = u∆t/δx69&2;D3a;869. . sr3(gmQB!&:[ #%'>#B69!/ &H9
. ,&_32Q] n &:,#TWVQ2,.;HWVS9g;z,2& xsr^]o 4, n ^ c ^_[k2^z$2H",. $gm
M∞ = 0.00169&2;
Re = 103569&2;gmgm,$7469. ,2"%9g
CFLu^
N|,2!/+6\RV"";D3`gh.^¯7469. ,2"9gCFLu
2v86;CsR"!"&2;C &2B;:"&2%!"&:7rrv &2;C,(69.` /69&_^ 2,&286;Cs\R"!"&2;C &2T,RV-gm#%S2@&(9&>R"!"&2;C &(";!86$7r&bghCFLu = 10−3 ^BN|
CFLu = 13t2%&2&C>R"!"&2;C 4&2";)!86%;C 71r" ^| .%2B86;CsbR"!"&C
;C &2pR7O;"T6H0:, $r:1;!"&:71r&2!"%/694^dQ^7183t2$,&286;CsbR"!"&2;C &2Tp . .2%#%4TB!&:8^HN|p71& 42
CFLu = 103t2B86;CsbR"!"&2;C &2@RV-gm#%@6 ^. .6
,&286;CsG!",&:R(698^X:,25!.869|(69 &~69. . !86/"2Q,&286;CsBR"!"&2;C & &2$R7O;"R #B69._RV-gm#B69&2!"S9go2iR2,2;9> #%p!"&:7rr&2!"4^ 2"i",. 69i &Hr1:;u694"#%&:i^z¥2M±K&uJQ,#B69&&\69&(69. sO/"z9g`!/(69R gO^ 2;C,(69.¥> /69&%!"&:71'r'&(!¬ #$RV/69&OWV"!869,2 o;C 71"¬256!"!,/6!sT9g22. ,&v¬6&2-!"&:71r";HR:!""i^,.;H.86;HB69&~ &(6!"!,/69p. ,&_^KXx&HR/6!!"432 |z&2|&2"!""//69sG86!/#B6!/ &2p´8d6986!/uR:sC!869._ #%'R_3 ^S2'ghT!/2:4p6%!"/69 &u.71.t9g`";C,(69.t$86!/~6M~'&(,$B!&2!sA69&2;6!"!,/6!s1^ 0 g5!",/43¬6!"!,/6!sb. "p69./H&k2$&:,#%!869.;C¥R(69&_3O2&2!"4326S&29>R"!"&2;C &2";H/!/2#% ^,.;dr&2/69i'e!"" 71[69¥c(!U69.D;C R(69&v.86;C &v6Q^|&v/. ,&_^SL:,2!/k69&\/,2T!869&k69.G:!"!, ^|2&b&2T,("Q2T,&286;CsuR"!"&2;C &&^| 6u#B69. . #%'R_ &2;"";D3 &)@!"&Cc24,/69&t3`2d,&286;CsAR"!"&2;C &2 &k @!.4/B2Q&29>R"!"&2;C &(";H!86[69&2;d !869&G2'gmQR7O;i69&G &(6!"!,/69i. ,&_^`[^71831gm2RW.#(69 ^$69@ &:"";b &_3 2@ #%'R2M!""RV&2;b
CFLu ≈ 13 :,2i2,&286;Cs
R"!"&2;C &2";~!/2#%['eCRV"!";GTWVi6|6!"!,/69M62[86;Cs%R"!"&2;C &28;H&24^VQ!"#R,";u2p86;Cs~4! . .U69&2[;C, & `BRV:;[7r2T!"r69S4;~ghQ86!/u#%2:;D^AS,2";<6 #%'R~"0:,(69.
π/20^|!/~!""RV&2;|
CFLu ≈ π/4^`]o 4, n ^ c ^ · 2^z52i71. ,&~9g
2zO &g¨!&B!":B!&:ghK2|.U6`RV:;B!"#$R,";d69&2;$2zK#B6;zW:s86!/%#%2:;B7r2p/69#%MRV:;D^ASW/69 &<2S69#%p6!"!,/6!s'^| uWV<86;Cs~69&2;~,&286;CsvR"!"&2;C &&"!/&0:,2"GT2B@!"#$#$ ";W:sA86!//!/2#%%T9gz2B;@9gz2%RV!"&:&2d!"r69
Zn/Z
Dual iterations
Log(
Resid
ual)
0 50 100 150 200-12
-10
-8
-6
-4
-2
0
Unsteady preconditioningSteady preconditioningNo preconditioning
M = 0.001 Re = 10 CFLu = 0.001
Mesh : 41 x 161
Dual iterations
Log(
Resid
ual)
100 200 300 400-12
-10
-8
-6
-4
-2
0
Unsteady preconditioningSteady preconditioningNo preconditioning
M = 0.001 Re = 10 CFLu = 0.01
Mesh : 41 x 161
Dual iterations
Log(
Resid
ual)
100 200 300 400
-12
-10
-8
-6
-4
-2
0
Unsteady preconditioningSteady preconditioning
M = 0.001 Re = 10 CFLu = 1
Mesh : 41 x 161
Dual iterations
Log(
Resid
ual)
0 500 1000-12
-10
-8
-6
-4
-2
0
Unsteady preconditioningSteady preconditioning
M = 0.001 Re = 10 CFLu = 10
Mesh : 41 x 161
]o 4, n ^ c ^_[C`#$R(69&29gV";C,(69.2!"&:71r&2!"]^| B69&2; ^| 2,K,&286;Cs@R"!"&2;C & &@69M∞ = 10−3 69&2;
Re = 1069&2;d&B2|c2&24;\
41×161'^ RC n 'g¨i
CFLu = 10−3 ^ RCz 4:[CFLu = 10−2 ^`5# n 'ghT
CFLu = 1^`5#z 4:T
CFLu = 10^
4;D^ 2$,&286;Cs<R2"!&(;C & &<&(69W."SH!869.RRV. s\2%69¥c(!69.;C R(69&kghS,2!/6$!86/^5";"83a6z R7:;"71sdg>6!"&:7rr&2!" ^ ^| . .¬#$R.sH |gmz2S&2'eOQ!"#$R(69&2WV ^"&<P\69¥e:]2"M#%2:;<69&2;~.:!/v/!/2#%4^
) ) * &143,#f# !2&! '%&?. / !$.f=; 14; 9< 7#f# #;H; !2.M="; 14;N["^z#%&:&2";d869. 83:R/6!!869.V!"#$R,/69 &(;S&2"0:, zS86!/d#B6!/ &2´8S &BR2,2;9!"&:71r&2!"p69z86!/~R:sC!869.D #%'R_^Xx5z!,2#B69sG$/69rQ6!"/69 &u;9g!"&:71r&2!"WV'gm%.8687: &<2B;C,(69.¥> #%B.:R &2%R4/69#~^HXx&AT"!&_3V^l^569&:T~#$R(6 ´8B(692!/2!"9g2;M9g!"&:71r&2!"[2@#B69 &bR(69/69#% ^|!/A;C 71i2B!&2!s\9g,Pb69¥e:]2"% #$R. !¥@!/2#%"8^dX&kc24, n ^ c ^ `23^$R"/&:S2%;,(69.¥> #%B!"&:71r&2!"%sb9g
Zn [
t / π
skin
frict
ion
coef
ficie
nt
0 0.5 1 1.5 2
-0.5
0
0.5
Unsteady preconditioningSteady preconditioningAnalytical solution
M = 0.001
Re = 10
∆t = π/20
Mesh : 21 x 81
qK"!"&2;C & &2 I`L:86;Cs
1.17 · 10−2
Z&286;Cs1.17 · 10−2
]o 4, n ^ c ^ · n 'g¨MKL:: &dgh!&H!":B! &Ogm2Q.U6RV:;v!"#$R,";D^M∞ = 10−3 3
Re = 103
∆t = π/2069&2;H#%"
21 × 81^` 4:S 69W.p,#$#B69 ´" &B2S|gmz86!/~#%2:;D^
2i";C,(69.t6g¨z&2iR:sO!869.D /69&Hgh| n N n SL Z 5!/2#%43P\]¬YLSL Z er!/('#%S69&2;HP\]¬q · erz!/2#%p69
M∞ = 10−3 3Re = 10
3∆t = π/20
69&2;~&~2[c(&2i4;41 × 161
'^
Dual iterations
Log(
Resid
ual)
100 200 300 400-12
-10
-8
-6
-4
-2
0
BL-ALGS(1) schemeMF-SGS(10) schemeMF-PJ(30) scheme
M = 0.001 Re = 10 ∆t = π/20
Mesh : 41 x 161
CFLτ = 106 Unsteady preconditioning";C,(69. P\2:; jQ,(69.tXx/69&2
n n N n SL Z / mP<]¬YLSL Z er
180
P<]¬q · er180
` n N n SL Z ' mP<]¬YLSL Z er
60
P<]¬q · er60
]o 4, n ^ c ^ `2 n 'ghM`j[,(69.¥> #%S!"&:71r&2!"MsG9go2i";C,(69.t6ghz&2[ROsO!869.D /69&<69M∞ = 10−3 3
Re = 103∆t = π/20
69&2;k&k2Tc2&24;D^T 4:$ 69W.$2^| &H2%69#%,&:S9g;C,(69.¥> /69&("01, ";~86!/\6!"/69 &<;9g!"&O7rr&2!"Mgmz86!/~#%2:;D^W271(69pP\69¥eO]2"$!/2#%"S"01, B69WV,
2.7 #%"T6M#B69&:s< /69&2@6i2%.:!/
!/2#%BH86!/#B6!/ &2B´8^ 0 &2B2T(69&2;D3¬2 Sc24,d;"!86"T2
~86!/ n ;!"&:71r&2!"A69&2;§69WV,
4/3)86!/ `/;~!"&:7rr&2!"4^[&2!"43[6!"!";C &+)2b;~9g
!"&:71r&2!"i^|!/B`/69r";D3:2|RV-gh#B69&2!"Q9g 2zP\69¥e$]2"z!/2#%"!869&%WV|71s;C &:8^ d#$R(6 ´8T(6983 ^5T!"#$R,@,2!"!"" 71. svgh,QRV:;i9gK2T86;Cs~4/! .¥.U69&2Qgm[;C &:
Zn4·
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106 Y / 8265
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106 mHYXH3075
8^0.58 8.8 · 10−5
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^|2m
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v¨"RV"! 71.¥s FVi,k
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1
Ωi
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k
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i − wni ) − 1
2∆wn−1i
∆t
d ^_[C^ `:
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1
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1
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),
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1
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(∂w
∂~n
)
i,k
+A~t,i,k ·(∂w
∂~t
)
i,k
≈ A~n,i,k ·(∂w
∂~n
)
i,k
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∂~n
)
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) d ^_[C^ n ^|2M2M#B69¥e
QVi,k
z;'c2&2";u6|gh. .^zp
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1
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i,k
g¨#m
m+ 1
sO. ;2S
∑
k
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i,k
)n,m · ~ni,kSi,k = −(∑
k
(QV
i,k
)n,m∆(wi − wo(i,k)
)n,mSi,k
).
d ^_[C^ d
Zd/Z
]o &(69. . sr3r &2/ &d d ^_[C^ g4`69&2;u d ^_[C^ d &:M2 n iL9gD"01,(69&< d ^_[C^ · '3f^5c2&2;$2| #$R. !¥/!/2#%gm|,&2,2!,";~4;p
Dn,mi · ∆wn,m
i +
(1
2Ωi
∑
k
(∆fE
o(i,k)
)n,m· ~ni,k Si,k
)
−(
1
Ωi
∑
k
(QV +
1
2QE
)n,m
i,k
· ∆wn,mo(i,k) Si,k
)= −Rn,m
i
d ^_[C^ Z er
^|
Dn,mi =
(1
∆τP−1
)n,m
i
+3
2∆tId +
(1
Ωi
∑
k
(QV +
1
2QE
)n,m
i,k
Si,k
)
Xx&b2@R(69!,.U69T!869g2B569"U69&k4;3_&2@"!"71M2&:,#%!869.!/2#%$;/69 .";k &k2c2zR(699go2M2"8^
" #1 & # #1 " # # 2P\69¥eO]2"`g¨/69#%h^S!869&SWV/69 4:-gm ^569;C. sp69RR. ";Sz2`,&(,2!,";T!/2#%Q d ^_[C^ Z er'^]o 83/^MR:!""";~2MRV!69._/6;C ,2 #$R. ¥c(!869&<&~2S;C R(69 71M#%z9g2 n ML\
QEi,k ≈ ρ(PAE)i,k · P−1
i,k = ρEi,k · P−1
i,k
QVi,k ≈ ρ(QV
i,k) · Id = ρVi,k · Id
2M,&2,2!,";u/!/2#%M2&~WV"!"#%"p
Dn,mi ·∆wn,m
i +
(1
2Ωi
∑
k
(∆fE
o(i,k)
)n,m· ~ni,k Si,k
)−(
1
Ωi
∑
k
Cn,mi,k · ∆wn,m
o(i,k) Si,k
)= −Rn,m
i
^|
Dn,mi =
(1
∆τP−1
)n,m
i
+3
2∆tId +
(1
Ωi
∑
k
(ρV Id+
1
2ρEP−1
)n,m
i,k
Si,k
),
Cn,mi,k =
(ρV Id+
1
2ρEP−1
)n,m
i,k
.
Xx&d;SR/1;,2!"[6pP\69¥eO]2" #$R. ! #%2:;D3C69. .a2&2&C;CU69r&(69. #%569.6e";B &d2#B69&&2Q9go2Sqo &: 6!"WtR:!"";C,S$s:.;A
Dn,mi ·∆w(l+1)
i = −Rn,mi −
(1
2Ωi
∑
k
(∆fE
o(i,k)
)(l)· ~ni,k Si,k
)+
(1
Ωi
∑
k
Cn,mi,k · ∆w(l)
o(i,k) Si,k
)
d ^ · ^ Z
Zd [
Xx&%2R(69!,.U695!869gD2Q!"#$R" W.i\(^z31 &Q^|!/d2R"!"&2;C & &&2"0:, ";D3:2R"!"&2;C &2
P"01,(69._(i;&: xsv#B69¥e~(69z2S!":B!&:
D69&2;
C";C,2!"SH
Dn,mi = (an,m
i + bn,mi ) Id ,
Cn,mi,k =
(ρV +
1
2ρE
)n,m
i,k
,
^|
an,mi =
(1
∆τn,mi
+1
2Ωi
∑
k
(ρEi,k)
n,m Si,k
) 69&2;bn,mi =
(3
2∆t+
1
Ωi
∑
k
(ρVi,k)
n,m Si,k
).
:,283M2A,&2,2!,";!/2#% d ^ · ^ Z H~,. sP\69¥eO]2"69&2; <;:"u&2u"0:, A&2 2#B69¥e &:7r&&2u#B69 e/69r4^ [^7r83 ^|2& ^!"#$R,2.^5P\6!/&:,#TWV \(^z832AR/!"&2;¥ &2 &#B69 e;C ~gh# 2A;&:¥Ysr^ &2"0:,2&:. s13M2;CU69r&(69.@!":B!&:D
WV"!"#%"\6g¨,. .S#B69¥eWVk &:71";D3p69&2;C
6969.W.:!/§!":B!&:83p(69</69r"0:, #%&:[69&2;uRV/69&2[!",&:Q9g2S #$R. ! 869#%&:[ 4&¥c(!869&:. sH &2!864^JQ712."836S2^|&k &k2B569"U69&!8643D MMRV4 W.v/691$6;C7469&:/69r%9g2RRV"p9g
PG1"R
2MR"!"&2;C &2";\69RRr6!/\6z #$R.S6z z!"#$R" W. \(^71&_^Xx&¯2¦c2+!/(69R83u2 ,1.HR"!"&2;C &2" ^| & &!"&2/769 71¦7469U69W." ^|6R'&:";A
P = Id+ (β2 − 1) ·Q^|
Q =γ − 1
c2
q2 = 12(u2 + v2) −u −v 1
uq2 −u2 −uv u
vq2 −uv −v2 v
Hq2 −uH −vH H
=
γ − 1
c2
1
u
v
H
·
(q2 ; −u ; −v ; 1
)
2$#B69¥eQ
S;#$RV&:83 Q2 = Q
3t2&2!"43¬2$R"!"&2;C & &2<#B69¥eA!869&AWV% &:7r";'eCR. !¥.¥sk
P−1 = Id+ (1
β2− 1) ·Q
Xx&$;KM'eOR. K`RRVxs13r &$25#B69¥eDn,m
i
3b^|769. ,(6969. .25#B69!""P
69`2|!"&:9g2
i>u!". .>3:,2836^MW/69 &
Dn,mi = an,m
i
(P−1
)n,m
i+ bn,m
i Id
2&_32M#B69¥eH| &:7r/;~ &H2S/69#% ^568sv9go2p569"U69&<!86%
(Dn,mi )
−1=
(1
a+ b
(Id+
a(β2 − 1
)
a+ bβ2Q
))n,m
i
Zd4·
5";"83( &\!":B!&:C
3 ^p!869&k69.B7469. ,(69T2p#B69¥eP
69Q2T!"&:[9gK2i>\!". .V^|!/
.86;z([gm. .^| &B'eCR"&
∆w(l+1)i = (Dn,m
i )−1 ·
[−Rn,m
i − 1
Ωi
∑
k
1
2
(∆fE
o(i,k)
)(l)· ~ni,k − (ρV
i,k)n,m · ∆w(l)
o(i,k)
Si,k
]
+ (Dn,mi )
−1 · (Pn,mi )
−1 ·[
1
2Ωi
∑
k
ρ Ei,k · ∆w(l)
o(i,k) · Si,k
]
d ^ · ^_[rN[|(^|&H &u!/(69R0`23C2MR:;C,2!
D−1 · P−1 z'eOR. ! . sv4 7r&~W:sb
(Dn,mi )
−1 · (Pn,mi )
−1=
(1
a+ b
(Id+
b(β2 − 1
)
a+ bβ2Q
))n,m
i[&2!"432M'eCR"& d ^ · ^_[r86;p
∆w(l+1)i =
1
(a+ b)n,mi
[∆w
(l)1 + ∆w
(l)2
]i
+(β2)n,m
i − 1
((a+ b)(a+ bβ2))n,mi
Qn,mi
[a∆w
(l)1 + b∆w
(l)2
]i
d ^ · ^ ·
^|
(∆w
(l)1
)i= −Rn,m
i − 1
Ωi
∑
k
1
2
(∆fE
o(i,k)
)(l)· ~ni,k − (ρV
i,k)n,m · ∆w(l)
o(i,k)
Si,k
(∆w
(l)2
)i=
1
2Ωi
∑
k
ρ Ei,k · ∆w(l)
o(i,k) · Si,k
#$R. ! Q869#%&:i&(69W."/%;"!",R.T&!". sH2p/69&2;269;<P\69¥eO]2"S #$R. ! Q869#%&:,2";dgm5!"#$R"/ W. \(^l
β = 1Kg¨#¯2M6;;";v869#%&:RV"!¥c(!i@.^5P\6!/H&:,#TWV \a^z
β 6= 1
'^oP\"7183 &25#B69&&2K9ga25!/(69R?`232869#%&:Q d ^ · ^ · ¬WV"!"#%"?^|2. . s@P\69¥eO]2"[¥gt2iRV"!¥c(!i'eCR"&H9gt#B69¥e
Q5#$R.s1";G!"#$R,S6p#B69¥eO>7r"!|R:;C,2!z,2!/
6Q ·X ,&2;2[gm#
Q ·X =γ − 1
c2
(q2X(1) − uX(2) − vX(3) +X(4)
)·
1
u
v
H
^| X(m) 2[!"#$RV&2&:|9gt7r"!
X^IK7r&:,(69. . s13
∆w(l+1)i
!869&GWV[!"#$R,";H69|6S.^¦!"4|69&2;^| <6@/;,2!";~#%#%sH/69rM,2 &$2igm. .^| &B'eCR"&
∆w(l+1)i =
1
(a+ b)n,mi
[∆w
(l)1 + ∆w
(l)2
]i+ ∆φ
(l)i ·
1
u
v
H
n,m
i
d ^ · ^ `:
Zd `
^|2M2M &2!#%&:∆φ
(l)i
|4 7r&~W:sb
∆φ(l)i = χn,m
i
[q2(a∆w
(1)1 + b∆w
(1)2
)− u
(a∆w
(2)1 + b∆w
(2)2
)
−v(a∆w
(3)1 + b∆w
(3)2
)+(a∆w
(4)1 + b∆w
(4)2
)](l)i^|
χn,mi =
[γ − 1
c2· (β2) − 1
(a+ b)(a+ bβ2)
]n,m
i
.
' & (1& :,283D2Pb69¥eO]2"@ #$R. ! Mgh/69#%h^\'eC&2;M86 . s<G2,&2,2!,";k4;\gh#T,.U69&_^ 2Bgm# 9g2Bc2&(69.|69. r #³#B69 &2@2G/69#%d(69&gmT2H569"U69&!864^ n @,2T&2!"(69%2vR7O,2B69. r # !869&WV~!",R.";¢69. ^| +2uL:sO#$#%!<M69,2YLO;.|.U6e269&R:!"";C,435^|!/%'eORV"!";\ #$R71v2H!"&:71r&2!"H/69H9gQ2v#%2:;D^J['eO$!/(69R69M;";C!869";H2i769. ;269&u9go2iPb69¥eO]2"i"!/&01,2M(69|(6871[WV"&H #$R.#%&:";~ &H2NML · P !":;4^<A ^| . .!"#$R(69$ ^| k"RV!SH2$J[h^|&C«is:.7A69. r # 69&2; ^+^| . .69.%!869sv,!"#$R(69&20^| ~2p6s:#$R!SR",'>W(6";u. 7rz9gNML · P^~
` ¯ ¯ ) i u b
#1 +#% 2TJQh^|&C«isO.7u69R2Rr6!/_3 ^|!/<(6WV"&uc2. sv #$R.#%&:";\ &u2@NML · P³!":;43(2"9!869. . sd69. .^zT!"#$R,[/69&(69sB. ,&25^| 2,5 #%QR2"!&G. &1";Bp2M] n!"&2;C &_^Q^71835 B"01, "d!"#$R, &69&2; & 6!"WU69&#B69!""v69&2;D3|!"&2"0:,2&:. sr3^|2&u#%"2"0^| \6$:,rS&:,#TWV[9gK.#%&:i69S,2";D32#%#%s~/69rpWV"!"#%"[69&u #$RV/69&:,24^ 2~P\69 eO]2"H #$R. ! B869#%&: ^|!/ ^56$R"/&:"; &+2HR/'7O,2B!/(69RB(6WV"&k;7r.RV";\d";C,2!"@2T#%#%s\/69r69&2;\ QM69.d2RV";\(69[W:su #$R. ghsO &G2T"-. ,&_3o169 &2T &qZ!869&)69.uWVd6!/71;_^ $(69RTT;71";~2%7469. ;269&)9gz2&2h^|. sd¥#$R2. #%&:/";H#%'2:;G &uNML · P 71][jQ · jI`,.|"-!86"^ 2[R#$ &$RRV"9gD2P<]k!/('#%"^|!/ ^'eC W ";B &%2[569"U69&B"69'eORV"!";BSWVzR"7r";d7r,&2,2!,";~4 ;28^
& " (%& A )F@)%A &!$# 7# & &)( 7=; 3 #& I ; 1NQ&% &:7:/!; \(^¢ &G6S!/(69&&2.a !"#$R,";D^ (zr"#%'s%9g 2!/(69&&2.V!"&29gt6i"!/69&4,.U69;#B69 &
(x, y) ∈ [0; 4] × [0; 1]3i^| 6/69 4:v,RRV ^569. .S69&2;§6!,71"; .^5 ^569. .>30^|24
!":;C &(69"[69M4 71&HW:sb
x < 1 y = 0
1 ≥ x ≥ 3 y = 0.1 (1 − cos[(x− 1)π])
x > 3 y = 0
](,z 4,.U694;Q69M#$R.s1";H%!869sv,z2S,2;C"p
• 6$!"r69M4;'^| n g n U69&4,.U69Q.#%&:T c24, Z eC^_[C^ Z '• 6#%";C ,#4;'^| · `fa n U69&4,.U69Q.#%&:8
Zdbc
Zd a
]o 4, Z eC^_[C^ Z AL:/69&(69s \(^71G6A &2uW,#$R_^§Xx4,.U69G!"r69u#%" ^| 878
U69&4,.U69.#%&:8^
• 6@c2&2i4 ;:^| g nbdbn U69&4,.U69Q.#%&:8
• 67rsdc2&2M4;'^| Z ` Z e2`%U69&4,.U69Q.#%&:8^](Q86!/<4;D3(2pI`,.i"01,(69&2i69T. 71";Hgh[69&<;869.¬1683 ^|
γ = 1.4^|](
M∞ ≤ 0.52
\(^#B69 &2[,W2&!43V69&2;~2&2!"S&:R2 !@69&2;~ /69&(69.|¨WV"!869,2Sgmz 42[P\6!/u&:,#TWV&2Br@6H/69&2/&!l\(^ ^| 6~2:!/2'369&2;A2BPb6!/. &2"69deCRV"!";uWVB!"#$R.. ssO#$#%!869./^| "RV"!Q2r"#%sv RV&:U69.\(^Q^oJ[&C;C #%&2&(69.7469. ,2"`69/691&65gm. .^zS
ρ0 = 1.43u0 = M∞
3v0 = 0
3P0 = 1
^L:,W2&!MWV,&2;269sH!"&2;C &2[69S!"&2;";
• 6952i &.'p¨.'gh^[ #$RV4i2i'eC&(69. /69._&:(69. R:s13C2i'eC&(69._&:R:sv69&2;G2 \(^69&4. ^| . ^S'eO/69RV.U69M2iR"/,[gh# &2;4
• 69p2B,.G¨ 4: ^%¥#$RV4/%2B/69!%R",l^| .l^B'eO/69RV.U69B2B;&2 xs69&2;2M7r.:! xsdgh# &2;4
• 69z2MRu69&(;~69z2MWV#~3H^i #$RV4/ ^|69.¥.tWV,&2;269sH!"&2;C &28^
A )F@)@ 2 #3]&V!; "S;C D&:z769. ,2"z9go2M &.zPb6!/H&:,#TWVQ69/M!"&2;";D32&(69#%. s
M∞ = 0.13M∞ = 10−469&2;
M = 0.5^ 0 ,R,RV4M|%769. ,(69M &~#%9gqKZ #%M2igm. .^| &d69RRr6!/2"T
• 2qo &: 6!"WOP\69¥eO]2"K#%2:; ^| α
&&2-> /69&2o(69^?^| . .:;&2P<]¬q α
'
• 2L:s:#$#%!S69,2-YLO;.:P\69¥eO]2"`#%2:; ^| α
&&2-> /69&(¬(69)^?^| . .:;&2P<]¬YLSL
α'
• 2J[h^|&C«is:.7@69. r #iP<QIL ^| X n Z R"!"&2;C &23r«isO.7pR(6!";C #%&2&"01,(69.O
50 ghV^|!/ ^!"&2;o(69o2!"&:71r&2!"9g(2 /69 71/!/2#%Kgm¬2. &2869
sO# Q6!/7r";u&2!"M2i. &2869";C,,#|.^z(69&10−α mJQ«%
α'^
i;C "!. sv. 71[2M/69&(69svI`,."01,(69&269&2;'^[,2/[2MN|ZMLOP hq!/2#%i &::;C,2!"";W:s\IK;/^569; n , _[ c 69&2;k!",R.";^| 6G&2&C>. #$ ";A. &2869. sb'e6!M"!"&2,2!&_^%N. .`2gm. .^| &B!"#$R,/69&2 ^MRV-gh#%";~,2 &BX&:/'.i"& · i´ ^| [Bi¦N[P P<#%s1^
Zd g
A )F@)*% , 214; #f# ./- #; ! 25 7! 7M = 0.1
N|¬,2!/6P\6!/T&:,#TWV832NzZiLOP hq_!/2#%R7:;"`6|71sM!""!/.¥,29& &2!"4346o&2!869&"i &~2ic24, Z eC^_[C^ · 322M. &2"9go2SPb6!/~W/69 &2";u&~2ic2&2"z4;H#B69 &usO#$#%!869.^| ~"RV"!z$2S &2W,#$R_^
0.085 0.1 0.115 0.13 0.145∆ M = 0.0015
]o 4, Z eC^_[C^_[CvL:/69&(69sW\(^¯71$6u &2BW,#$R+69M∞ = 0.1
^\XY. &2"9g2GPb6!/)&:,#TWVW/69 &2";u&~2[c2&(|4;D^Xx&k;MGRV-gm#!"#$R(69&2SWV ^"&A2$;C &:S#%2:;S;"! WV";A869. 83_2$! &^|!/b[,2";<G;#$ &22!"&:7rr&2!"/69@c2i;'c2&2";D^ T/69 &b2
L2&2#9g2
";C,(69.
Res(n) =
√√√√NBEL∑
i=1
(∆wni )2
√√√√NBEL∑
i=1
(∆w1
i
)2
^|2 J[5I n T[2@&:,#TWVM9g.#%&:p69&2;w
;&2"M 2M2@;&2 xs13V2@#%#%&:,#2S&24sr^Xx&/69W. Z eC^_[C^ Z 3^BR"&:@2B",. T(69 ^dW/69 &2"; ^| )2"d;C D&:@#%2:;8^ n T,2R"!d2B#%869& &k9g#%G6!&:sO#%8^ JQ5I n Bp(B&:,#TWV9g.#%&:3 X/69&2 G69&2;8qKZ 69"RV"! 7r. sp2&:,#TWV9g(&2&C>. &2869 /69&2K69&2;p2|qKZ+ #%"01, ";Ti6!/7r!"&:71r&2!"438qKX R//&:t2K!"4tRV_ /69&D^| . 8qKqKqKX ;&2"t2!"4_RVt /69&69&2;~RV[.#%&:83(c2&(69. . sr3 P\I 0 S2Mg¨/6!&\9gqKZ #%T!"&2,#%";uW:sH2pP\4IeCRV&2 710 RV/698^~X&2B!86B9g|2BPb69¥e:]2"d!/2#%"83¬#%4T9g|2GqZ® #%BTRV&:T;C, &<2,WC> /69 71|R:!""/K9g 25.U6e269&BR:!"";C,Thqo &: 6!"WaL:sO#$#%!QS69,2-YLO;.£j^| . &~2S!86S9gJ[h^|&C«isO.7~69. r #~322TqKZ #%Mz#B69¥&2.¥sH!"&2,#%";~c2&2;H2S. ,&9g2i &:71. 71";v. &2869QsO#~^N[p'eCRV"!";D3`JQh^|&«is:.769. r #"01, "pgmh^ /69&2@\6!/71B2d!"&:7rr&2!"4^<Xx&%/69W.435&2~!869&+&2!"~(6983gh%2~!"r69H4;D32~J[«!/2#%~%/69&2r'. s."BB!&:(69&\ghQc2&2i4;M &2!"T ["0:, "
120&2&C>. &2869M /69&2Md86!/b286;Csu/69D^| .@
/691"Q&. s45
375
69&2;65
&<2T2c2&2Q4;8^ 2p86&2[9g,2!/\6%WV(687O,i69p&2 ^. .,&2;::;D^`N|69&:s!86/43r2zJQ«!/2#%|R7O;"6[71s@r::;$ &: &2!B!&2!sr3:2^7r83O6`
Zdbn
!"4RV /69&d 4. s%'eCRV&2 7r43124.W(69.V!"#$R,/69&(69.D!"4&2p#B69. .>^P\"71831 5&2!"869W.S(69|2M!"4|RV|RV &:zRV| /69&>qKqqKX-5"z6|2i4;vWV"!"#%"|c2&28^ !"4p!869&WV$";C,2!"";)6v. .%W:sb;C #$ & &<2d69#%,&:T9g|.¥ &2869T";C,,# "0:, ";Au6!/7r2@ /69 71R:!"/8^pX&2;"";D3D2J[«% · Q!/2#%@R/7:;"i2$/69#%B!&2!s<(69&k2J[«% Z er!/2#%gm56p";C,2!"";v!#$R2,/69&(69._!"48^`NQ56S",. 83M^[;C;d&2RV-gh#¯2[!"#$R,/69& ^z¥2SJQ«% Z er5/!/2#%S&H2ic2&2"z4;D^0 &2v2$(69&2;D3K2vPb69¥e:]2"H!/2#%"&2"";6<.U69r~69#%,&:%9gQ¥/'69&k6!/7rv2!"&:71r&2!"69&2;D3D6gh"!86";<W:sH2T±K&\J[,#B69&&k69&(69. sC83ai69#%,&:[ &2!866Q2T4;WV"!"#%"$c2&28^X&+;$b#B69 &:/69 &6\r::;;269#$R &9g[2v.^5mg¨/0:,2&2!s%#%:;"83 $&2"!""/69sBT &2!86[2Q&:,#TWVz9g¬,WC> /69&283"RV"!U69. . s%gm2[qo &:- 6!"Wt69. r # !/(69R c '^ Q^5'7183|6$2u!"4%RV% /69&9gi2~Pb69¥e:]2"u!/2#%"d69u;C/69#B69!869. . s#B69. .>32"S!/2#%"Q69M4.W(69. . sv#%SB!&:(69&~2 JQh^|/&C«is:.7H!",&:R(698^Xx&c24, Z eC^_[C^ · 3[2k!"&:7rr&2!"ks ^| §"RV"!H2\&:,2#TWV~9g@ /69&2<69&2;4^z¥"RV"!Bk2<qKZ #%~$R"&:/;gm%2vc2&2"%4;D^ A~!869&¢!.869. sW(7rH2u& &: &2!iB!&2!sG9g¬2[J[«!/2#%[W,83O(69&CT(' 571sB!/2869RH!"4RV /69&_32[P\]!/2#%"Q69M#%S!"#$RV 71S &H#%9gqKZ #%4^
Iterations
Resid
ual(
L 2)
0 100 200 300 400 50010-13
10-11
10-9
10-7
10-5
10-3
10-1
Block - NK(3)MF - SGS(40)MF - PJ(120)
AUSM+ (P) scheme
M = 10-1
Mesh : 14104 triangular elements
Flow over a sine bump
CPU Units
Resid
ual(
L 2)
0 500 1000 1500 200010-13
10-11
10-9
10-7
10-5
10-3
10-1
Block - NK(3)MF - SGS(40)MF - PJ(120)
AUSM+ (P) scheme
M = 10-1
Mesh : 14104 triangular elements
Flow over a sine bump
]o 4, Z eC^_[C^ · L:/69&(69sl\(^¦7156p &2W2,#$RH69M∞ = 0.1
^K&O7rr&2!"[sl^| G"RV"!%2p&:,#TWV[9g`&2&C>. &2869[ /69&2 n 'ghz69&(;:^| <"RV"!B2@qKZ #%Ghz 4:gm2c2&2"z4;D^
A )F@) * 5./; %% 7# &V ; # #& # 3 #f& 5%; 15'gh\,2;s: &)2<RV-gm#B69&2!"\RRV"H9gT86!/#%2:;§69
M = 10−4 3z.G,2G;869. ^z¥2vRW.# 9g[2v!869&2!". .U69& $(69%:!"!,+^|2&2vPb6!/&:,#TWV%7rs)#B69. .>^N[2kP\6!/§&:,#TWVu;"!86"83 ^bW2/7rb(69H2k!"&:71r&2!"A9gT2bR"!"&2;C &28;/!/2#%;:"$&2$86!/2v´8b#B6!/ &24^X&)c24, Z eC^_[C^ `23^H!.869. s)&2!"H,2!/+6\WV(687:,8^]o RRV4 &25@! !,#T7r&:5,2Q!869&GWVgm,&2;B & Z ` W,gh,25'eOR.U69&(69&2z69&2;G/. ,&2^d4 71&)W:sP. .@ & cbn 69&2;W:s)LO"2&& & g$e ^bX&gm6!83ogmT.^¯Pb6!/)&:,#TWVRW.#~3D2$687r/69r@R2",@i9g2;M9g,& xs ^| .@ MR(69U69.769U69&ki9g2@;9gT0:,(69\9gS2\P\6!/&:,#@WV'^ 2&_3z2<#%/<2\Pb6!/&:,#TWVH;"!86"3z2<#%\2
Zdbd
JQ5I n P\2:; X/69 &( qKZ qKX qKqKqKX P\I 0
n g n P\]¬q c er200 14
^0.07 8 · 10−5 79
P\]¬YLSL Zc 190
/XG0.06
NX ·10−5 76
J[«% Z er /X 71
8^0.6 67 · 10−5 72
J[«% · /X 51
8^0.43 48 · 10−5 62
· `fa n P\]¬q n er [ d e127
8^0.44 12.6 · 10−5 90
P\]¬YLSL [ber300
/ XG0.31
·10−5 86
J[«% Z er ' m188
8^4.2 120 · 10−5 88
J[«% · ' m120
8^2.7 77 · 10−5 82
g nbdbn P\]¬q Z eber380 474
8^1.25
/m ·10−5 93
P\]¬YLSL F`fer370
' 6XG1.28 16 · 10−5 93
J[«% Z er Hm1156
8^15 195 · 10−5 93
J[«% · Hm682
8^9.1 115 · 10−5 93
Z ` Z e2` P\]¬q Z [ber430 1120
8^2.6 18.5 · 10−5 94
P\]¬YLSL F`fer430
/ /LG2.27
8 ·10−5 93
J[«% Z er − − 8^ − − −
J[«% · m1707
8^26 184 · 10−5 93
69W. Z eC^_[C^ Z $L:/69&(69s\(^71T6~ &2%W,#$R_^M = 0.1
^vJQ5I n S2B&:,#TWV@9gz. #%&:3Xx/69&252i&:,#TWV|9g¬&2&C>. &2869z /69&2|$6!/7ri2i!"&:7rr&2!"43aqKZR"&:|2qKZ #%432qXK2i!"4RV5 /69&_3aqKqKqKX2i!"45RV5 /69&u69&2;GRV|.'#%&O83P<I 0|2ig¨6!' &<9gKqZ #%S!"&2,#%";~W:sd( P\4zIeCRV&2 71 0 RV/69^
R(69U69.V7469U69&H9gt2QR",QWV"!"#%"|#B69. .^| G"RV!5T 56871/69rQ7469. ,243C69&(;d2Q#%i6&:,#%!869.69. r # &:71. 7: &~6d &4.@R",@WV"!"#%"i &: &2!869. . s< &(6!"!,/69WV"!869,29g2!869&2!". .U69&+'8^\]2@ &2/69&2!"43 &,!86432d!"#$R,";)7469U69&9g2BR"/,B69WV,10−8 69
M = 10−4 ^ 2&_3¥gQ2GR",H!&O7rr"$\2G´8b#B6!/ &243&(69#%. s10−15 3K2
R(69U69.DR",[7469U69&H|&. sG6!"!,/69M . . 2i/7r&C>~;C 4 8^ 5,&2;Ob6 D"!52"0:,(69&2T69&2;D3o!"&2"01,2&:. s13t2B!"&:7rr&2!"dT!869&&2p86!/2"p2$´8H#B6!/2¥&(^GXx&A;
[bebe
@! !,#T7r&:5,243: &2"!""//69s%@;"!",R.[2Q2#%:;CsO&(69#$![R"/,g¨#® R(69U69.;CsO&(69#$!Ra69`M6i. #$ &(69| og¨#2#%#%&:,#"0:,(69&_^HA &::;C,2!"";%,2!/%!""!&2 &\,Q]o & 'j[ D&2!"@!":;@69&2; ^T,2!"!""";";u &<86!/<2p#B6!/ &2T´8%ghQPb6!/<&:,#TWV[9g69WV,
10−8 ^ 2\!""!&2~9gp2ANiL · P !":;\"0:, b6.U69rb69#%,&OH9gp#%:;C¥c(!869&269&2;bR"!869,&2^$5";"83_2 \(^lRV"";M(69 ^%69$¥&O"";k &69,2,(69. . sb9g2$;p9gM = 10−4 gmI^|!/H2M6!"!,/6!sd|, B!&:8^ :,23^i;C;G&2|R:!"";H@2"[#%:;C¥c(!869&28^J[7r2."3C i^,.;HWVMR9c2/69W.S #$R.#%&:Q2S!""!&2 &H2igh,,4^Xx&25c24, Z eC^_[C^ `23f^z!869&B69.M&2!"|(6983:69. 2,4B2|#$ & #T,#.71'.(9gV2|";C,(69.2 &2!86/"WV"!869,2p9go2p!"69&(!. .U69&\/832S!"&:71r&2!"S/69S;:"z&2z7469s,^| ~"RV"!z$2SPb6!/&:,#TWV8^ o#$R(6 ´8"`6i^. .¥>:&2^|&RRVxsp9g22R"!"&2;C &2 &i"!/&0:,2"V^|!/@#B69 &27469. ;Ggm|2SP\69¥eO]2"M #$R. ! 869#%&:8^
Iterations
Resid
ual(
L 2)
100 200 300 40010-14
10-12
10-10
10-8
10-6
10-4
10-2
100
AUSM+ (P) schemeMesh : 3468 triangular elements
Flow over a sine bump
M = 10-1
M = 10-4
M = 10-2
M = 10-3
MF - SGS(20)
]o 4, Z eC^_[C^ `2`L:/69&(69s \(^+7rK6Q &2W,#$R_^K&:71r&2!"5sT9ga25P\]¬YLSL [ber¬/!/2#%&%2z"!"&2;$4;u · `fa n .#%&:oghK/7r/69.(P\6!/%&:,#TWV8^`I`#$R(69g 2z!869&2!". .U69&dRW.#~^
[be Z
A )F@) , 214; #f# ./- #; ! 25 7! 7M = 10
−4
Xx&A2%#B69&&29g|2%R7O,2pR(69/694/69R_3^%R"&:T &Ac24, Z eC^_[C^ c 2B. &2"T9g|2BPb6!/&:,#TWV`W/69 &2";$&@271sMc2&24;D^ 2. ,&#B69 &2`sO#$#%'!"69.Cghoo.^5P\6!/&:,#WVM!86/!"&Cc2#$ &~2@r::;bRRV"p9g2NzZiLCP hq[!/2#% &b20:,(6¥> &2!"#$R"¥W2. 4 #%4^
8.50E-05 1.00E-04 1.15E-04 1.30E-04 1.45E-04∆ M = 1.5 . 10-6
]o 4, Z eC^_[C^ c TL:/69&(69s \(^7rp6v &2W,#$R69M∞ = 10−4 ^@XY. &2"M9g2$Pb6!/k&:,#TWV
W/69 &2";u&~2[c2&(|4;D^ 2B &2;RV&2;&2!"v9gz2d!"&:71r&2!"d/69Q^| "RV"!Tu2dP\6!/&:,#TWV83(69D^5B2^";869. 83¬M7469. ;bgmp69. .K2% #$R. ! T!/2#%"8^d" ;23t^| &uH2%!869&2!". .U69&83t S/69r"."[ /69&2ighi71s~#%2:;bH6!/'71@2!"&:7rr&2!"4^p](i¥&(/69&2!"43 &<c24,2 Z eC^_[C^ `23 [&2!"869W.i(6983:gh2[P\]¬YLSL [ber/!/2#%[&G2["!"&2;G4;D3O2Q";C,(69.D/694&(69"|6gh
160 /69&2S69M = 10−4 ^|2¥.@ i86!/b2T#B6!/ &2%6!"!,/6!s\6gh
300 /69&2p69
M = 10−1 ^ 2"%",. T69d6.~!"&Cc2# &A/69W. Z eC^_[C^_[C^GN[p6v",. 83qKZ® #%B!"&2,#$R&p#B69. .8^J[7r2."3t&(%!869&&2!"%(69MgmS2%J[«!/2#%"p2%!"4pRV'pRV &:@69&2;ARVS /69& &C!86"M6B. .D^| b"RV"![d2@!8669
M = 0.1^[Xx&2;"";D3^@W271";<(69[2@J[«#%2:;
WV"!869#%5. "/¬/W,2j^|2&$2Pb6!/@&:,#TWVK;"!86";$69&2;@"0:, ";@#%. &2869 /69&2`Q86!/2B/69#%%.7r.9g|. &2869p";C,,#~^vXp'eCR.U69 &2 ^|:sb2GqKqKqXiSW 4rT(69&AgmS2%R7O,2!864^5P<"7183gmz2Mc(&2"4;D32.U6!/H9g`W,2&2"Q!869&&2WVp! !,#T71&:";<W:sv&.U694 &2| ´8|9gV2|X n Z #B69¥eWV"!869,25 `2&$"0:, "`:i#T,2!/BN[P#%#%sr^ :,2342|0:,2"&9g2i#%#%sv"0:, #%&:6@W %,2M71&vgm6$!.6/!869.V[j"-!864^0 &2b2v(69&2;D32\!"4GRVvRV &:GRVv /69&§;21"v&2G7469s+#T,2!/¦ &2\!86/\9gp2Pb69¥e:]2"!/2#%"8^ 2s#B69 &dW,269#B69. .aPb6!/%&:,#TWV569&2;D3:6K&(9 !"";B &%2zR7O,2"!&_31P<]¬q 69&2;$P\]tYLpLT!/2#%"R7:;|01,2¥/2569#%5B!&2!sr^`P<"718342s69#%!"#$RV 71B(69&2 TJ[h^|&C«isO.7A!",&:R(6969&2; SS&2 !"869W.d(69p p/69r"p."@qKZ #%Q@¥#@,.U69Q20\(^§7r2c2&2"54; ^| G2[Pb69¥eO]/i!/2#%"(69&vT!"#$R,2[2/69#%]\(^¢71K2z ;%4;l^| B2JQ«% · K69. r #~^`]2,2#%43^z;C;%&2K#%"69&Os@,2^| ~2M#%#%sv"01, #%&:^
[beb[
JQ5I n P<2:; X/69&2 qKZ qKX qKqKqX P\I 0
n g n P\]tq c er100
LG0.076
·10−5 79
P\]tYL2SL Zc 100 7.8
^0.078 8.9 · 10−5 76
J[«% Z er ' 44
8^1.1 125 · 10−5 84
J[«% · ' 32
8^0.8 91 · 10−5 78
· `fa n P\]tq n er Zc e69
8^0.46 13.3 · 10−5 90
P\]tYL2SL [ber160
m X G 0.33
/ ' ·10−5 85
J[«% Z er X X184
8^8.4 241 · 10−5 94
J[«% · X X146
8^6.6 191 · 10−5 92
g nbdbn P\]tq Z eber180 228
^1.27
·10−5 93
P\]tYL2SL F`fer175
X X G1.3 16.4 · 10−5 93
J[«% Z er X X1058
8^48 610 · 10−5 98
J[«% · X X590
8^27 340 · 10−5 96
Z ` Z e2` P\]tq Z [ber225 585
^2.6 18.4 · 10−5 93
P\]tYL2SL F`fer X X m ' G2.34
·10−5 93
J[«% Z er − − ^ − − −
J[«% · J[IKP − 8^ − − −
69W. Z eC^_[C^_[CiL:/69&(69s \(^71M6d &2TW,#$R_^M = 10−4 ^MJQ5I n [2T&:,#TWVS9g.#%&:83
Xx/69&252i&:,#TWV|9g¬&2&C>. &2869z /69&2|$6!/7ri2i!"&:7rr&2!"43aqKZR"&:|2qKZ #%432qXK2i!"4RV5 /69&_3aqKqKqKX2i!"45RV5 /69&u69&2;GRV|.'#%&O83P<I 0@2Bgh/6!&9gMqKZ #%G!"&2,#%";)W:sA2 P\4IeCRV&2 7r 0 RV/69^AJ[IKP #%869&2$JQI`&2,4<P\#%sr^
[be ·
A )F@) , 214; #f# ./- #; ! 25 7! 7M = 0.5
Xx&ui"!&_3 ^@,2;CsH2TRV-gh#B69&2!""M9g,iP\69¥eO]2"T¥#$R2.¥! i869#%&:i &<2@!86T9g,W2&!+\(^z8^ A$2&S2BP\6!/&:,#TWV@69
M = 0.569&2;^B!"#$R(69B2%#%2:;L^|
2GJ[h^|&C«isO.7)69RRr6!/_^ 2d",. $69d4 71&) &)/69W. Z eC^_[C^ · 69&2; ^G!869&W27rd(6983!"#$R(69Hb2v.^5Pb6!/+&:,#TWVB!86"832HP\69¥eO]2"H!/2#%"%"01, u6<.U69rd69#%,&:B9g /69&2$86!/u2S86;Csv/694^ z| &~r::;<694"#%&: ^| ~2S±K&uJ[,#B69&&\69&(69. sO(69?^5RV-gh#%";$ &d(69R c ^](`,W2/&!5\(^z3425 r&:7469. ,2"`69r57rs@;CR(69/6957469. ,2"`/(692pRV"!/69.t/6;C ,2Q #$R. ¥c(!869&\z71sGRV&(69. ´" &Bgmz2p!"&:71r&2!"p/694^ 0 &~2S2(69&2;D3 ^|2&<2pR"!"&2;C & &v#B69¥eu[69RR. ";\69[#B69. .oP\6!/<&:,#TWVQ4 #%"[2p!"&2;C &&:,#TWV[Q71s~!.4pd,& xs:^|!/b&2,"Q(69[2TPb69¥eO]"@ #$R. ¥c(!869&k;:"[&2Q869. . s;/69[2[B!&2!sr^KP<"7183M^Q&2(692[JQ«69RRr6!/~69.p"0:, "#%Q /69&2
J[I n P\2:; Xx/69&2 qKZ qKX qKqKqKX P\I 0
n g n P\]¬q c er390
[ d 8^0.074 8.5 · 10−5 78
P\]¬YLSL Zc 340
X X G 0.065
Y ·10−5 75
JQ«% · 6X X49
8^0.4 46 · 10−5 60
· `fa n P\]¬q n er ` Z e184
8^0.45 12.9 · 10−5 90
P\]¬YLSL [ber420
/YLG0.33
/ ' ·10−5 85
JQ«% · m158
8^2.4 70 · 10−5 80
g nbdbn P\]¬q Z eber480
' G1.26
6m / ·10−5 93
P\]¬YLSL F`fer480 624
^1.3 16.4 · 10−5 93
JQ«% · / 714
8^8 100 · 10−5 87
Z ` Z e2` P\]¬q Z eber650 1455
8^2.24
/m / ·10−5 93
P\]¬YLSL F`fer610
' Y XG2.34 16.6 · 10−5 93
JQ«% · Hm1443
8^19 136 · 10−5 91
69W. Z eC^_[C^ · $L:/69&(69s\(^71T6~ &2%W,#$R_^M = 0.5
^vJQ5I n S2B&:,#TWV@9gz. #%&:3Xx/69&252i&:,#TWV|9g¬&2&C>. &2869z /69&2|$6!/7ri2i!"&:7rr&2!"43aqKZR"&:|2qKZ #%432qXK2i!"4RV5 /69&_3aqKqKqKX2i!"45RV5 /69&u69&2;GRV|.'#%&O83P<I 0|2ig¨6!' &<9gKqZ #%S!"&2,#%";~W:sd( P\4zIeCRV&2 71 0 RV/69^T!"&:71r4^`[^7r &2!86/."¥#$RV/69&:(69&dgh2QP\]A!/2#%"^Xx&2;"";D3M^|2&G2/69i &#%K9ga /69&2`WV ^"&%25P\]¬YLSLT!/2#%z69&2;@2|JQ«§69. r #l`9g 69WV,
6.6gh
[be2`
2Qc2&2"5#%"~69M = 0.1
3O 586!/2"8.1
69M = 0.5
^`Q^7183O2"/[.459go &: &2!MB!'&2!s*-,2z, B!&:z$#B691i2SJ[«l69RRr6!/\6z!"#$RV 71T6| zPb69¥e:]2"S!",&:R(69^ 2c24, Z eC^_[C^_aTR"&:2Q7469. ,2"59gt2[Pb6!/d&O,#TWVgh,W2&![!864^ 2[. ,&z . .ts:#$#%!p &2!"i2SPb6!/~&:,#TWVz#B69 &2Q#B69. .z(69&~,& xsH69z2M&2´"´".4^
0.42 0.675 0.93∆M = 0.015
]o 4, Z eC^_[C^_aCvL:/69&(69sW\(^¯71$6u &2BW,#$R+69M∞ = 0.5
^\XY. &2"9g2GPb6!/)&:,#TWVW/69 &2";u&~2[c2&(|4;D^
A )F@) * &V.6 25!$# &5!](|69. .V2[!"&Cc24,/69&2|(69I^[,2;C";D3O2[Pb69¥e:]2"Q #$R. ! 5869#%&:z;" 4&2";dgh2,&2,2!,";T4;V^56¬2^|&pWV6¬B!&:`6t 569"U69&T71&_^ ¬&2h^¥#$R2.¥! !/2#%43!",R."; ^| T 2LSLitq .U6e269&pR:!"";C,43&(69W."t|";C,2!";C/69#B69!869. . sM2!"4t9gC2!"#$R,/69&2^| @"RV"!o2Kgm#%oJ[h^|&C«isO.7S #$R. ! K69. r #"RV"!'U69. . sSgh¬2. ^5Pb6!/v&:,#TWV \(^S^KP<"71837r&v¥g¬2M!86i!"&2;"; ^56|&. s [j3^[(6;G$!RV ^| H2,2[9g¬2[#%#%sd"0:, #%&:5gh52iJQ«69RRr6!/_^ 0 &v2i25(69&2;D3C2i #$R. ! xsG9g¬2Pb69¥e:]2"|#%2:;d69. .^zKiRV-gm#g>6` #T,.U69&2"^| %4,.U69!"#$R,8^`N[ogm"!86";$W:s2±&JQ,#B69&&%69&(69. sC69&(;T2|569"U69&"-!86/"8392P<]¬YLSLp!/2#%5&2,"`671sMr::;!"&:71r&2!"p/69T7r& ^|2&<2S#%"uWV"!"#%"Q7rsGc2&2p(69&C% r:1;<;269#$R &GRRV"71i2@.^5mg¨"0:,2&2!s<4&_3^z . 2P\]¬q !/2#%"01, "M#%@ &&2-> /694&2pv6i. #$ &(69HB!&:. s69. .52v#%:;"^A[^7r83K2d.U69/'$#B69 &2%6\!"#$RV 7ru69.¥/'&(6969RRr6!/~gmz!"#$R, &G69. .tRV""; \(^z7rz,&2,2!,";~4;8^&:71r&2!"SRV-gm#B69&2!""0^| . .WVQ&2^¦,2;C";Bgm · j!"&Cc24,/69&25gh5^|!/v2[0:,2"&v9g_2Q#%#%sB"0:, #%&: ^| . .VWV6B!,2!U69.o,24^ 2T&2'eC["!& ^| . .`69 # 69[,2. ¥&2¥&2,^|22i2TP\69¥e:]2"@!/2#%"[&(69W.%!"#$R, \a^7rz4;| &:71. 7O &B6.U69rp69#%,&:Q9goRV &:z&28^
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(x, y, z) ∈ [0; 4] × [0; 0.5] × [0; 1]^| 6G/69 4:M,RRV ^569. .K69&2;A6G!,71
.^i^569. .
x < 1 z = 0
1 ≥ x ≥ 3 z = 0.1 (1 − cos[(x− 1)π])
x > 3 z = 0
2u.U69/69. ^569. .G9gM2<!"#$R,/69&(69.i;2#B69¥&¦!""RV&2; A2<,-g>6!"u9gM2 [j;#B69 &_^[&2!"43agm,i;C &:i4;i,2 &v¥eOmg>6!TR#%S69687469 .69W2. %6gm[2 [j"-!86vQ6d!"r69&2 ^| ` ·bd ev.#%&:p69&2; Z [ Z e · gm6!""% c24, Z eC^ · '3¬6d#%";C ,# &2 ^|
34, 780.#%&:
69&2;91, 428
gm6!""83(6@c2&2S&2 ^| 118, 470
.#%&:[69&2;306, 323
g>6!"3269&2;<67rsdc2&2S&2 ^| 282, 080
.#%&:[69&2;723, 304
g>6!""8^
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.#%'&O¨R#%|69&2;
12, 103gm6!""^
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M∞ = 0.1^
&:,#%!869.T. ,&!"#$R,2";S^| §2kN|ZMLOP hqv69RRr6!/H &§r::;694"#%'&O:^| 22"!869.¬/. ,&_^
[beba
]o 4, Z eC^ · ^_[CQL:/69&(69s:\(^ &b6 · jl &2W,#$Rb!/(69&&2.`69M∞ = 0.1
69&2;\71(Sc2&2"Q4; [ n [be n e%.#%&:'^ R`XY. 4&2z9g2MPb6!/~&:,#TWV8^K5# `XYW(69!M!,71"8^
[beg
2|",. 69z,#$#B69 ´8";% &$25/69W. Z eC^ · ^ Z ^`NQK'eORV"!";%2|J[h^|&C«is:.7` 4. s@. #$ ";W:sT `R2¥W2¥/;%#%#%s"0:, #%&:8^`X&2;"";D34gh#2z"!"&2;4 ;+^| B&. s
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JQ5I n P\2:; X/'69&2 qKZ qKX qKqKqKX P\I 0
4, 390P\]tq c er
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0.71 ·10−4 83
P\]¬YLSL [ber220
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·10−4 83
JQ«% · 6XH17500
8^146 332 · 10−4 95
34, 780P<]¬q n er Y X m /LG
8.3X ' ·10−4 85
P\]¬YLSL F`fer340
·/Zn e%8^9.3 2.7 · 10−4 88
JQ«% · J[IKP − 8^ − − −
118, 470P<]¬q Z eber ' /mH LG
37Y F ·10−4 85
P\]¬YLSL c er460 18800
8^41 3.5 · 10−4 89
282, 080P<]¬q Z [ber m X ' LG
107 3.8 · 10−4 86
P\]¬YLSL c er a · e a c ebebe$8^103
Y ·10−4 89
69W. Z eC^ · ^ Z L:/69&(69s \(^7r6 · j &2iW,#$R_^M = 0.1
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Iteration
Resid
ual
200 400 600 80010-13
10-11
10-9
10-7
10-5
10-3
10-1
MF-SGS(50)MF-PJ(120)
Flow over a 3D sine bumpAUSM+(P) scheme
Mesh : 282080 elements
M = 0.1
CFL = 106
CPU units
Resid
ual
0 20000 40000 60000 8000010-13
10-11
10-9
10-7
10-5
10-3
10-1
MF-SGS(50)MF-PJ(120)
Flow over a 3D sine bumpAUSM+(P) scheme
Mesh : 282080 elements
M = 0.1
CFL = 106
]o 4, Z eC^ · ^ · L:/69&(69s \(^ &¦6 · j &2W,#$R!/(69&&2.M69M∞ = 0.1
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10, 000.#%&:Q¥& · j^ 0 &u2p2(69&2;D3(,2 &dPb69¥e:]2"
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L3
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P21 P22 P24
d
d L2
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[ ZbZ
[ Z [
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(.# " &n S,2MR"&:S2%/69&(69s\RW.# ^$/. 7r4^<A$,RRV4$H;869."^| 6v!869.!869. . sbRV-gm"!16%£ghi &2/69&2!"%69 '^pN[i2^|&k &\c24, ZbZ ^ Z ^ Z 3_,2!/\16M&:M2@ &.S ^| k#$RV/69,TH
69&2;k#%#%&:,#mH
69&2;k2$ &. ^| A#$RV/69,TC
69&2;k#%#%&:,#mC
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mH(s) =6 ¯mH
d2
(d2
4− s2
), mC(s) =
6 ¯mC
d2
(d2
4− s2
),
sWV &<2B;C/69&2!"Q^| "RV"!p~2%,WVd6eC83 ¯mH
69&2; ¯mC2d6871/69r%#%#%&:,#%8^HN|
2S,. 0 3C2MR",Mz!"&2/69&:T¨ &~R(6!"S69&2;H #%|69&2;~"01,(69._P0
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& ! " #%& (M!"&2;z2SJi687:-YL:1""01,(69&2|gmz!"#$R" W.432!869.!869. . sHRV-gh"!z163C&(69#%. sk
∂ρ
∂t+ ~∇ · (ρ~u) = 0
∂ρ~u
∂t+ ~∇ · (ρ~u⊗ ~u+ pI7 ) = ρ~g + ~∇ · τ7
∂ρE
∂t+ ~∇ · (ρ~uH) = ρ~g · ~u+ ~∇ · (τ7 · ~u) + ~∇ · (λ ~∇T )
p = ρRT = (γ − 1)ρ
(E − 1
2~u · ~u
)
ZbZ ^ · ^ Z
^|2
τ7 = µ
(~∇⊗ ~u+
(~∇⊗ ~u
)T
− 2
3
(~∇ · ~u
)I7)
n |,2z!869sG,|2[&2&;C #%'&(&(69. ´6&<9go2i"0:,(69&2p ZbZ ^ · ^ Z '^]o |9g`69. .>3/^i;'c2&2M2gm. .^| &B!":B!&:T
∆T = TH − TC , ¯T =TH + TC
2, ε =
∆T
2 ¯T, ¯mt = ¯mH + ¯mC , αH =
¯mH
¯mt.
ZbZ ^ · ^_[r
[ Z8·
2&_32M'gm&2!"M7469. ,2"69S!/24&<65gm. .^zp
lref = d , Pref = P0 , ρref =P0
R ¯T, uref =
¯mt
ρref, tref =
d
uref,
λref = λ(ρref , Pref ) , µref = µ(ρref , Pref ) .
ZbZ ^ · ^ ·
Xx&H2MsO# ZbZ ^ · ^ Z '3(;C #%&2&(69.t7469. ,2"69MR.U6!"";~W:sb
~r =~r
lref
, t =uref
lref
t , u =u
uref, p =
p
Pref
, ρ =ρ
ρref, ρet =
ρet
Pref
,
T =T − ¯T
∆T, λ =
λ
λref
, µ =µ
µref, τ =
lref
µref urefτ .
ZbZ ^ · ^ `:
Ng¨z/69 4O-gh ^569;u!869.!,.U69&283(2[gm. .^| &dsO#9g`"01,(69&25gh,&2;A
∂ρ
∂t+ ~∇ · (ρ~u) = 0
∂ρ~u
∂t+ ~∇ · (ρ~u⊗ ~u) +
1
γM2∞
~∇ · (pI7 ) =1
Frρ~eg +
1
Re(~∇ · τ7 )
∂ρE
∂t+ ~∇ · (ρ~uH) =
γM2∞
Frρ~eg · ~u+
γM2∞
Re~∇ · (τ7 · ~u) + 2ε
γ
γ − 1
1
Pr Re~∇ · (λ~∇T )
p = ρ(1 + 2εT ) = (γ − 1)ρ
(E − γM2
∞
2~u · ~u
)
ZbZ ^ · ^ c
^|2
M2∞ =
ρref u2ref
γPref
=¯m2
t
γP 2ref
R ¯T , Pr =µref
λref
γ
γ − 1R , Re =
¯mt lref
µref, F r =
(¯mtR
¯T
Pref
)21
glref
.
ZbZ ^ · ^_arn M,2[ &::;C,2!"2@WV,&2;269sb!"&2;C &2M,&2;M;C #%&2&."/[gm#~^SN[[2sO# iRV&2";D3P (t)
%4 71&+W:s2HWV,&2;269s+!"&2;C &2d &+2u,. 0 3 P (t) = PO(t)/P0
^ 2H &.#$RV/69,"i69$
TH =TH − ¯T
∆T=
TH − TC
2(TH − TC)= 0.5 , TC =
TC − ¯T
∆T=
TC − TH
2(TH − TC)= −0.5 .
2M &.Q6871/69rS;C #%&2&."#%#%&:,# 69M4 7r&~W:sb
mH = ρH · uH = αH · ¯mt , mC = ρC · uC = (1 − αH) · ¯mt .
2[&2&C;C #%&2&(69.t. ,&H9g¬RW.#;RV&2;|&G2Qgm. .^| &&2&C;C #%&2&(69._R(69/69#8
L1/d3L2/d
3C69&2;L3/d
F^|!/H69&2&v&2 W. s$ &/\2,2&: &d2#$¥eC &T4&G¥g_2sB69QW &2,4a'3O2QRV"!¥c(!Q2869/69
γ312QQ'sO&2.;&:,#TWV
Re312Qq`/69&2;.V&:,#TWV
Pr312Q]2,2;
&:,#TWVFr
325Pb6!/&:,#TWV83r2/694αH
69&2;ε^oXogh. .^z`(69834¥g ^/691|6o&2&C;C #%&2&(69.
R(69/69#%
[ Z `
L1/d L2/d L3/d γ ε αHq` Q P ]
Z [ d g Z ^ ` eC^_[ eC^ n eC^ g Z ebe Z · ebe Z d69&2;'^iceG2M&:,#%!869.t7469. ,2"
d = 1#, ρ =
P0
RT= 1
O$#−3 , mt = 1
O$#−2
−1 , R = 288 O
−1 «−1 ,
^S!869&u!"#$R,p69. ._2S;C #%&2&(69.¬0:,(69&: "z &:71. 7r";H &~2iRW2. # >LOX,& '
L1 L2 L3 u P0 T TH TC mH mC µ0 λ0Z [ d g Z a2`f[ d e [b[ · ^_[ [bagO^ n Z g n ^_a eC^ n eC^_[ eC^_e Z Z `2^ `B VGUM 6E/F Kj2 K"UfFUfG 2S/69&(69sH. ,&u9g|R/W.'##T,2z/69-ghsd2S;C 7rr&2!"p!"&2;C &
~∇ ·(ρ~v)
= 0 . ZbZ ^ · ^ g4
Xg)^[&24."!|2[!"&: W,&u9gt2[2869|;C ,2&~692[ &.z69&2;H692i,.Sm69RR8eC #B69&^|!/~z!"/!¥g2M/"!&269ig>69z&2,4~g¨#2M#$¥eO &%4&a'32 |#T,(|WV%
∫
H
;S(ρH~v · ~n
)+
∫
C
;S(ρH~v · ~n
)= −
∫
O
;S(ρH~v · ~n
),
ZbZ ^ · ^ n 3CW:sd&24."! &d2M#%
O(M2)
∫
H
;S(ρh~v · ~n
)+
∫
C
;S(ρh~v · ~n
)= −
∫
O
;S(ρh~v · ~n
).
L: &2!" ^T69T;869. & ^| b6%!869.!869. . s~RV-gm"!Q16[69&2;u2p9!869. .";<2#%:;Cs:&(69#$!@R",S!"&2/69&:832 5gm. .^z|(69p
∫
H
;S(~v · ~n
)+
∫
C
;S(~v · ~n
)= −
∫
O
;S(~v · ~n
),
∫ ;
V(~∇ · ~u
)= 0 .
ZbZ ^ · ^ d ]2#"01,(69&) ZbZ ^ · ^ n '3H^M!869&<69.%;";C,2!"M(69p
mH TH SH + mC TC SC = −∫
O
;S(ρH~v · ~n
)
Xg¬(M,.Q"!& 0 |gm69z&2,4~gh#2M"5*Y,&2!&_322S,.T
z!"&2/69&:832&(69#%. sb
TO =mH SH
mOSO
TH +mC SC
mOSO
TC = αH TH + (1 − αH)TC = 250«.
ZbZ ^ · ^ Z er
[ Zc
" " " (A)A*) )%A 2 ./&143 #; !$!$# I ; !2& -; # 2|!"&27469 7rz!#$R2" W.z"01,(69&2 ZbZ ^ · ^ Z `69|. 71";@,2 &i2|!"#$R" W.|;&2 xs1>W(6";. 719g NML · P ^|!/%#$R.sOK69&,&2,2!,";$]o & ±K. ,#%z69RRr6!/_^N[o;"! WV";$869. 832i &:7:/! ;,\2,Ce"z69M!"#$R,";H,2 &%!.U6!869.D,RM^| &2;H/!/2#%3'eO&2;";H@ 42;|,2 &2uR #$ 71<7469U69W.<"!&(,2!&§9gp|69¦69&2; "RV&_3;"! WV";¥& _[ 69&2;¢WV.&4 &v2%P<ZML n gm69#$ . s t#%"7rp2&:,#%!869.;C R(69&9g2 &:7O!;k!/2#%"SM!""!";/69O &S &:p6!"!",&: n ^¢Pb6!/%&:,#TWVR"!"&2;C & &22^`±Q/!,(\2,Ce"69Q69RR8eO #B69";B,2 &T6. &2869. s:'e6!5'eC&2&G9g_2Q;CU69#%&2;d#%2:;G9gtJQ _abe ^K&2!"& &@( #%Q;C! ´69&_32QI`,. #$R. ! 5!/2#%Q,2";D^ 2&2&C>. &2869RW.#69 &pg¨#2 #%[;C! ´69&G. 71";B7:U6S2[Pb69¥e:]2"Q #$R. ! #%2:;G(69I^ &::;C,2!"";G &G2"8^`";"83M^Q!",R.2SP<]!/2#% ^| \6%L:s:#$#% !@M69,(-YLO;._.U6e269&~R:!"";C,4^A)A*) )@ i& h9 -.M= !"143"7b&7# .L./&143 #; !$!$# 5 ; !2& -; #Xx&A;pHR7O;%!"#$R(69&23t2 "i*-,&2!&ARW2. # p69./H!"#$R,";,2 &<6vR"/,'W(6";. 71~9g@2NML · P !":;4^X&,2!/. 71H2A!#$R2" W."01,(69&2<69 &gh#&2'> #%i!869.[69&2;G&2'R(6!"i!869..^5Pb6!/G&:,#TWVz6sO#$R!"|69Q. 71";D3:&(69#%. sr3C&2Q(6!"#$R,2[h!"&2/69&: &@R(6!"_2#%:;Cs:&(69#$!5R",
P (t)392RV""; ~u(~r, t) 392#$RV/69,
T (~r, t)3269&2;H2S;Cs:&(69#$!SR",
p′((~r, t))32. ,&u9g2M¥&2¥69.t769. ,2MRW.#
1
γP
;P;t
+(~∇ · ~u
)=γ − 1
γP
~∇ ·(λ ~∇T
)
∂~u
∂t+(~u · ~∇
)~u =
RT
P
(− ~∇p′ + ~∇ · τ
)+ ~g
∂T
∂t+(~u · ~∇
)T =
γ − 1
γ
T
P
( ;P;t
+ ~∇ ·(λ ~∇T
))
ZbZ ^ `2^ Z
2R(6!";C/! ´69&kiRV-gh#%";b,( &~6vS69.: &k]o & $I`.#%&:p69RRr6!/_^p±.:! xsb69&2;#$RV/69,B(69RVTgh,&2!&2p69$01,(6;C/69!u¨U69&4,.U69p.#%&: ^| gd;4"%9ggh"";#%p69&2;0:,(6;C/69&4,.U69@&2" ^| d ^| .BR",B(69RV$g¨,&2!&2@69%. &2869G¨U69&4,.U69@.#%&: ^z¥· ;4"G9g|gh"";#%69&(;0:,(6;C/69&4,.U69$&2"D^| `:'^b&2!"& &b2d #%d;C! ´69&_3K25ji] Z #$R. ! @!/2#%%S,2/";D^ 2%&2&C>. &2869TRW.#69 &~gh# 2% #%B;C! ´69&. 71";v7OU6$6Tce";O>RV &:Q!/2#%%
1
γPn
( ;P;t
)n
+(~∇ · ~u n+1
)=γ − 1
γPn
~∇ ·(λ ~∇Tn
)
(∂~u
∂t
)n
+(~u n · ~∇
)~u n+1 =
RTn
Pn
(− ~∇p′ n+1 + ~∇ · τ n+1
)+ ~g
(∂T
∂t
)n
+(~un · ~∇
)T n+1 =
γ − 1
γ
Tn
Pn
(( ;P;t
)n
+ ~∇ ·(λ ~∇T n+1
))
ZbZ ^ `2^_[r
N[[2^|&\ & _a Z 3 S69. r #(6M69.GWV"&b7469. ;269";\ &k!"#$R, &H2. ,&k9g56d/01,(69!8687O xs ^| u.U69rM#$RV/69,p;C D&2!""8^
[ Z a
:8JLUGFVCFUMGG jVVE6 VjF VGZQ&. r[2Qg¨,. . s:!"#$R" W.M!86Qgh ^|!/ ^[ #$RV4[2i;C #%'&(&."687r/69r[#%#%&:,2#6986!/~ &.83243H^i #$RV4M2M;C #%&2 &2. "/Q6871/69rM7r.:! "[65gh. .^zp
uH =¯uH
uref=αH
¯mt
ρH
ρref
¯mt= αH(1 + 2εTH)
1
P= αH(1 + ε)
1
P,
uC = (1 − αH)(1 − ε)1
P.
5";"83|69%2u,.83 ^H #$RV4~(69B2~71.:!' YsR"&:G6bR9c2.:^|!/ %R(69/69WV. !\69&2;sO#$#%! ^| k"RV"!MG2@,WV%6eO^ 2!"&2/69&:S!":'B!&:S &:71. 71";\ &k,2!/bR(69/69WV.U6G;#$ &2";~,2 &$2S;C 71r&2!"p0:,(69&)£c2z"0:,(69&u9gz ZbZ ^ `2^ Z '^
" (1 & #%&'(A)A*) )%A * &V -; # 3); ./; " ?!$# !]o T9g69. ..p,(SRV-gm#6~!"&:7rr&2!"v69&(69. sO ^| "RV"!T~2B#$";C #%&2&_^~X&2!86@9g2TR",'>W(6";\/. 7r83V2@!"r69"[(6 ·/Z8·bc .#%&:% Zc .#%&:iRV[,WV@/"!& c24, ZbZ ^ c ^ Z ?^| .z2z#%";C ,##%"B(6 cbcbn eS.#%&:i [beM.#%&:RVK,2WVz/"!&aK69&2;2$c(&2"p#%"(6 n g$[ c .#%&:v [ c .#%&:TRVp,WVB"!&a'^~X&2B!86d9g|2B;&2 xs:
]o 4, ZbZ ^ c ^ Z "5*Y,&(!' &t^ 2S!"r69M#%" ·/Z8·bc .#%&:'^W(6";. 7183]^</691u#%"2" ^| ¦#%\.#%&:A¨&2<(69v&2 2v2<;2'4""H9gSg¨"";#&2M2%6!"!,/6!sb9g2%69RRr6!/692$/69#%%6i &b2@R"/,'>W(6";k/. 7r'^ 2$!r69/"4;u(6 Z [ba · eB.#%&: · e%.#%&:[RV,WVT"!&a'^ 2p#%";C ,##%"u(6 [b[2`febeB.#%&:F`fe~.#%&:TRVp,WVB"!&a ^| .%2$c2&2"p#%"(6 ·bc ebebe~.#%&:v c e~.#%&:@p,WV"!&a'^Xx&¢c24, ZbZ ^ c ^_[ ^<R"&:v2\869#$. &2"~69&2; 2<#$RV/69,b. &2"8^N[G&2\!869&¦"432$2p69&2;k2$!".; \2,;A;v&2S#$¥e 3 22869S/69&2#$&69 71"MW:s<2#B69.;C ,2&_^Xx&+c(4," ZbZ ^ c ^ · 30^uR"&OG2\. ,&¦& 2\"!&2vq
23q
333zq
22q
2369&2;q
22q
32(69
^vW/69 &2";,2 &b2H6sO#$R!vR",'>W(6";!"#$R" W.H. 718^ 0 &2v2$(69&2;D3K &c24," ZbZ ^ c ^ `23 ^5HR"/&:B2u/.¥,2& &¢2u"!&2dq
23q
3335q
22q
2369&2;¢q
22q
32(69Q^
[ Z g
185 212.5 240 267.5 295∆ T = 2.5 K
]o 4, ZbZ ^ c ^_[C "T*-,&2!&_^)L:/69&(69s. ,&_^)L:869#$. &2"u¨Ra69&2;)#$RV/69,H. &2"¨WV#d54 71&HW:sG2S;&2 xs:>W(6";~!"#$R" W.p. 718^
!"#$R,";u,2 &%2p;&2 xs1>W(6/";~!"#$R"/ W.p. 718^NQ&2S!869&<"43(69. ._2MR2"&:";~",. 69S01, M#%"C> &2;RV&2;"&:8^
[ Zn
210
220
230
240
250
260
270
0 0.2 0.4 0.6 0.8 1
finemediumcoarse
T
uxuys
180
185
190
195
200
205
210
215
0 0.2 0.4 0.6 0.8 1
finemediumcoarse
T
uxuys
180
190
200
210
220
230
240
250
260
270
0 0.2 0.4 0.6 0.8 1
finemediumcoarse
T
uxuys
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0 0.2 0.4 0.6 0.8 1
finemediumcoarse
T
ux
uys
-0.16
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0 0.2 0.4 0.6 0.8 1
finemediumcoarse
T
ux
uys
-2
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0 0.2 0.4 0.6 0.8 1
finemediumcoarse
T
ux
uys
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0 0.2 0.4 0.6 0.8 1
finemediumcoarse
Tux
uy
s
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8 1
finemediumcoarse
Tux
uy
s
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 0.2 0.4 0.6 0.8 1
finemediumcoarse
Tux
uy
s
]o 4,
ZbZ^c^ ·
"p
*-,&2!
&_^kN[
sO#$R
!dR"
,'>
W(6";
!"#$R"
W.H
. 7r8
^LO. ,
&
&2
"!
&\q23q 3
3¨.'gh
'3aq 2
2q 2
3¨#$
;;C.Q
69&2;<q 2
2q 3
2¨ 4
:'^ s
2T
!,7O .
&2869i69
W2/!/6C
^|NQ. .t
20:,(69
&: "
Q69S
'eOR"
/";v &~
2pLOX
sC#~
^
[ Zd
210
220
230
240
250
260
270
0 0.2 0.4 0.6 0.8 1
finemediumcoarse
T
uxuys
180
185
190
195
200
205
210
215
0 0.2 0.4 0.6 0.8 1
finemediumcoarse
T
uxuys
180
190
200
210
220
230
240
250
260
270
0 0.2 0.4 0.6 0.8 1
finemediumcoarse
T
uxuys
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0 0.2 0.4 0.6 0.8 1
finemediumcoarse
T
ux
uys
-0.16
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0 0.2 0.4 0.6 0.8 1
finemediumcoarse
T
ux
uys
-2
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0 0.2 0.4 0.6 0.8 1
finemediumcoarse
T
ux
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0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0 0.2 0.4 0.6 0.8 1
finemediumcoarse
Tux
uy
s
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8 1
finemediumcoarse
Tux
uy
s
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 0.2 0.4 0.6 0.8 1
finemediumcoarse
Tux
uy
s
]o 4,
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4~69RR8eC #B69%2869.K&:,#TWVp9g|!"#$R,/69&
RV &:8^pXx&\/69W. ZbZ ^ c ^ Z 3^@RVMS69RR8eO #B69$&:,#TWVighM7rsu4;D3D &b;MGRV-gh#2S!"#$R(69&28^
P<" J[5I n P\2:; Xx/69&2 qKZ qKX qKqKqXr69 Z [ c `fe N[sO#$R!
41858^
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Z [ba · e P<]¬YLSL Zc 1050
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P<";C ,# [b[ · [be N[sO#T! X XH7775
8^35 158 · 10−5
[b[2`febe P<]¬YLSL Zc 1380
X /LG2
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·bc ebebe P<]¬YLSL Zc 2200
/ Y 8^3.15
/ ·10−5
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0
0.2
0.4
0.6
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1
210 220 230 240 250 260 270
Fully-compressibleAsymptotic-compressible
T
uxuy
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180
185
190
195
200
205
210
215
0 0.2 0.4 0.6 0.8 1
Fully-compressibleAsymptotic-compressible
T
uxuys
0
0.2
0.4
0.6
0.8
1
180 190 200 210 220 230 240 250 260 270
Fully-compressibleAsymptotic-compressible
T
uxuy
s
0
0.2
0.4
0.6
0.8
1
-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2
Fully-compressibleAsymptotic-compressible
T
uxuy
s
-0.16
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0 0.2 0.4 0.6 0.8 1
Fully-compressibleAsymptotic-compressible
T
ux
uys
0
0.2
0.4
0.6
0.8
1
-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
Fully-compressibleAsymptotic-compressible
T
uxuy
s
0
0.2
0.4
0.6
0.8
1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
Fully-compressibleAsymptotic-compressible
Tux
uy
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0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8 1
Fully-compressibleAsymptotic-compressible
Tux
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0
0.2
0.4
0.6
0.8
1
-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Fully-compressibleAsymptotic-compressible
Tux
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mH69&2;L^B!/2:4d6H #%';RV&2;"&:@#$RV/69,
TH65gh. .^zpTH(t) = TH(t = 0)(1 +A sin(ωt))
^|2A
69&2;ω
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241.02
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1
2
3
4
5
6
7
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t
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!869&~WVM'eCR"";u65gm. .^zS
tcon,ref =d¯u
= 1
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αref =λref
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−1
[&2!"432M!/(69/6!!S #%igm|2M#$RV/69,p;C ,2&u!869&~86;A
tdif,ref =d2
αref= 70
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cref =
√γP0
¯ρ= 300
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2S!/(69/6!!p #%igh6!",2! ^56871i|2&~2igm. .^| &u
tac,ref =d
cref= 3.33 · 10−3
69W. ZbZ ^_aC^ Z ",#%"o2"!/(69/6!! #%"69&(;p2 `6:!U69";TR,./69 &(52π/t
'^oX&@;oW271i 4&¥c(!869&:z#$RV/69,M7469U69&2| &v2 \(^c(.;H;C, &B&2[RV:;D36^[(6871[!/2:4/6$R,.69&\#B69. .Q(69&<,& xsu69&(;<6%&2&C>&24. 4 W.$69#$R. ,2;4^)S2'ghT/69rp2Sgh. .^| &7469. ,2"S
A = 0.1 , ω = 0.1−1
2|RV:;%9gV2|#$RV/69,]\2,2!,(69&G69K25 &.¢`2&2π/0.1 = 62.8
c24, ZbZ ^_aC^ Zgm|2MR"&:/69&<9g2M#$RV/69,
TH;C, &d&2iRV:;('^
trefh
2π/trefhQ´
Nz!",2!3.33 · 10−3 1890
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69W. ZbZ ^_aC^ Z "5*-,&2!&_^(69/6!!S #%"8^
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St = 31.4
8^ %!.869. skW2/7r%2$RR(69169&9g52%#$RV/69,Q\2,2!,(69&28^Xx&);@< &:71" 169Bgh,2@2G "!9g2"d76969 &(83^Bgm:!,2@&)2G"!&q
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t = 7.85T¨.U69WV.[r'3H^pW2/'7rT6$.U69rS7469U69&<9g`2
71.:!¥Ysux
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~∇ · ~u =γ − 1
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~∇ ·(λ ~∇T
).
Xx&B2Q2,WV43O(69WV ^"&vq23
q33
69&2;dq24
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185 212.5 240 267.5 295∆T = 2.5
185 212.5 240 267.5 295∆T = 2.5
185 212.5 240 267.5 295∆T = 2.5
185 212.5 240 267.5 295∆T = 2.5
]o 4, ZbZ ^_aC^_[C "i*-,&2!&_^dJ[&C/69&(69sA. ,&_^d]2,. . s:!"#$R" W.d. 7r8^ #$RV/69,. &2"G
«Si &2%2p,WVd69
t = 7.85d¨RC>.'g¨'3`69
t = 15.7d¨RC> 4:'3`69
t = 23.55
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p¨WV#> 4:^
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ux &2!86"3O2Q.:!869. #%#%&:,#&G2i"!&G &2!86/"8^ ";C,2!""
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t = 7.85M¨.U69WV.)[r69&2;H69
t = 15.7M¨.U69WV. · '^`NQ2 \2,;
.#%&:[!"#$ &Bgh#2S2 &.Q(687rS86!/2";u(p"!&) c24, ZbZ ^_aC^_[%Ru 4:g¨/69#%'32769U69&9g2@#$RV/69,2$i71su #$RV/69&:8^JQ!"(69M2@ &2!86%9g#$RV/69,$ 42M &b2#$;;C.$9g2,WV ^|22RV"";b[2.U69r"83 ^|22@2869S'eC!/(69&r"W:sb!"&:7r"!&)69$2#%4M #$RV/69&:8^d";"3_2$RV"";A69.&
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3(69&2;u69./$$2Mg>6!z2M(MW,2s:69&2!sGgm!"M(6¥&(!'/"6/";~WV"!869,2S2#$RV/69,p &2!86"8^]2#
t = 15.7p
t = 23.6G¨.U69WV. · 69&2;W`< &c24, ZbZ ^_aC^ · '3`2B7r.:! xs
ux;"!86"D^z¥
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t = 23.6@
t = 31.483`2G#$RV/69,~;"!86"8^ 0 &2!"~69169¥&t3K $;"!86"#%~69
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567
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789
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567
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CFLu = u∆t/δx(,.;<WV@9g;i,& xs)h[."G6[G&2,
#$RV/69.Q6!"!,/6!sr^)]2#³2~qZ #%vRV¥&O%9gQ7O h^S32^7r83 $!,2#B69s)\"4869i7469. ,2"i9giR(69/69#%8^SXx&\R/6!!"43 4869i #%'R69pc2i!/(14/&b2&\2¥i7469. ,2[;"!'/"6/";<,&: .¬2p&:,#%!869.`. ,&b;:"Q&2Q769sH#T,2!/_^ :,283a2TR #B69.o #%'R2Qgh
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∆t =π
ω· 1
N^|2N
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23q
3369zR.";Bgh2zgh. .^| &T #%'>.71.[
t = 15.769&2;
t = 31.4[£c24, ZbZ ^_aC^ c '3
t = 47.1T69&2;
t = 62.8B£c(4, ZbZ ^_aC^_ar'^GNQp&2%!869&&2!"43¬2B;&2 xs:>W(6";. 7rp!",R.";
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βu =δx
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c24, ZbZ ^_aC^ g4'^ 2G'e6!T7469. ,2"@9gz(d!,- ^dR"!. s'e269#$ &2";) &2!"[email protected]@r7r&2T&(@&. sA2!"&:71r&2!"$/69$W,p69.v2%6!"!,/6!sk9g2$. ,&_^$Xx&2;"";D3¬6i ^56M#%&:&&2";A &A!/(69RC g$69&2; n 3(69&(;u6z i^56zW,4:,R~ & _ab[ 3(6:$W B!,-A!869."z,&RRV. sH2M&:,#%!869.;C R(69&d.86;C &@T69&B &(6!"!,/69[. ,&_^ 2Q,2;C"71869.";B(69 &d,!864312#%:;C¥c(";,&286;Csu!,-L^|6 xs~ #%"i6#T,2!/b6Q2p'gh&2!"@Pb6!/u&:,#TWVi9gK2pRW.#~^[&C"0:,2&:. sr3¬/u6S~&2,%WVW,2&2"T69&2;6!"!,/6!s13V^%R";AghS2$gm.¥.^z¥&2</694sc283-^5S6d4869i7469. ,2Tgm[2$!,-)2&^;C #$ &k M;C, &H2;C,(69.¥> #%%!"&:71r&2!"R:!""z,&: ._86!/ &v6@7469. ,2S9go2S;Q9g2SP\6!/~&:,#TWV8^ :,283 i^56|. 1. sG$#B69 &:/69 &6$r::;<!"&:7rr&2!"T/69T69&2;u%"!"71'Q2T/69#%T6!"!,/6!sH(69& ^| <2SR",'>W(6/";u/. 7r8^J[&22."839 ¬¬&2"!""/69sMQ,2. ¥&((69o,2!/6/694sSR7:;"V\2,2!,(69 &iRV-gh#B69&2!""j^z¥"RV"!p2RW.# ^|!/G,2;C";D^`P\"7r83: &2569. ^568sCRV4/ W.S";C,2!"Q2Q!,-@6T/!869.RRV. s%2Q;C/ R(69&,^| 2,5;/69[2W,2&2"/9g_2Q!/2#%4^`N¦. 1. s. ,& ^,.;TWVQ/R(69/692R"!"&2;C &2 &igm#T,.U69&,2";pgm¬2 #%'; 7469 71MF^|!/
[b[ n
0
0.2
0.4
0.6
0.8
1
200 220 240 260 280 300
Fully compressible
Asymptotic compressible
T
ux
uy
s
0
0.2
0.4
0.6
0.8
1
200 220 240 260 280 300
Fully compressible
Asymptotic compressible
T
ux
uy
s
0
0.2
0.4
0.6
0.8
1
-1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2
Fully compressible
Asymptotic compressible
T
ux
uy
s
0
0.2
0.4
0.6
0.8
1
-1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2
Fully compressible
Asymptotic compressible
T
ux
uy
s
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3
Fully compressible
Asymptotic compressible
Tux
uy
s
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3
Fully compressible
Asymptotic compressible
Tux
uy
s
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ux71.:! xs¢¨#$;;C.%gh/69#%"S69&2;
uy71.:!¥Ys+¨WV# g¨/69#%"'M69%R"&:";&"!&q23
q33
69&2;kgmS2$gm. .^| &u #%%.71.Bt = 15.7
z69&2;t = 31.4
^NQ. ._0:,(69&: "Q69S'eCR"";v &~2TLOX,& 8^
p"RV&2 W.%gh@!"&:7rr&2!"GB!&2!s2T69&2;(69T,2";kgmp2G69¥c(!U69.5;C R(69&)#%v¨'RV&2 W.[ghz6!"!,/6!s'^L:,2!/u6pgh#T,.U69&H#B68sdWV[ #$R.#%&:";'^| &~6T#T,. R.iR2/,2;9> #%gh/69#%h^6p"!"&:. s;C!,2/";A & n [ ^HXx#$R.#%&:/69&9g,2!/6H#T,. ¥>.7r.R"!"&2;C &&gm#T,.U69& ^| . ._(687rMWVM2S,WC*"!'9gog¨,, ^ ^ji"R b6!"/69 &§.4G9g@B!'&2!s ^| ¦"RV"!v2\86;Cs¢!8643z2b;&2 xs:>W(6";¦. 7r!",R.";4^| 2\P<]¬YLSL+ #$R. ! v869#%&:vR7O;"v69&§6!"!,/69b. ,& gmv686&(69W.
[b[ d
0
0.2
0.4
0.6
0.8
1
200 220 240 260 280 300
Fully compressible
Asymptotic compressible
T
ux
uy
s
0
0.2
0.4
0.6
0.8
1
200 220 240 260 280 300
Fully compressible
Asymptotic compressible
T
ux
uy
s
0
0.2
0.4
0.6
0.8
1
-1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2
Fully compressible
Asymptotic compressible
T
ux
uy
s
0
0.2
0.4
0.6
0.8
1
-1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2
Fully compressible1
Asymptotic compressible
T
ux
uy
s
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3
Fully compressible
Asymptotic compressible
Tux
uy
s
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3
Fully compressible
Asymptotic compressible
Tux
uy
s
]o 4, ZbZ ^_aC^_aC "S*Y,&2!&_^uJ[&C/69&(69s. ,&_^\#$R(69&WV' ^"&2BR",'>W(6";69&2;A2d;&2 xskW(6";O. 718^ #$RV/69,b¨Rg¨/69#%"''3
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q33
69&2;kgmS2$gm. .^| &u #%%.71.Bt = 47.1
z69&2;t = 62.8
^NQ. ._0:,(69&: "Q69S'eCR"";v &~2TLOX,& 8^
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[ · e
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3
Fully compressible
Asymptotic compressible
Tux
uy
s
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uy &A"!&q
23q
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t = 15.7gh56pW @769. ,2[9gt2,&286;CsB!,-`^`N. .V2Q0:,(69&: "|69['eCR"";d,2 &%LOX,& 8^
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• &. s6k &4.'RV"!"B71&¢9gi2vgh"'>#B69¥e/"69#%'&OB(6%WV"&¢ #$R.#%&:"; ^| &NiL · P)^ 2¢¨,. s¦/69 4:-gh ^569;(~'eO&2&9g269RRr6!/#T,. ¥RV"!":\(^RW.#%i[9g!",T&2"";";<gmQ/6!/O. &G:sC;Cr&k!"#TW,2&bRW.#%S69&2;\,2!/k6B/62,.;HRW(69W. svWVi2[c2zR~/691[gm. .^| &%2S!"&2!. ,2&<9g2iR//&:z2"8^
• 7r&:,(69. . s13 ^'^569&OBA"%(69B BB . .[6k!/(69. .&4 &)/669RR. sR"!"&2;C &28;!"#$R" W.,RM^| &2;B. 71KM2z". ,&d9gD,&286;Cs.^5Pb6!/B&:,#TWV\(^z8^Xx&2;"";D32M&2h^;'c2& &<9g2M,&286;CsH!,-A;" 4&2";vgm|2SPb69¥e:]2"M #$R. ! 869#%&:!869&&2|&2,i #T,. /69&2",2. sGB!&2!sH69&2;H6!"!,/6!sG;#B69&2;^ :,23C.U69WV/69M/69/'4"(687r|MWVz;71.RV";$gh86!/B!86/|/S6KM,&¥g¨s@2"|WVB"01, #%&:/8^`N¢. 1. s. ,&v! !,#T7r&:S,2!/\,2"83^,.;bWV@G/R(69/69@2@R"!"&2;C &2 &Hgm#T,.U69&k,2";<gh2p #%'; 769 71vF^|!/\"RV&2 W.SgmQ!"&:71r&2!"@B!&2!s[69&2;u(69Q,2";~ghQ269¥c(!U69.5;C R(69&#%HF^|!/!"&:.|6!"!,/6!s'^kL:,2!/6~R#$ &\gh#T,.U69&#B68sWVi #$R.#%&:"; ^| &<6@#@,. R.MR2/',(;9> #%igh/69#%h^ n [ 69&2;'^,.;HWVS!"/69 &. sv2,WC*"!'9gog¨,,S;71.R#%&:8^
[ ·bc
A)`§ u
! "$#%& ()%A*) A j./& 5#%- 9<7#f#%./; ! ( &&# 7=; #% -"#%!$.6#%> 5 2 ; ! Y) 2S;C #%&2&."QsC#9go2SI`,.Q"01,(69&2!869&~WV ^| &u,&2;z2igh. .^| &%gh#
wt + (fE(w))x + (gE(w))y = 0hNT^ Z ^ Z
^|2t
2[;C #%"&."5 #%3x
69&2;y
69/2[R(6!U69. !":;C &(69"3w = w(x, y, t)
2Q7r"!9g`!"&27r";H7469U69W."83(69&2;
fE 69&2;gE 69M2M &:7O!;,\(,CeG71"!|g¨,&2!&23
w =
ρ
ρu
ρv
ρE
, fE =
ρu
ρu2 + p
ρuv
ρuH
, gE =
ρv
ρuv
ρv2 + p
ρvH
.
hN@^ Z ^_[r
^|2Q69. .(2z;C #%&2&."01,(69&: "(6871|WV"&G69. 86;Cs$;'c2&2";B &%2|c2!/(69R8^ 2sC#z!.4";HW:sG2S;C #%&2&.""0:,(69&u9g`/69%
p = ρT = (γ − 1)ρ
(E − 1
2|~u|2)
= (γ − 1)ρe .mNT^ Z ^ ·
2S;C #%&2&."Q,&2;~RV"";Hz;'c2&2";u6|gh. .^zp
c =√γ T =
√γ p
ρ
2SRV"!¥c(!M/69.t&24s~69&2;~&:(69. R:sH!869&~2&~WVM;2c(&2";u6S
E =c2
γ(γ − 1)+
1
2(u2 + v2) , H =
c2
(γ − 1)+
1
2(u2 + v2) .
[ · g
[ ·bn
J[^S3/^S!869&u86 . sv!"#$R,S2i &:7O!; 6!"WU69&u#B69!""T
AE =
0 1 0 0
γ−32 u2 + γ−1
2 v2 (3 − γ)u −(γ − 1)v γ − 1
−uv v u 0
−γuE + (γ − 1)u(u2 + v2) γE − γ−12 (v2 + 3u2) −(γ − 1)uv γu
hN@^ Z ^ `:
BE =
0 0 1 0
−uv v u 0
γ−32 v2 + γ−1
2 u2 −(γ − 1)u (3 − γ)v γ − 1
−γvE + (γ − 1)v(u2 + v2) −(γ − 1)uv γE − γ−12 (u2 + 3v2) γv
hN@^ Z ^ c
)%A*)F@ ! #f-5!2( & #f14 7#%& 1 7#f# ./; ! Y)Xx&%;pW/69 &B2z r&2sO#; 71";gh#l2I`,."0:,(69&283f^|&2"";%2z/69&2-gm#B69&#B69!""Q,2!/H(69z2 6!"WU69&
AE z;CU69r&(69. ´8";\65gm. .^zS
P−1A ·AE · PA =
u
· u
· · u+ c
· · · u− c
2Mr&2/69.¬'eOR"&~9g2"M/69&2-gh#B69&u#B69 !""|4 71&HW:sb
Pκ =
1 0 ρ2c
ρ2c
u ρκyρ2c(u+ cκx) ρ
2c(u− cκx)
v −ρκxρ2c(v + cκy)
ρ2c(v − cκy)
u2+v2
2 ρ(uκy + vκx) ρ2c(H + c~V ·~1κ) ρ
2c(H − c~V ·~1κ)
69&2;
P−1κ =
1 − γ−12 M2 (γ − 1) u
c2(γ − 1) v
c2−γ−1
c2
1ρ(vκx − uκy)
κy
ρ −κx
ρ 0
cρ(γ−1
2 M2 − ~V ·~1κ
c ) 1ρ(κx − (γ − 1)u
c ) 1ρ(κy − (γ − 1) v
c ) γ−1ρc
cρ(γ−1
2 M2 +~V ·~1κ
c ) − 1ρ(κx + (γ − 1)u
c ) − 1ρ(κy + (γ − 1) v
c ) γ−1ρc
[ ·bd
2&κ = A
2&κx = 1
3κy = 0
69&2; ~1κ = ~1x^ 0 &\2T2[(69&2;D3^|2&
κ = B2&
κx = 03
κy = 169&2; ~1κ = ~1y
^
" & #% ' ( ! "$#%& ()@) A W# !2./& 2! V./& I# 147#f#%./; ! 2u;C #%&2&."dsC# 9gi2uJ[687O-YL:1d"0:,(69&2B!869&¢WV'^z &¢,&2;B2Hgh. .^| &gm#
wt + (fE(w) − fV (w,wx, wy))x + (gE(w) − gV (w,wx, wy))y = 0hNT^_[C^ Z
^|2fV 69&2;
gV ;&2M(i7O!",2 \2,Ce"83
fV =1
Re
0
τxx = 43µux − 2
3µvy
τxy = µ(uy + vx)
uτxx + vτxy + γµPrex
, gV =1
Re
0
τyx = µ(uy + vx)
τyy = 43µvy − 2
3µux
uτyx + vτyy + γµPrey
.hN@^_[C^_[r
NQ. .2d;C #%&2&."0:,(69&: "$69d;'c2&2";) &(69R Z ^ 2d7O /!",2 6!"WU69&#B69!""$69'eCR"";~6|gh. .^zp
AV0 (w,wx, wy) =
∂fV
∂w(w,wx, wy) ; BV
0 (w,wx, wy) =∂gV
∂w(w,wx, wy)
AV1 (w,wx, wy) =
∂fV
∂wx(w,wx, wy) ; BV
1 (w,wx, wy) =∂gV
∂wx(w,wx, wy)
AV2 (w,wx, wy) =
∂fV
∂wy(w,wx, wy) ; BV
2 (w,wx, wy) =∂gV
∂wy(w,wx, wy)
Xx& ;d/&2,u2<;CU69r&(69.i;#$ &(69&2!"b9gS2u #$R. ! G,RM^| &2;!/2#%43]^u&24."!";2!"&: W,&u9go2[/69&r&:U69. 6!"WU69&2z &v2i #$R. ! zRV/69T (69R Z '3C#%iR"!. s &%2
x;C "!& ^&. s/69 &2";B2 6!"WU69&
AV1
^| .z &%2y;C "!& ^z/69 &2";
BV2
h"69. Z a '^ 2" 6!"WU69&u#B69!""Q!869&~WV ^| /'&\65gm. .^zp
AV1 =
µ
Re ρAV
1 =µ
Re ρ
0 0 0 0
−43u
43 0 0
−v 0 1 0
−αu2 − βv2 − γPrE αu βv γ
Pr
[2`fe
BV2 =
µ
Re ρBV
2 =µ
Re ρ
0 0 0 0
−u 1 0 0
−43v 0 4
3 0
−αv2 − βu2 − γPrE βu αv γ
Pr
^|
α =4
3− γ
Pr
69&2;β = 1 − γ
Pr
%;'c2&2M2S'eCR. !' QRV/6983H^S69./&2"";H2M#B69¥eSV ^|!/~|4 71&~W:s\
SV = AV2 +BV
1 =µ
3Re ρSV =
µ
3Re ρ
0 0 0 0
−v 0 1 0
−u 1 0 0
−2uv v u 0
)@)F@ W# !2./& 2!:!3i; .7# #># 2!N[
AV1
69&2;BV
2
69[.^5U69&4,.U69|#B69!""83C2 |RV"!/69. /6;C ,2|!869&vWV[!"#$R,";H/69 4:-gh-^569;C. s1^%P<"71S2sb(692%/69#%%;CU69r&(69.#%83_ &:71. 7: &~(69S2 S"RV"!'¥71%RV"!/69./6;C ,2Q69S"01,(69.>^ RV"!/69._6;C ,2|z4 7r&~W:sb
ρV =µ
Re ρ· max
[γ
Pr,4
3
]
](|2p69 Q69z#%1;2'69S#$RV/69,43 ^i2&~(6871$
ρV =µ
Re ρ· γ
Pr
k`
#% +# & LO71/69.¬#%2:;i(687rpWV"&b;71.RV";< &\;[G,2;CsH2@B!&2!s<9gK&:,#%!869.`!/2#%"8^QXx&#%4z9go2"/i#%2:;832&2i&24."!z2S "!9go2iWV,&2;269sH!"&2;C &2zW:sv6/,#$ &$(69z2!"#$R,/69&(69._;#B69 &v. #$ ."|W:s%,2 &@RV:;C![WV,&2;269sd!"&2;C &28^ 2Q±&GJ[,#B69&&#%2:;),2"; & ^A\,2;CsA2G/69W . xs9gQ71/69.&:,#%!869.z!/2#%"@TW(6";)&)2](,S/69&2-gm# 69&2;k2$RV"!/69.69&(69. sOS9g2"%!/2#%"8^%Xx&kR/6!!"43¬&2%!/2"!/OS(69S27469. ,2z9g 2Q69#$R. ¥c(!869&Bg>6!9g 2!/2#%"K#B69 &2#B69. .K(69&B,& xs$&%2i^|2.;#B69 &_^ 2\±K&¦J[,#B69&&69&(69. sCv&(69W."~69.)" #B69\2^ g>6d2b,^| . .iWV\;269#$RV";lh69#$R. ¥c(";('3_R7O;C &u69&b;86G9g2^(!"&:7rr&2!"$/699g2%69. 4¥#71. 7r"8^ 0 g!",!"432K",. tW2/69 &2";pW:sM,2!/@6|#%2:;p(6871KzWV!"#$#%&:";@!869'g¨,2.¥. s1^oX&2;"";D392K±&pJ[,#B69&&2"sd"0:, "z6T. &2869zRW.#<^|!/H#%869&2|(69|2i!":B!&:9g¬(i"0:,(69&2z69/i!"&2/69&:69&2;D36B#%&:&¦869. 83|RV:;C!<WV,&2;269s¢!"&2;C &2H69\6,#%"; ^|!/&(69W."G2u,2u9g](,"8^<Xx&)R d9g2"B. #$ /69&23K/69W . xsA",. ^G2^|&)uWVd#%869& &g¨,.ghR/6!!869.!"#$R,/69&(8^d5";"S2$±&AJ[,#B69&&#%2:;!869&AWV$,2";k71sb86 . s\,4)6!"#$R,R4/69# &u;z.86;~sO#B69!S,2;C"8^
" (%& (Xx&G;5@,&2;/69&2;v2gh,&2;269#%&:/69.DR &2! R."z9gt2[±K&vJ[,#B69&&~69&(69. sC83O.,25!"&2;2M. &2869!869.U69Q6;C71"!&~RW.#~^
)@) A #%"; #!2./% # > -; .7b#%& ;102"7# &V
∂u
∂t+ a · ∂u
∂x= 0
hi^_[C^ Z n
uWVi2M'e26!. ,&u9g2M"0:,(69&hi^_[C^ Z '3a6;C!S/. ,&u9go"0:,(69&~86;p
uni = un
i + εnih[^_[C^_[r
[2` Z
[2`f[
^|2εni
;&2"M2$T69M #%$.7r.tn = n · ∆t 69&2;A69M#%"ARV &:
xi = i · δx ^ O,283D(!"&:";u!/2#%M86;p
un+1i − un
i
∆t+εn+1i − εni
∆t= − a
2δx· (un
i+1 − uni−1) −
a
2δx· (εni+1 − εni−1)
hi^_[C^ ·
εn+1i − εni
∆t= − a
2δx· (εni+1 − εni−1)
hi^_[C^ `: 2S
ε Q69.%6$. ,&u9g2M; /!p6;C7r"!&<"0:,(69&_^
)@)F@ ?& 2 #f# ; # j; #M# ; !Xg2WV,&2;269s\!"&2;C &2T69$6,#%";\GWVRV' :;C!43_2$
εni!869&bWV$;"!"#$RV4";k &:H6
](,S"i &kR(6!"$69M86!/k #%.71.>^L: &2!"2R(6!";#B69 &kM9g|6Bc2& .&4A&2 ^| . .(6871S6;C/!S](,zR"&:/69&\,#$#%";~716@c2& M&:,#TWVQ9go(69#%&!"8^Xx&H6@&2';C #%&2&(69.¬;#B69 &~9g¬.&4
2L3O2[#B6eC #T,#<^5687r.&4~(69|!869&HWVi!869R,"; ^|
6@#%" c24,Si^_[C^ Z S
λmax = 2Lhi^_[C^ c
min xδ= 2
max = 2L
xδ
L0
−L
]¬ 4,p[^_[C^ Z KPb6eC #T,#69&2;H#$ & #T,# ^5687r.&42Q!869R,";~W:sH6Tc2& M#%"0 &G2[2(69&2;D3O2Q#$ & #T,# ^56871.&4G(695!869&dWV[W/69 &2";G. "5&G2[ ´8Q9gt2Q#%"δx
λmin = 2δxhi^_[C^_ar
687r&:,#TWV[6:!U69";'^| ~2 ^56871.&42!869&~WVS;#$ &2";<65gm. .^zp
k =2π
λ
h[^_[C^ g4[&2!"432&2ir
kmin =π
L
kmax =π
δx
[2` ·
n N
WVK2K&:,#TWVo9gR(6!"'R2δx
"0:,";S;C! ´8.&4L
32K(69#%&!"¬R"&:";&~2i#%"u69M4 71&HW:sb
kj = j · kmin = j · πL
∀j ∈ [|0;N |] hi^_[C^ n 2S
εniz;"!"#$RV4";~ &:B6$](,"6p
εni =N∑
j=−N
Enj exp(ı · kj · iδx)
hi^_[C^ d
~R:;C,2!"kj · δx
;&2"Q6R(6p69&4.432z";C,2!"";'^567r&:,#TWV832 &~2S]2,2'QR(6!"%
φ = kj · δxhi^_[C^ Z er
:,2832S](,"|/"6;
εni =N∑
j=−N
Enj exp(ı · iφ)
hi^_[C^ ZbZ
2M";C,2!"";:^5687r&O,#TWV!"71|(M;#B69 &[−π; π]
&uR2z9gπ/N
φmin =δx
L· π
φmax = π
2~4&69,&2;φ = 0
!""RV&2;BA.^gh"0:,2&2!Q^| .~2~4& !.4/Hφ = π
6:!U69";'^| H2i 4Cmgh"01,2&2!sH9go2MRV"!,#~^KXx&vR(69!,.U698322i7469. ,2
φ = π6:!U69";
^| u2S 42"zgh"01,2&2!sv"/. 769W.p&u2S#%"_32&(69#%. sH2Mg¨"0:,2&2!sH9g^56871.&4λmin =
2δx^
)@) % 1435 # I./7# & ( .7b&&# 2[RW.#|6,#%";GTWV[. &28698386!/v(69#%&!M9gt2i
εni2'ghS6T. ,&H9g¬2
;C!S"0:,(69&
En+1 − En
∆t· exp(ı iφ) +
a
2δx· (En exp(ı (i+ 1)φ) − En exp(ı (i− 1)φ)) = 0
hi^_[C^ Z [rj[ 7:;C &%WOs
exp(ı iφ).86;z~
En+1 − En +σ
2En(exp(ı φ) − exp(−ı φ)) = 0
hi^_[C^ Z8·
^|2σ =
a∆t
δx
^ /69W . xs !"&2;C &§/69-c("; ¥g)269#$R. ,2;l9g69&:s§(69#%&!
En ;:"§&24^ &~ #%43(69z83C2M/69
|G| =
∣∣∣∣En+1
En
∣∣∣∣ ≤ 1 ∀φ . hi^_[C^ Z `:
[2`b`
G2|69#$R. ¥c(!8694&$g>6!834(69`K6gh,&2!&%9ga #%'/'R
∆t34";C,2!""; ^56871&:,#TWV
φ69&2;#%"
´8δx
^K](|2S!"&:";u!/2#%432&2M(6p
G = 1 − ı σ sin(φ)h[^_[C^ Zc
2M#%:;C,. ,2Q9gG
83C &~z!86/43C4869(69& Z gm69. ._";C,2!"";:^5687r&:,#TWVT
|G|2 = 1 + σ2 sin2(φ) ≥ 1 ∀φ hi^_[C^ Z ar&2"0:,2&:. sr322M!"&:";u/!/2#%M|,&2!"&2;C &(69. . sH,&2/69W.4^
(.# "$#% ( )*%) A #%"; #M#67# & &)( 7="; !"!7b; 1n z,2!"&2;z2M:sORVWV. !SsC#9g`!"&2769&u.U6^zp
wt + f(w)x = 0hi^ · ^ Z
^|2w(x, t)
z6T7r"!z9gm
!"#$RV&2&:Q69&2;f
|6 \2,Cedg¨,&2!&u6:!U69"; ^| v2 6!"WU69&#B69¥e
A = ∂f/∂w^ 2dsC# h[^ · ^ Z p!869&)WVG69RRr6!/2";)W:s6Hc(-;$6!"!,/69G,RM^| &2;
'eCR. !¥Q!/2#%%
wn+1i − wn
i = −∆t
δx
(δµfn
i − 1
2δ(|An|δwn)i
) h[^ · ^_[ri"!869. ._(69z2SR(6!"S;C! ´69&< Q6!/7r";~,2 &%;C D&2!"pRV/69p
δ[ψ(x)] = ψ(x+δx
2) − ψ(x− δx
2) , µ[ψ(x)] =
1
2ψ(x+
δx
2) +
1
2ψ(x− δx
2)
h[^ · ^ · ^|2
ψM;'c2&2";A69S86!/k#%"bRV &:
xi = i · δx 69&2;δx
i2$!"&2/69&:M#%"A ´84^T]o 83 2sC#. &2869 ´8";GW:sd6,#$ &@(692 6!"WU69&H#B69¥e
A!"&2/69&:^ :,220\2,Ce%g¨,&2!&
86;p
f(w) = A · w hi^ · ^ `:LO"!"&2;D322M. ¥&("69Q!/2#%M|#B6;S #$R. ! T
∀n ∈ N H · ∆wn = −K · wn h[^ · ^ c ^|2 H 69&2; K 69M. &2869QRV/69;'c2&2";<6S
∀i ∈ Z
H · ∆wni = (Id+ A µδ − 1
2|A| δ2)∆wn
i
K · wni = (A µδ − 1
2|A| δ2)wn
i
^| A =
∆t
δxA
69&2;δ2 = δ δ
^
h[^ · ^_arXx&;%k,2;Cs2u69#$R. ¥c(!869&+gm6!B9g[2v #$R. ! %,RM^z¥&(;+!/2#%~6/:!69/; ^| +2sC#9g!"&27469&u.U6^z832&2i&2"";~,2M2M](,/69&2-gh#~^
[2` c
)*%)F@ ?& 2 #f# ; # 7#!2( &&#f1L: &2!"(&:,#%!869.. ,&Ai&2M&2"!""/69 . s\RV:;C!43t&2$,2"i2$](,M/69&2-gh#~^]o 83&2$'eO&2;M2. &2869S!/2#%~hi^ · ^ c QGg¨,&2!&2S;'c2&2";k&2S&. sb69i2$R(69U69.`#%"kRV &:x = i ·δx 34W,gh69. .
x ∈ R^]2
ψ ∈ L2(R)m 34.ψ = Φ(ψ)
;&2|2|R(69U69.2](,`/69&2-gm#~3(69z|2igh,&2!&u9g
L2(R)m ;'c2&2";<6S
Φ(ψ) = ψ(ξ) =1√2π
∫ +∞
−∞
exp(− 7 ξ
δxx) ψ(x) dx .
hi^ · ^ g4^|2
ξ;&2"[2S";C,2!""; ^56871S&:,#TWVM69&2;u7 2 = −1
^z]2i69&:svg¨,&2!&ψ ∈ L2(R)m 3a!"#TW¥
&(69&29g`R(6!"S;C &2!"pRV/6986;A
(δµψ)(x) =ψ(x+ δx) − ψ(x− δx)
2,
(δ2ψ)(x) = ψ(x+ δx) − 2ψ(x) + ψ(x− δx) ,NQRR. s: &%2S](,z/69&2-gm#2"/[gh,&2!&2z.86;zH
Φ(δ1µ1ψ)(ξ) =7 sin(ξ)ψ ,
Φ(δ21φ)(ξ) = −2(1 − cos(ξ))ψ , :,28342/69&2-gm#%";R(6!"5;C D&2!"|RV/6969&2;@2 `!"#TW &(69&2K/69156Q7rsT #$R.gm#~^sd/69&2-gm#$ &d86!/u;M9g`"01,(69 &)hi^ · ^ c 5&2SW/69 &2p
H · ∆wn+1i = −K · wn+1
i ,h[^ · ^ n
^|2H
3K
69&2;wn+1 ;&2%2$/69&2-gm#%";bgh,&2!&2p9g H 3 K 69&2;
wn+1 "RV"! 7r. s1^ 2RV/69
H69&2;
K86;A
K = |A|(1 − cos(ξ)) +7 A sin(ξ)
H = Id+ |A|(1 − cos(ξ)) +7 A sin(ξ)
)*%) % 1435 # I./7# & ( .7b&&# 2k69#$R. !869&#B69¥e9gT2\. &2869v #$R. ! H,RM^| &2;¦/!/2#%k6:!U69";4^| ¦2bsC# 9g!"&27469&~.U6^zzz;'c2&2";<65gm. . ^zS
∀n ∈ N wn+1 = G · wn h[^ · ^ d Xx&H2iR"/&:!8643&(irp
G = Id−H−1 ·K hi^ · ^ Z er 2M±&~JQ,#B69&2&u/69W . xsv!"&2;C &<z;'c2&2";HW:sb
∀ ξ ∈ [−π, π] ρ(G(ξ)) ≤ 1hi^ · ^ ZbZ
^|2ρ(G)
$2HRV"!/69.z/6;C ,2%9g[2u69#$R. ¥c(!869&¢#B69¥eG
^[ 71&b2v.U69rH&:,#TWV9g[R(69/69#%%(69B&2v(6$b/69rv &:A6!"!",&:%A69&(69. sCv2H/69W . xs9gi,2!/¢6b!/2#%436&:,#%!869._,2;CsB,2 &$ ^"R2|7r52iR(6!"i9g¬2[R(69/69#%|"0:,";D^ 2 ^568sB$69&(69. sC2p69#$R. ¥c(!869&ugm6!zz'eCR.U69 &2";vgh,28^
[2`fa
" (%&( 4# " & "$#%& " #% ) ) A ; 1435%;+7=; 1435 # ./7b# & ( .7b&&# &)( 7="; W# #; .7 j&V -; #n 5,2!"&2;52 #$R. ! 5,R^| &2;G!/2#%Qgh2[I`,.5"0:,(69&2 &,^|!/v2['eOR. ! |/69r9ga ;O;"^| .52 #$R. ! K/69r9g(c2-;8^]( ^9;C #%&2&(69.2R2W.#~34 ^5[";C,2!"";^56871&:,#TWV5(687rQTWVQ;'c2&2";G &v86!/v;C "!&k
ξ &G2
x;C "!&u69&2;
η &G2
y;C "!&_^
Ng¨|2S](,z/69&2-gm#~32M. &2869z #$R. ! Q!/2#%M86;p
H · ∆wn+1i,j = −K · wn+1
i,j ,h[^ `2^ Z
^|
K =2
3
(D1 · z1 + D2 · z2
)+
7(AE s1 (1 − 1
3z1) + BE s2 (1 − 3
2z2)
)
H = Id + 2(D1 · z1 + D2 · z2
)+
7 (AEs1 + BEs2
)
:,2z2p69#$R. ¥c(!869&u#B69¥e~6/1!U69";:^| ~2SjQ "![LO. 71z86;p
G∗ = Id−H−1 ·K hi^ `2^_[r 2%±&J[,#B69&&)69&(69. sCp pW(6";&A2B,2;Csb9g|2BRV"!/69./6;C ,2T9g|@69#$R. ¥c(!869&
ξ / π = η / π
ρ(G
)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
CFL = 103
CFL = 102
CFL = 10
High FrequenciesLow Frequencies
]o 4,Si^ `2^ Z KNQ#$R. ¥c(!869&~g>6!5gm|2Sj[ "![LO. 7183M = 0.5
3u/v = 1
^
#B69¥eD^~Xgρ(G) < 1
2dT &2dsC# 69d;269#$RV"; ^|286T¥gρ(G) > 1
2d6969#$R. ¥c(";.86;C &<~ &2/69W . xs1^ 2%,2/%9g6H!"#$R,TR/4/69# &(69W."p,2Su;C/6^2%7469. ,29g2p69#$R. ¥c(!869&ugm6!i^| u"RV"!z$2 ^56871&:,#TWV
ξ69&2;
η^K]2z2S/691M9g` #$R. ! xsr3
& v69RRV&2;C¥eD3i^ ^,.;!"&2;G(69ξ = η
^](M = 0.5
3u/v = 1
69&2;71/69.M] n
[2`Mg
&:,#TWV@CFL = 10
3CFL = 102 69&2;
CFL = 103 '3C2p69#$R. ¥c(!869&ug>6!9go2S;C "!Q. 7r5R"&:";~&Gc24,Mi^ `2^ Z ^ Ai!869&H&2i(6983Ogm586!/<] n &:,#TWV832S69#$R. ca!"69 4&Hg>6!#B69 &2p#B69. .M(69&A,& xsugmS69. .?^56871&:,#TWV ^|!/k#%869&2S(69M2$j[ "!TLO. 71S!/2#%/69W.@gmM,2!/A!&(;C &28^ 0 W:7:,2.¥s13D2$ &2!86$9g2d] n &:,#TWVp&(69W."Sv #$R7r$2B!&2!sd9g_2[!/2#%4^`X&(;";_3M^Q!869&G&2Q(692Q769. ,2[9g_2i69#$R. ¥c(!869&Gg>6!;"!86"gmz 4b] n &:,#TWV8^5[^7r83H^T69.%&2S(692p;"!86Sz&2,&¥gm#,RV&u2 ^|2.R(6!"T9gK2 ^567r&:,#TWV8^](Q 4ug¨"0:,2&2!"83a(69Qzgm0^5687r&:,#@WV'i!.4p
π3( &2!86 &
2$] n &:,#TWVM;:"[&2[R7:;$6%4869i #$R7r#%&:M &2!"p269#$R. ¥c(!869&bg>6!i;:"[&2!/(69&[email protected]^ 0 &H2M2|(69&2;D3C2iz6T4869| &:"|/691[ 4<] n &:,#TWV| &~;|W/69 &<6WV;269#$R &B &~2M.^§g¨"0:,2&2!sH6986~¨(69z5gm|2 ^5687r&:,#TWVQ!.4M$´8'/1'^ ) )F@ #% / # 7= 7="; ./&V -; # 3); ./; #7b;
Iterations
Log
(Res
idua
l)
50 100 150 200 250-12
-10
-8
-6
-4
-2
0
CFL = 103
CFL = 102
CFL = 10
]o 4,Si^ `2^_[C&:71r&2!"SsG &~W,#$RC!/(69&&2.V^| 3600
.#%'&O83M = 0.5
^
Xx&S¬69RRV&2;C¥eD32^"^569&:t2^(69t2K",. _g¨#2K±K&TJQ,#B69&&69&(69. sC¬69K#%869& &gh,.gmR/6!!869.D!"#$R,/69&28^CA[!#$R2,";G20\(^¦7r|6SW,#$Rv &v6T!/(69&&2._69
M = 0.5,2 &$6
.:!/ n &2 6!"WV #$R. ! 869#%&:I^| Z ep,WC> /69&2^ 2R.9g_2[69#$R. ¥c(!869&Gg>6!9ga,2!/$6#%2:;@¬71sp!.4Q2&2W/69 &2"; ^| @2j[ "!KLO. 71/!/2#%4^JA,2";6#%"^|
3600.#%&:p69&2; ^!/24@2/69#%%] n &:,#TWVM(69&\ghi2@±K&kJQ,#B69&&69&(69. sC8^
2H",. d69HR"&:";+ &c24,ui^ `2^_[C^+N[@gm!86";W:s)2H±K&¢J[,#B69&& 69&(69. sC832T6v4869T;C D&2!"%WV ^"&2B!"&:7rr&2!"B/69BW/69 &2";W^|
CFL = 1069&2;
CFL = 103 ^5";"832M!"&:71r&2!"S/69"i^|
CFL = 102 69&2;CFL = 103 69i71sG!.4/S6z'eCRV"!";D^
T!869&k!"&2!. ,2;Tgh# 2"/T",. [(69i2@!"&:71r&2!"/69@9g6d#%2:;\[#B69 &. s<;C 7r&bW:s [B!&2!su &<2p.^g¨/0:,2&2!s<6986C^QX&(;";_3(ghQ 4<g¨/0:,2&2!"83a269#$R. ¥c(!869&\g>6!Q69RR8eC #B69 7r. sd2[/69#%zgm569. ._] n &:,#TWV ^| .Q 71s%;C D&:gm.^ gh"01,2&2!"^ 2R/6!!869.¬!"#$R,2/69&2Q&\69&<6!86;#$!S"-!86S9gK6 \(^71Q6W,#$Ru &<6!/(69&&('.o,. &2S2
[2` n
&H &/\2,2&2!"B9g2$!/2!"$9g2B] n &:,#TWVp&k2$!"&:7rr&2!"%/6943t69&2;kiWV(687O,!869&~WVM&2.¥sH;C,2i/2S;C D'/'&(!"QW271";v &~2M.^§g¨"0:,2&2!sv;#B69 &_^ ) ) % i; .7 &)( 7=; 14; !$= !$#6;n ,2!"&2;6S;#B69 & ^| v6M.&4G9g
2L69&2;d6S!"&2/69&:#%"d ´8
δx^Hz(^";% &d"!&
i^_[%(69[2 ^56871.&42 ^|!/b!869&<WVT!869R,2";\&\,2!/b6d;#B69 &\/69&rpg¨#2δx
2L
^ 2!""RV&2;C &k";C,2!""; ^56871&:,#TWV/69&rdgh#
πδx/Lπ
^ :,283`2G ´8G9g2vRV"!,#. "@&2B#%" ´84^ B/6^(69T2%4&69,&2;
πp.U69";~2d2 ^56871.&42
^| .v2G4&¢69,&2;0
R"&:(G.& ^56871.&428^ n ,2R"!G(^¯2G&2&+9g2|.&l^|687r.&4u|#%"u;RV&2;&:8^n ¢,2¢!"&2;¢"§#%"2"83\6!r69/§#%"_3\6#%";C ,# #%"69&2;6c2&2#%";'c2&2";6S
δxc = 2 · δxm = 4 · δxfh[^ `2^ ·
2M#$ & #T,# ^56871&:,#TWV@k = 2π/λ
W/69 &2";u&~2"M"M#%"2"z|2S/69#%43&(69#%. sb
kminc = kmin
m = kminf =
π
L.
h[^ `2^ `:0 &~2S2z(69&2;D32M#B6eC #T,# ^56871&:,#TWV[69$
kmaxc =
π
δxc, kmax
m =2 · πδxc
, kmaxf =
4 · πδxc
.h[^ `2^ c
J[^S3C.z,2z!"&2;2i";C,2!"";:^56871&:,#TWV0^|24S;'c2& &u|4 71&HW:sbφ = k · δx hi^ `2^_ar
Xx69RRV869[!.869. sv(692S";C,2!"";:^56871&:,#TWVπ
^,.;u!""RV&2;~d6$;C D&:0^56871&:,#TWV^| ~"RV"!z$2M#%"u ´84^`](z2[c2&(|#%"_3/^M(6871$
kmaxc · δxf =
π
4, kmax
m · δxf =π
2, kmax
f · δxf = π ,hi^ `2^ g4
^| .igm|2S!"r69M#%"_3kmax
m
69&2;kmax
f
69i&2zR"/'&O";~/ ^i(6871$kmax
c · δxc = π .hi^ `2^ n
:,2831gm2[!"r69#%"_3:2 ^5687r&:,#TWVkmax
c
WV.&rp2 4dg¨"0:,2&2!sB69/"6 ^|286gh2|c2&2|#%"t3r KWV.&rM2z.^5mg¨"0:,2&2!s%6986C^ 2z6M¥ghK^569;2z.^5mg¨"0:,2&2!s%625#%"% ´8|;C #$ &2"8^ 25 4$gh"01,2&2!"9gD6i!"r69|#%"$#%71";i2|.^5mg¨/0:,2&2!sR(699g¬2MRV"!,# c24,Mi^ `2^ · ^|2&H2[#%"H ´8Qg>69. .5`^`NQ|6T",. 83O2M/69#% ^56871.&4^,.;\WV@;269#$RV";b; D&:. s ^| b"RV"!id2T#%"k ´84^i]([ &2/69&2!"43Vgh[2jQ "!pLO. 71!/2#%G,2 &
CFL = 103 32,^5687r.&4.U69";)kmax
c
^,.;)WVG;269#$RV";+6!"!/;C &b";C D&:Q69#$R. ¥c(!869&ugm6!p
ρ (G(kmaxc ))|c = ρ (G(π)) = 0.33 ,
ρ (G(kmaxc ))|m = ρ (G(π/2)) = 0.6 ,
ρ (G(kmaxc ))|f = ρ (G(π/4)) = 0.57 .
[2` d
φ / π
Am
plifi
catio
nFa
ctor
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
Shift towards the low frequencies
as δx diminishes
]o 4,Si^ `2^ · n ^5mgh"0:,2&2!sH¥gh
2v2$D"!$9g2G#%" ´[email protected]";\2v6!"!,/6!s1^NQ@2G#%"WV"!#%"c2&283K2R"&:/69&%9g(2.&r"^5687r.&4$o#%56!"!,/69z69&2;T(#$ & #T,#";C,2!""; ^56871&:,#TWVWV"!"#%"[71s~!.4pB´8^ 2T#%2:;<WV"!"#%"[."[B!&:[WV"!869,2T269#$R. ¥c(!869&\g>6!&2;@<,& xskgm@,2!/)";C,2!""; ^56871&:,#TWV8^bP\"7r83?^G2^";(69@2GB!&2!s,RV&.^5mgh"01,2&2!s)6986<#B69 &. s;C 7r"@2G!"&:71r&2!"G/6943`2&2!"436<.^z;^|&9g2G!"&:71r&2!"/69,^,.;)WVv'eORV"!";+6@2G#%" ´8dWV"!"#%"@c2&28^X&)R/6!!"43KR2&2#%&(6\@,2.¥sW271";D^
`
Z ^(NQ#B69.U6;26Q69&2;u@^«S69#B69_^Xx#$R. ! Q69&(;H#T,. 4;uR:!"";C,"gh86;Cs:/69S!"#$R,2/6&2i^| ~,RM^| &2;<69. r #%8^ 3H[ n Zn g[ Z [C3 Zdbdbd ^
_[ ^O|69H69&2;Gj^ "RV&_^ 2[;" 4&H69&2;v69RR. !869&H9gt,RM^| &2;v!/2#%"&G,&2,2!,2";#%"2"^ 3R(69r" nbd e · abaC3 Zdbnbd ^
· T^5869#69&2;%p^#69#$ &2^ 0 &%2|!"&2,2!&G69&2;B69RR. !869&d9gV #$R. ! `g>6!";%!/2#%"gmz!"&27469&~.U6^z^ 3 ZbZ nbc Z [ d 3 Zd g n ^
` N@^"!"!869&: &>3(]5^jM69WWV&243aLV^«i,2;CU6917V3 ^2P\694&(69,2;_32T^q69 . .43a69&2;HI^(L:,2;8^L: #T,.U6&H9g_:sC;Cr&v.86i69&2;G!"#TW,2&v &G.U69ri!869.Qr"#%"8`P\:;.|69&2;d#%2:;8^¬Xx& 32q69832]2/69&2!"43VLOR8^ [b[[2` [bebe · ^
c N@^15"!"!869&: &D69&2;d«$^CLV^DZ&2,2!,";%c2&2¥/7r. ,#%Q/!/2#%"gh(z.^ #B6!/G&:,#TWV!"#R"/ W.BJi687:-YL:1"T"0:,(69&28^vXx&:&(69.RV@9gzjiP [rLLO]P\I nO P\] 9 $eb[ e2` n NT3IKN@3aLC6!.U68s132]/69&(!43 [bebeb[C^
_a N@^1&:/69°mWt3 ^r±&2;.>3OT^OL: #%&_3 n ^1. ,#%&Cgm.;D3OX'^ 469!/2&13O69&2;d@^rq`69 . . f4^tN -869#"/ &G2Q#$/6Sg>6! . xsuIeCRV #%&:/69.D",. |69&2;d769. ;269&H9gt2Q. ,#$RV";O>R(69/69#B !'gm;G&:,2|!":;4^oXx& ! #"$ &% '% )( 3NQ¥e:&CqK7r&2!"43a]/69&2!"43(JQ7 ^ Z ` Z al[bebe c ^
g @^O *Y.D69&(;Gq^ A"/. ¥&22^¬N§,2&¥c(";G#%2:;Ggh|!"#$R, &$ &2!"#$R" W.p69&2;v!"#$R" W.\(^z &BWV,&2;269s:mc2";G!":;C &(69"8^+* -,.(!/ 0 !1 23(!24 5,.(5/ '46 3 Z ` Z Zc4· Z g · 3 Zdbdbn ^
n ^( .su69&2;<T^(P<!"ji&(69.;D^LO. ,&b9g`2S#T,2.¥ ;¥#%&2&(69.!"#$R"/ W.TJi687:-YL:1""0:,(69&2W:sB6Sr'&('69.¥ ´8";v¥#$R2.¥! 5#%2:;D^* ',.(5/ 70 5123(!24 5,.(5/ -486 3/[2`2 · g$[ ·bd gO3Zd gbgO^
d ^| .s§69&2;T^|P<!"ji&(69.;D^NQ&§717Oh^ 69&2;¦r&2/69. ´694&9g@ #$R. ! u&(687:-1"69. r #%[69&2;u69RR8eC #B69Mg>6!'/ ´869 &t^ 512:9 /4 3 · eC n eg n [ n 3[bebe Z ^
Z e qo^t,2.^S3¬j^¬LO!/M^83 ^t]2&2369&2;[^_P\O.4^BNR"!"&2;C &2";;C,(69.¥> #%43;CU69r&(69. ´8";6;Ct!/2#%igm|,&286;CsH!"#$R,/69&28^ ( ;9 3R(69r" d g[ Z e Z 3 Zdbd gO^
ZbZ qo^(,2.^S3VLV^±K&469" ^569/69&_3a69&2;<[^2P\O.4^KID"!Q9g`6RV"!z/69B&u!"&:7rr&2!"4^<* ',.(5/ 3 · [C_[2`fe Z [2`fe n 3 Zdbd `2^
Z [ qo^C,2.^S32LV^:±&69" ^569/69&_3269&2;H[^OP\:.4^L:/69W . xsG69&2;G!"&:71r&2!"M69&(69. sC59gt #$R. ! ,RM^| &2;u!/2#%"8^= 512:9 / 4 3 · eC d a Z dbnbn 3[bebe Z ^
[ c/Z
[ c [
Z8· LV^S(69R,. 83d[^Si,;C &_3 ^iq`68sr&_3 ^iPb694&(69,2;D3$69&2;N@^[P\&(6871&_^ QsO;C/9>2#B69.¥#%"!/(69&!869.z69&(69. sO@9g|2#B69.gm69 4,2B &6H#$¥eO &<"4^ 6 / 9)( , 54 , 9 9 '4 , (5, 9 -4 5, 3·bd _a Z ga3 [bebe c ^
Z ` ¶$^ @^a2o69&(;\[^(P<O.4^ 2T69RR. !869&b9g`R"!"&2;C & &2G &u7:/!",2]\(^z8^* ',.(5/ 70 5123(!24 5,.(5/ -46 3 Z e c _[beg[b[ · 3 Zdbd4· ^
Zc N@^2 &_^rN&:,#%!869.1#%2:;Mgm_. 7O &z2` &2!"#$R" W.7:!",2\(^ARW.#%8^;* ',.(5/70 5123( 24 5,.(!/ -486 3 [C Z [C3 Zd agO^
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