adv ection
TRANSCRIPT
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. ,
. :
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u(t, x)t
+ Au(t, x)x
= f, u=u1...un
, A= a11 . . . a1n
... . . .
...an1 . . . ann
,
(A)
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(t, x)
t +c
(t, x)
x =f
(, (t, x)) c.
2E(t, x)
t2 c2
2E(t, x)
x2 =f
(, E(t, x)) c .
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2E(t, x)
t2 c2
2E(t, x)
x2 =f
u= E
x
, v= E
t:
v
t c2
u
x =f
u
t
v
x = 0
t
u
v
+
0 c2
1 0
x
u
v
=
f
0
, (A
) = c .
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, , . , , . , (t, x) ( )
, . unm u n m
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un+1m u . (n+ 1,m) . , :
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(n+ 1,m). , un+1m . . .
, (n+ 1,m)
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.
.
tn+1, xm ,
,
ut
+cux
= 0
x ct= const. x ct=xm ctn+1, t
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--
,
., , , . ,
- . ( h), .
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, , y(t, x)
unm = [y]n
m
nm. [y]n
m
(tn, xm).
, , .
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un+1m u
nm
+c
unm unm1
h =fnm
un+1m unm
+c
unm+1 u
nm
h =fnm
un+1
m un
m
+cu
n
m+1 un
m12h
=fnm
un+1m 1
2 unm1+u
nm+1
+c
unm+1 u
nm1
2h =fnm
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[y]n+1m = [y]n
m+[yt]n
m+2
2[ytt]
n
m+O(3)
[y]nm1= [y]n
m h[yx]n
m+h2
2[yxx]
n
m h3
6[yxxx]
n
m+h4
24[yxxxx]
n
m+O(h5)
[yt]n
m+
2[ytt]
n
m+O(2) +c
[yx]
n
m h
2[yxx]
n
m+O(h2)
=fnm+
([yt]nm+c[yx]nm fnm) + 2 [ytt]nm+O(2) c
h2 [y
xx]nm+O(h2)
=
=
2[ytt]
n
m ch
2[yxx]
n
m+O(2 +h2) =O(+h)
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[y]n+1m = [y]n
m+[yt]n
m+2
2[ytt]
n
m+O(3)
[y]nm1= [y]n
m h[yx]n
m+h2
2[yxx]
n
m h3
6[yxxx]
n
m+h4
24[yxxxx]
n
m+O(h5)
[yt]n
m+
2[ytt]
n
m+O(2) +c
[yx]
n
m+h2
6[yxxx]
n
m+O(h4)
=fnm+
([yt]n
m+c[yx]n
m fn
m) +
2 [ytt]
n
m+O(2) +ch2
6[yxxx]
n
m+O(h4)
=
=
2[ytt]
n
m+ch2
6 [yxxx]
n
m+O(2 +h4) =O(+h2)
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[y]n+1m = [y]n
m+[yt]n
m+2
2[ytt]
n
m+O(3)
[y]nm = [y]n
m h[yx]n
m+h2
2[yxx]
n
m h3
6[yxxx]
n
m+h4
24[yxxxx]
n
m+O(h5)
[yt]n
m+
2[ytt]
n
m+O(2) +
h2
2[yxx]
n
m+O
h4
+
+c[yx]nm+ h2
6[yxxx]
n
m+O(h4) =fnm+
([yt]n
m+c[yx]n
m fn
m) +
2[ytt]
n
m+O(2) +
h2
2[yxx]
n
m+O
h4
+
+ch2
6
[yxxx]n
m+O(h4) =
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=
2[ytt]
n
m+h2
2[yxx]
n
m+ch2
6[yxxx]
n
m+O
2 +
h4
+h4
=O
+
h2
+h2
=O
+
h2
h., =O(h), , =O(h2), O(1), . ()
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. , c
h >1 c
h
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un+1m =un
m+c
h
unm1 u
n
m
+fnm, m= 1,M
un+1m =
1 c
h
unm+
c
hunm1+f
n
m, m= 1,M
|un+1m | 1 ch |unm| + ch |unm1| +|fnm|, m= 1,M
un = maxm=0,M
|unm|, u = maxn=0,N
un,
un+1 max
|un+1
0 |,
1 c
h
maxm>0
|unm| +c
h maxm>0
|unm1| +fn
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un+1 max
|un+1
0 |,
1 c
h
maxm>0
|unm| +c
h maxm>0
|unm1| +fn
maxm>0
|unm1| u
n, maxm>0
|unm| un
un+1 max{|n+1|, un +fn} max{|n+1|, un} +f
max{|n+1|, |n|, un1} + 2f max{, } +Tf
u + +Tf
,
.
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,
un+1m =
un
m+
>0, . . , .
un+1m =
1
2+
c
2h
unm1+
1
2
c
2h
unm+1
ch
1,
.
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. . , .
, , . , 0. u0m =e
im. , .
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O(+h2)
un+1m unm
+c
unm+1 u
nm1
2h = 0
u0m =eim
u1m
u1m =u0m+
c
h
u0m+1 u
0m1
2 =u0m
1 +i
c
h sin
,
unm =
1 +ic
h sin
neim
unm =neim.
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O(+h2)
unm
u = maxm
maxn=0,N
|neim| = maxn=0,N
|n| = max(1, ||N)
|| 1, (C = 1),
, .
|| = 1 +D, (C= (1 +D)T/ eDT), .
, D, , .
||1 , .
|| >1,
.
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O(+h2)
= 1 +ic
h sin
|| =
1 +
c22
h2 sin2 >1
=O(h), ||1 =O1
. .
=O(h2), || =
1 +csin2
1 + csin2
2 1 + c
2. C=e
cT
2
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!