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Computational Fluid Dynamics


http://www.nd.edu/~gtryggva/CFD-Course/

Grétar Tryggvason

Lecture 8February 6, 2017

Computational Fluid Dynamicshttp://www.nd.edu/~gtryggva/CFD-Course/

Diffusion versus dispersion

Godunov Theorem

Godunov Method

Upwind and Flux Splitting

Higher order schemes

Artificial viscosity


Modified Equation


Looking at the structure of the error terms—by deriving the Modified Equation—can often lead to insight into how the approximate solution behaves


Derive modified equation for upwind difference method:

0)( 1

1

=−+Δ−

−

+nj

nj

nj

nj ff

hU

tff

Using Taylor expansion:

+Δ∂∂+Δ

∂∂+Δ

∂∂+=+

62

3

3

32

2

21 t

tft

tft

tfff n

jnj

+∂∂−

∂∂+

∂∂−=− 62

3

3

32

2

2

1

hxfh

xf

hxf

ff nj

nj


Substituting

⎭⎬⎫

⎩⎨⎧

−⎥⎦⎤

⎢⎣⎡ +Δ

∂∂+Δ

∂∂+Δ

∂∂+

Δnj

nj ft

tft

tft

tff

t

621 3

3

32

2

2

062

3

3

32

2

2

=⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ +

∂∂−

∂∂+

∂∂−−+

hxfh

xf

hxf

ffhU n

jnj

Therefore,

+−Δ−+Δ−=∂∂+

∂∂

xxxtttxxtt fUh

ft

fUh

ft

xf

Utf

6622

22

It helps the interpretation if all terms are written in xxxxx ff ,


Taking further derivatives (first in time, then in space):

+−Δ−+Δ−=+ xxxtttttxxttttxttt fUh

ft

fUh

ft

Uff6622

22

++Δ+−Δ=−− xxxxtttxxxxttxxxtx fhU

ftU

fhU

ftU

fUUf6622

22222+

ftt =U2 fxx + Δt − fttt

2+U2fttx +O(Δt)

⎛⎝⎜

⎞⎠⎟

+Δx U2fxxt −

U 2

2fxxx +O(h)

⎛⎝⎜

⎞⎠⎟


Similarly, we get ),(3 htOfUf xxxttt Δ+−=

),(2 htOfUf xxxttx Δ+=

),( htOUff xxxxxt Δ+−=

Final form of the modified equation:

( ) ( ) xxxxx fUh

fUh

xf

Utf 132

61

22

2

+−−−=∂∂+

∂∂ λλλ

+O h3,h2Δt,hΔt 2 ,Δt 3⎡⎣ ⎤⎦

htUΔ=λ


Numerical dissipation (diffusion)

( ) ( ) ++−−−=∂∂+

∂∂

xxxxx fUh

fUh

xf

Utf 132

61

22

2

λλλ

By applying upwind differencing, we are effectively solving:

Also note that the CFL condition 1<Δ=htUλ

ensures a positive diffusion coefficient

Dissipation( ) xxf

Dispersion( ) xxxf


�

∂f∂t

= α ∂ n f∂xn

�

f (x, t) = a(t)eikx

�

da(t)dt

= (ik)nαa(t)

�

a(t) = aoe( ik )nαt

For even n we get diffusion, for odd n we get dispersion

Consider:

Look for solutions of the form:

Substitute to get

solve


First-Order Methods and Diffusion

Upwind:∂f∂t+U ∂f

∂x=Uh21−λ( ) fxx −

Uh2

62λ 2 −3λ +1( ) fxxx

Lax-Friedrichs:

∂f∂t+U ∂f

∂x=Uh2

1λ−λ

#

$%

&

'( fxx +

Uh2

31−λ 2( ) fxxx

Dissipative = Smearing

htUΔ=λ


Second-Order Methods and Dispersion

Lax-Wendroff:∂f∂t

+U∂f∂x

= −Uh2

21− λ2( ) fxxx − Uh

3

8λ 1− λ2( ) fxxxx

Beam-Warming:

∂f∂t

+U∂f∂x

=Uh2

61− λ( ) 2 − λ( ) fxxx −

Uh3

81− λ( )2 2 − λ( ) fxxxx

Dispersive = Wiggles

htUΔ=λ


dt=0.25*h

20 40 60 80-0.5

0

0.5

1

1.5Lax-WendroffUpwind


First-Order Methods and Diffusion

Upwind:∂f∂t+U ∂f

∂x=Uh21−λ( ) fxx −

Uh2

62λ 2 −3λ +1( ) fxxx

Lax-Wendroff:∂f∂t+U ∂f

∂x= −

Uh2

21−λ 2( ) fxxx −Uh

3

8λ 1−λ 2( ) fxxxx

Dispersive = Wiggles

Dissipative = Smearing

�

λ = UΔth


Godunov Theorem


Question: How can we avoid oscillatory behavior?(How can we preserve monotonicity)

Godunov Theorem (1959):

“Monotone behavior of a numerical solution cannot be assured for linear finite-difference methods with more than first-order accuracy.”

Godunov’s Theorem

Computational Fluid DynamicsGodunov Theorem - 1

0=∂∂+

∂∂

xf

Utf

For linear equation

we have:2

22

2

2

;xf

Utf

xf

Utf

∂∂=

∂∂

∂∂−=

∂∂

A general linear scheme can be written as:

∑ ++ =

k

nkjk

nj fcf 1

Expanding in Taylor series aroundnkjf +

njf

)(2

32

222

hOxfhk

xfkhff n

jnkj +

∂∂+

∂∂+=+


Substituting

Also, can be Taylor expanded in time around

∑ ∑∑ +∂∂+

∂∂+=+

k kk

kkk

nj

nj hOck

xfhkc

xfhcff )(

232

2

221

(a)=(b) (2nd order accurate):2

2;;1 ⎟⎠⎞⎜

⎝⎛ Δ=Δ−== ∑∑∑ h

tUckhtUkcc

kk

kk

kk

1+njf

njf

)(2

32

221 tO

tft

tftff n

jnj Δ+

∂∂Δ+

∂∂Δ+=+

)(2

32

222 tO

xftU

xftUf nj Δ+

∂∂Δ+

∂∂Δ−=

(a)

(b)


Condition for monotonicity:11

11 then,If ++++ >> n

jnj

nj

nj ffff

Since ∑ ++ =

k

nkjk

nj fcf 1

( )∑ +++++

+ −=−k

nkj

nkjk

nj

nj ffcff 1

111

> 0 The above should be valid for arbitrary

00 111 >−⇔> ++

+nj

njk ffc

njf

Is it possible?


Hence we get:

22;;1 ⎟

⎠⎞⎜

⎝⎛ Δ=Δ−== ∑∑∑ h

tUckhtUkcc

kk

kk

kk

ckk∑⎛⎝⎜

⎞⎠⎟

k2ckk∑⎛⎝⎜

⎞⎠⎟= kck

k∑⎛⎝⎜

⎞⎠⎟

2

ek2

k∑⎛⎝⎜

⎞⎠⎟

k2ek2

k∑⎛⎝⎜

⎞⎠⎟= kek

2

k∑⎛⎝⎜

⎞⎠⎟

2

Define: )0(2 >= kkk cec

This violates Cauchy inequality a ⋅a( ) b ⋅b( ) ≥ a ⋅b( )2

Monotone 2nd Order scheme is impossible!

a

ba = e1,e2,e3,…,ek,…( ); b = 1e1, 2e2,3e3,…,kek,…( )

a ⋅a( ) b ⋅b( ) = a ⋅b( )2or

where


The key word in Godunov’s theorem is linear. To overcome its limitations, look for nonlinear scheames:

1.  Use central differencing and introduce numericalviscosity where needed (Artificial viscosity)

2. Limit the fluxes in such a way that oscillations are avoided (high-order Godunov methods, flux-corrected transport, TVD shemes, etc.)


Godunov’s Method


Godunov Method – Finite Volume Method + Riemann solver

∂f∂t

+∂∂x

F( f )[ ] = 0

Consider a 1-D Conservation Eq:

Integrating across the j-th cell:

Godunov Method

Fj-1/2 Fj+1/2

1+jff j−1

jf

xj j+1 j-1

0)()( 2/12/1

2/1

2/1=−+

∂∂

−+∫+

−jj

x

xxFxFfdx

tj

j

Define the cell-average: fav =1Δx j

f dxxj−1/2

x j+1/2∫Δx j

∂fav∂t

+ F(x j+1/2 ) − F(x j−1/2 ) = 0

Δx j = x j+1/2 − x j−1/2


Integrating over time

where the time integration is done by solving Riemann Problem for each cell boundary:

favn+1 = fav

n −ΔtΔx j

1Δt

Fj+1/2 dt −1Δt

Fj−1/2 dtt

t+Δt

∫t

t+Δt

∫⎡⎣⎢

⎤⎦⎥

tΔ

[ ] 0)( =∂∂+

∂∂

fFxt

f

⎪⎩

⎪⎨⎧

>≤

=++

+

2/11

2/1)0,(jj

jj

xxf

xxfxf 1+jf

jf

xj j+1

t

2/1+= jcdtdx

t+Δt

Godunov Method


Wave Diagram:

1+jff j−1

jf

j j+1 j-1

ShockShockRarefaction

Wave

Godunov MethodComputational Fluid Dynamics

Godunov Method: The Procedure

1. Construct a cell average value

fav =1Δx j

f dxxj−1/2

x j+1/2∫2. Solve a Riemann problem to find the time integration of fluxes

3. Construct new

4. Go to 1.

avf

ttt Δ+=

Godunov Method


Special Case: Linear Advection Equation

0,0 >=∂∂+

∂∂

Uxf

Utf

for which the Riemann problem is trivial and we get

( )nj

nj

njav

njav ff

htUff 1,

1, −+ −Δ−=

jj UfF =+ 2/1

1st Order Upwind Scheme


fi+1

Consider the following initial conditions:

1f j−1 fi

Fj−1 / 2 = Ufj−1n = U

Fj+1 / 2 = Ufjn = 0

f jn+1 = fj

n −Δth(Fj+1 / 2

n − Fj−1 / 2n )

Godunov Method

Computational Fluid DynamicsGodunov Method

Advect—Average—Advect—Average—etcThe averaging leads to numerical diffusion, even though the advection is exact!

Computational Fluid DynamicsGodunov Method

Numerical versus exact solution


For nonlinear equations and systems the full Rieman problem must be solvedSince only the fluxes are needed, often approximate solvers are used


Flux Vector Splitting


For upwind schemes, it is necessary to determine the upstream direction. For systems with many characteristics running both left and right, there is not one “upstream” direction


Generalized Upwind Scheme (for both U > 0 and U < 0 )

0),( 11 >−Δ−= −

+ UffhtUff n

jnj

nj

nj

0),( 11 <−Δ−= +

+ UffhtUff n

jnj

nj

nj

Define

�

U + = 12U + U( ), U− = 1

2U − U( )

The two cases can be combined into a single expression:

�

f jn+1 = f j

n − ΔthU +( f j

n − f j−1n ) +U−( f j+1

n − f jn )[ ]

Upwind Scheme - Revisited

Where we have split the flux in an “upwind” and “downwind” part


For a system of equations there are generally waves running in both directions. To apply upwinding, the fluxes must be decomposed into left and right running waves


Steger-Warming (1979)

A system of hyperbolic equations

Flux Splitting

�

∂f∂t

+ ∂F∂x

= 0

can be written in the form

�

∂f∂t

+ A[ ] ∂f∂x

= 0; A[ ] = ∂F∂f

The system is hyperbolic if

�

T[ ]−1 A[ ] T[ ] = λ[ ]; T[ ]−1 =q1T

qNT

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥


�

F = A[ ]f = T[ ] λ[ ] T[ ]−1fThe matrix of eigenvalues is divided into two matrices

�

A[ ] = A+[ ] + A−[ ] = T[ ] λ+[ ] T[ ]−1 + T[ ] λ−[ ] T[ ]−1Hence

[ ]λ

�

λ[ ] = λ+[ ] + λ−[ ]

Define

�

F = F + + F−

�

F + = A+[ ]f, F− = A−[ ]f( )Conservation law becomes

�

∂f∂t

+ ∂F+

∂x+ ∂F−

∂x= 0

Flux SplittingComputational Fluid Dynamics

Example: 1-D Hyperbolic Equation

�

∂2 f∂t 2

− c 2 ∂2 f∂x 2

= 0

xf

wtf

v∂∂=

∂∂= ;

vtwt

⎛⎝⎜

⎞⎠⎟+0 − c2

−1 0⎛⎝⎜

⎞⎠⎟vxwx

⎛⎝⎜

⎞⎠⎟= 0

Leading to

f =vw

⎛⎝⎜

⎞⎠⎟; F =

−c2w−v

⎛⎝⎜

⎞⎠⎟; A[ ] = ∂F

∂f⎡⎣⎢

⎤⎦⎥=

0 − c2

−1 0⎡

⎣⎢

⎤

⎦⎥

Flux Splitting


Leading to:

�

T[ ]−1 =q1T

q2T

⎡

⎣ ⎢

⎤

⎦ ⎥ =

1 − c1 c⎡

⎣ ⎢

⎤

⎦ ⎥

�

λ+[ ] =c 00 0⎡

⎣ ⎢

⎤

⎦ ⎥ ; λ−[ ] =

0 00 − c⎡

⎣ ⎢

⎤

⎦ ⎥

�

F + = 12cv − c 2w−v + cw

⎡

⎣ ⎢

⎤

⎦ ⎥ ; F− = 1

2−cv − c 2w−v − cw

⎡

⎣ ⎢

⎤

⎦ ⎥

�

A+[ ] = T[ ] λ+[ ] T[ ]−1; A−[ ] = T[ ] λ−[ ] T[ ]−1

�

F + = A+[ ]f; F− = A−[ ]f

Flux Splitting

For a nonlinear system of equations such as the Euler equations, there is some arbitrariness in how the system is split


Solve using first order upwinding with flux splitting

�

∂f∂t

+ ∂F+

∂x+ ∂F−

∂x= 0

�

F + = 12cv − c 2w−v + cw

⎡

⎣ ⎢

⎤

⎦ ⎥ ; F− = 1

2−cv − c 2w−v − cw

⎡

⎣ ⎢

⎤

⎦ ⎥

vw⎡

⎣⎢

⎤

⎦⎥j

n+1

=vw⎡

⎣⎢

⎤

⎦⎥j

n

− Δth12

cv − c2w−v + cw⎡

⎣⎢

⎤

⎦⎥j

n

−cv − c2w−v + cw⎡

⎣⎢

⎤

⎦⎥j−1

n

+−cv − c2w−v − cw⎡

⎣⎢

⎤

⎦⎥j+1

n

−−cv − c2w−v − cw⎡

⎣⎢

⎤

⎦⎥j

n⎛

⎝⎜

⎞

⎠⎟

xf

wtf

v∂∂=

∂∂= ;

vtwt

⎛⎝⎜

⎞⎠⎟+0 − c2

−1 0⎛⎝⎜

⎞⎠⎟vxwx

⎛⎝⎜

⎞⎠⎟= 0

Finite volume upwind approximation:


Higher Order Schemes


Even though the fluxes were computed EXACTLY, the solution was very inaccurate due to numerical diffusion. Since the advection is exact, the problem must lie with the averaging of the solution and the assumption of a constant state in each cell

To generate more accurate schemes, we need to reconstruct a more accurate representation of the function in each cell.

Higher Order Methods


For the first order spatial fluxes, the value in each cell was assumed to be a constant. For higher order computations of the spatial fluxes, the slope inside each cell is found

�

Fj+1/ 2n+1/ 2 = F f L( ) j+1/ 2

n+1/ 2, f R( ) j+1/ 2

n+1/ 2( )

Once the value at the cell boundary has been found, the flux is found using these values

Higher Order MethodsComputational Fluid Dynamics

For the Lax-Wendroff scheme the time and space discretization are closely linked.

In the Beam-Warming method, on the other hand, the time integration is independent of the spatial discretization



The separation of space and time discretization is generalized in the Method of Lines, where we convert the PDE into a set of ODEs for each grid point by writing:

df jdt

=−1Δx

Fj+1/2n − Fj−1/2

n( )The time integration can, in principle, be done by any standard ODE solver, although in practice we often use second order Runge-Kutta methods


To get a second order time integration, use a half-step predictor to give second order accuracy in time:

f jn+1/2 = f j

n −Δt2Δx

Fj+1/2n −Fj−1/2

n( )

f jn+1 = f j

n −ΔtΔx

Fj+1/2n+1/2 −Fj−1/2

n+1/2( )

Then correct by a midpoint rule:

where the fluxes are found at the half step

j-1 j j+1



For a method that is second order in time we therefore must find the fluxes twice. For the method on the last slide we find

�

Fj+1/ 2n = F f j

n, f j+1n( )

at the beginning of the step.And

at the half step. For a first order in space methods we take the values on either side to be the cell averages but for higher order methods the variables are extrapolated to the cell boundaries to give higher order fluxes.

�

Fj+1/ 2n+1/ 2 = F f L( ) j+1/ 2

n+1/ 2, f R( ) j+1/ 2

n+1/ 2( )


∂ f∂t

+ ∂F∂ x

= 0 fi (t) =1Δx

f (x, t)dxVi∫

ddtfi (t) =

1Δx

F( f (xi+1/2 ,t) − F( f (xi+1/2 ,t)( )

=1Δx

Fi+1/2 − Fi−1/2( ) = L( f )i

For high order methods the time integration is often done using high order Runge-Kutta methods, such the following third order method:

f (1) = f n + Δt L( f n )

f (2) = 34f n + 1

4f (1) + 1

4Δt L( f (1) )

f n+1 = 13f n + 2

3f (2) + 2

3Δt L( f (2) )


A fairly general family of higher order methods can be constructed by weighing the slopes in each cell in different ways:


j j+1 j-1 j+2

1+jf 2+jf

jf1−jf

Ljf 2/1+

f j+1/2L = f j +

h2

1−κ2

⎛⎝⎜

⎞⎠⎟S− +

1+κ2

⎛⎝⎜

⎞⎠⎟S+⎡

⎣⎢⎤⎦⎥

Higher order fluxes

�

f j+1/ 2L = f j + h

2S

�

S+ =f j+1 − f j

hS− =

f j − f j−1h

Define slopes:

Define a weighted average:

�

k = 0 use 12 S

+ + S−( )k =1 use S+ only

k = −1 use S− only



f j+1/2L = f j +

h2

1−κ2

⎛⎝⎜

⎞⎠⎟S− +

1+κ2

⎛⎝⎜

⎞⎠⎟S+⎡

⎣⎢⎤⎦⎥

= f j +h2

1−κ2

⎛⎝⎜

⎞⎠⎟

f j − f j−1h

⎛⎝⎜

⎞⎠⎟+1+κ2

⎛⎝⎜

⎞⎠⎟

f j+1 − f jh

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

= f j +1−κ4

f j − f j−1( ) + 1+κ4 f j+1 − f j( )

Using

�

S+ =f j+1 − f j

hS− =

f j − f j−1h

We find


Rjf 2/1+

�

f j+1/ 2R = f j+1 −

1−κ4

f j+1 − f j( ) − 1+ κ4

f j+2 − f j+1( )

Similarly

�

f j+1/ 2R = f j+1 −

h2S

�

S+ =f j+2 − f j+1

hS− =

f j+1 − f jh

j j+1 j-1 j+2

1+jf 2+jf

jf1−jf



j j+1 j-1 j+2

1+jf 2+jf

jf1−jf

Ljf 2/1+

Rjf 2/1+

�

f j+1/ 2L = f j + 1−κ

4f j − f j−1( ) + 1+ κ

4f j+1 − f j( )

�

f j+1/ 2R = f j+1 −

1−κ4

f j+1 − f j( ) − 1+ κ4

f j+2 − f j+1( )

1for −=κ

Evaluation of the function at the cell boundary:

Different values of k give different second order schemes


Selecting k differently allows us to recover a number of standard schemes:

�

f j+1/ 2L = f j + 1−κ

4f j − f j−1( ) + 1+ κ

4f j+1 − f j( )

κ = −1κ = 0κ = 1 / 2κ = 1 / 3κ = 1

Second order upwindFromm’s schemeQUICKThird order upwindCentered scheme



�

f jn+1/ 2 = f j

n − Δt2h

Fj+1/ 2n − Fj−1/ 2

n( )

�

f jn+1 = f j

n − Δth

Fj+1/ 2n+1/ 2 − Fj−1/ 2

n+1/ 2( )

�

Fj+1/ 2n+1/ 2 = F f L( ) j+1/ 2

n+1/ 2, f R( ) j+1/ 2

n+1/ 2( )

Predictor step

Final step

where is a higher order flux

Can use first order fluxes for the predictor step

10 20 30 40 50 60 70 80-0.5

0

0.5

1

1.5

upwind

K=-1

K=0

K=1Example: Linear Advection


Notice that we can also write the flux as

�

f j+1/ 2L = f j + 1−κ

4f j − f j−1( ) + 1+ κ

4f j+1 − f j( )

f j+1/ 2

L =12

f j + f j+1( ) − 1−κ4

f j+1 − 2 f j + f j−1( )Where the fluxes appear as centered approximations with a stabilizing artificial dissipation



An even more general family of schemes can be constructed by writing

Where

f j+1/ 2

L = f j +12

B f j − f j−1, f j+1 − f j( )

f j+1/ 2

L = f j +12Ψ(r) f j − f j−1( )

r =

f j+1 − f j

f j − f j−1

Where B is a function of the slopes to the left and the right. Often we take

B f j − f j−1, f j+1 − f j( ) = Ψ(r) f j − f j−1( )

Giving

Later we will consider adjusting this function. Sometimes called limiter


The k scheme can be written in terms of the limiter

Ψ(r) =

1+κ2

r + 1−κ2

f j+1/ 2L = f j +

12Ψ(r) f j − f j−1( )

= f j +12

1+κ2

r + 1−κ2

⎛⎝⎜

⎞⎠⎟

f j − f j−1( )

= f j +12

1+κ2

f j+1 − f j

f j − f j−1

⎛

⎝⎜

⎞

⎠⎟ +

1−κ2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

f j − f j−1( )

= f j +1+κ

4f j+1 − f j( ) + 1−κ

4f j − f j−1( )

r =

f j+1 − f j

f j − f j−1

As can be seen by substitution:



For the k scheme we have:

Ψ(r) =

1+κ2

r + 1−κ2

r =

f j+1 − f j

f j − f j−1

Ψ(r) =12

; κ = −1

Ψ(r) =r2+

12

; κ = 0

Ψ(r) = r; κ = 1

Second order upwind

Fromm’s scheme

Centered scheme

f j+1/ 2

L = f j +12Ψ(r) f j − f j−1( )


We can also write other schemes in this way. For example:

r =

f j+1 − f j

f j − f j−1

First order upwindLax-WendroffBeam-Warming

f j+1/ 2

L = f j +12Ψ(r) f j − f j−1( )

Ψ(r) = 0Ψ(r) = 1Ψ(r) = r



But what about the oscillations?


Artificial Viscosity


Use centered difference method (e.g. Lax-Wendroff) - Second order accuracy- Oscillation near discontinuity

In order to “damp out” oscillation, we can either- Use implicit numerical viscosity (upwind) or- Add an explicit numerical viscosity

Artificial Viscosity Computational Fluid Dynamics

Von Neumann and Richtmyer (1950)

0=∂∂+

∂∂

xF

tf

xf

Dhxf

FF∂∂=

∂∂−=′ 2; αα


Replace the flux by:

�

∂f∂t

+ ∂F∂x

= − ∂∂x

−α ∂f∂x

⎛ ⎝ ⎜

⎞ ⎠ ⎟ = ∂

∂xDh2 ∂f

∂x∂f∂x

⎛ ⎝ ⎜

⎞ ⎠ ⎟ Giving:

O(1) coefficient

→ 0 as h → 0 -  Simulates the effect of the physical viscosity on the grid scale, concentrated around discontinuity and negligible elsewhere.

- h2 is necessary to keep the viscous term of higher order.

Consider:

Computational Fluid DynamicsArtificial Viscosity

�

∂f∂t

+U ∂f∂x

= ∂∂x

Dh2 ∂f∂x

∂f∂x

⎛ ⎝ ⎜

⎞ ⎠ ⎟

Example: Linear Advection Equation:

f jn+1 = f j

n −UΔt2h

f j+1n − f j−1

n( ) + 12UΔth

⎛⎝⎜

⎞⎠⎟2

f j+1n − 2 f j

n + f j−1n( )

+Δth

Dh2∂f∂x

∂f∂x

⎛⎝⎜

⎞⎠⎟ j+1/2

− Dh2∂f∂x

∂f∂x

⎛⎝⎜

⎞⎠⎟ j−1/2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Discretization by Lax-Wendroff


Dh2∂f∂x

∂f∂x

⎛⎝⎜

⎞⎠⎟ j+1/2

= D fj+1n − f j

n f j+1n − f j

n( )Approximate:

f jn+1 = f j

n −UΔt2h

f j+1n − f j−1

n( ) + 12UΔth

⎛⎝⎜

⎞⎠⎟2

f j+1n − 2 f j

n + f j−1n( )

+ D Δth

f j+1n − f j

n f j+1n − f j

n( ) − f jn − f j−1

n f jn − f j−1

n( )⎡⎣ ⎤⎦

gives

1+jff j−1

jf

xj j+1 j-1


% Artificial Viscosity by LaxWendroff n=161; nstep=250; length=4.0;h=length/(n-1);dt=0.25*h; y=zeros(n,1);f=zeros(n,1);f(1)=1;time=0; for m=1:nstep,m hold off;plot(f,'linewidt',2); axis([1 n -0.5, 1.5]);hold on; plot([1,dt*(m-1)/h+1.5,dt*(m-1)/h+1.5,n],[1,1,0,0],'r','linewidt',2);pause; y=f; time=time+dt; for i=2:n-1, f(i)=y(i)-(0.5*dt/h)*(y(i+1)-y(i-1))+... (0.5*dt*dt/h/h)*(y(i+1)-2.0*y(i)+y(i-1))+...

+0.5*(dt/h)*(abs(y(i+1)-y(i))*(y(i+1)-y(i))-... abs(y(i)-y(i-1))*(y(i)-y(i-1)) );

end; end;


20 40 60 80 100 120 140 160-0.5

0

0.5

1

1.5

20 40 60 80 100 120 140 160-0.5

0

0.5

1

1.5

20 40 60 80 100 120 140 160-0.5

0

0.5

1

1.5

upwind

Lax-Wendroff D=0

Lax-Wendroff D=0.5



20 40 60 80-0.5

0

0.5

1

1.5Upwind Lax-Wendroff

Nonlinear advection equation


Artificial viscosity can be used with most other centered difference schemes and was, for a while, THE way aeronautical computations were done. Other types, including higher order, have been used.


What we know so far:

First-order methods result in a low accuracy and dissipative solution, particularly at a shock. The solution does, however, not oscillate (preserves monotonicity).

Second-order methods capture shocks better but produce wiggly solutions.

For the nonlinear advection equation, the first order captures the solution reasonably well, since the characteristics terminate in the shock


The “Modern” view


The formulation of second order schemes in terms of slopes allows us to understand the reason for oscillations near discontinuity. The construction of slopes leads to overshoots:

j j+1 j-1 j+2

1+jf 2+jf

jf1−jf

OscillationsComputational Fluid Dynamics

To prevent oscillations near shocks when using high order schemes, we have already talked about artificial viscosity where we attempt to smooth out oscillations around the shocks.

The more modern approach is to prevent the apparence of oscillations by either:

Limit the slopes when the variables are extrapolated to the cell boundaries

Limit the fluxes near shocks to prevent under or over shoot

Oscillations


j j+1 j-1 j+2

1+jf 2+jf

jf1−jf

Near shocks the linear reconstruction leads to over and undershoots.

To prevent oscillations, apply LIMITERS to reduce the slopes where they will cause oscillations but leave then where they do not

Many possible limiters can be designed


Example: Linear Advection Equation

∂f∂t

+ U∂f∂x

= 0

�

F =Uf

�

Fj+1/ 2 =Uf j+1/ 2L

�

Ψ r( ) =r + r1+ r

rj+1/ 2 =

f j − f j−1

f j+1 − f j

�

Ψ = 2 if f j+1 − f j = 0

j j+1 j-1

1+jf

jf1−jf

Oscillations


10 20 30 40 50 60 70 80-0.5

0

0.5

1

1.5

upwind Second order

Limited


The realization that we could enforce monotonicity in otherwise high order schemes using nonlinear limiters revolutionized the computation of flows with shocks. The idea originated with Bram van Leer and Jay Boris but has since been taken further by a large number of researchers. It is still an active area of research.

A large number of limiters have been proposed and we will examine a few of those in more detail.

Oscillations

Computational Fluid Dynamicshttp://www.nd.edu/~gtryggva/CFD-Course/

Diffusion versus dispersion

Godunov Theorem

Godunov Method

Upwind and Flux Splitting

Higher order schemes

Artificial viscosity

computational fluid dynamics - university of notre …gtryggva/cfd-course2017/lecture-8-2017.pdf ·...

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