2011-Lecture-15.pdf

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Computational Fluid Dynamics

Advection

Grtar Tryggvason!Spring 2011!

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


Godunov Theorem (1959):!

Monotone behavior of a numerical solution cannot

be assured for linear finite-difference methods with

more than first-order accuracy.!

Godunovs Theorem!

5 7- .

.

.


Flux LimitersReview


The separation of space and time discretization is

generalized in the Method of Lines, where we convertthe PDE into a set of ODEs for each grid point bywriting:!

dfj

dt=1

hF

j+1/2

nF

j1/2

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!

Higher Order Methods!

Computational Fluid DynamicsFor the Linear Advection equation we have:!

r=f

j+1 f

j

fj f

j1

(r) =1

2;

(r) =r

2+

1

2;

(r) = r;

Second order upwind!

Fromms scheme!

Centered scheme (Beam-Warming)!

F

j+1/2

L= f

j+

1

2(r) f

j f

j1( )

Higher Order Methods!

Lax-Wendroff!

First order upwind!

(r) = 1

(r) = 0


For the linear advection equation, it is possible to

do the advection step exactly, since the solution

simply moves with uniform velocity. This allows usto see the central role played by the reconstructed

slopes in each cell!

Higher Order Upwind!

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Computational Fluid DynamicsHigher Order Upwind!

j!j-1!

fj!1

jf

U!t

hfjn+1

= fjn h !U"t( )+U"tfj!1n

j!j-1!

fj!1

jf

U!t

fjn+1

= fjn+ sj x ! x j( )

hfj

n+1= f

j

n sj

2Ut

h Ut( )

+ fj1n+

sj1

2h Ut( )

Utfj1

n

fin+1

= fjn1! "( )+ "fj!1

n

f

j

n+1= f

j

n(1 ) + fj1

n

h

2(1 )(s

js

j1)

!=U"t

h


j j+1!j-1! j+2!

1+jf 2+jf

jf1jf

Near shocks the linear reconstruction leads to over and

undershoots.!

To prevent oscillations, apply LIMITERS to reduce theslopes where they will cause oscillations but leave thenwhere they do not!

Many possible limiters can be designed!



In Gudunov like methods, the solution process consists

of three steps:!

1. Reconstruction of the solution in each cell!2. Advection of the solution, using the fluxes from one

cell to the next one!3. Construction of the average solution in each cell!

Only the reconstruction of the solution can lead tooscillations!



Designing Limiters:The Sweeby

diagram


f

j+1/2

L= f

j+

1

2(r) f

j f

j1( )

r=f

j+1 f

j

fj f

j1

r

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It can be shown that for a scheme to be second order

and TVD, the limiter must lie in the shaded region.!

1 2 3

1

2

r

! r( )

! = 2r

r=f

j+1 f

j

fj f

j1

! = r

Lax-Wendroff

Beam-Warming!

=1

Limiters!


It has been found that it is also best to take the limiterto be between Lax-Wendroff and Beam Warming!

0 (r) r r 0

r=f

j+1 f

j

fj f

j1

(r)

r

= (1

r

)

Generally we also require thelimiters to be symmetric!

In many papers, r is defined as the inverse of the one used here!

Limiters!


Using the limitations given by second order accuracy, TVD and

the requirement that the limiters lie between Lax-Wendroff andBeam-Warming, gives the Sweby-Diagram. !

The limiters must lie in the shaded region!

1 2 3

1

2

r

! r( )

! = 2r

! = r

Lax-Wendroff

Beam-Warming!

r=f

j+1 f

j

fj f

j1

Limiters!


The Sweby region is not the only one that is used todesign limiters, but it is by far the most widely used. A

very large number of limiters have been proposed thatfall within this region. See, for example: !

N. P. Waterson and H. Deconinck. Design Principles

for bounded high-order convection schemesaunified approac. JCP 224 (2007), 182-207!

This paper treats only steady state problems.!

Limiters!


The Minmod Limiter (Roe and others) !

1 2 3

1

2

r

! r( )

!=

2r

! = r

minmod!

r=f

j+1 f

j

fj f

j1

(r) = max[0,min(1,r)]

Simple and

generally workswell. Fairlydissipative!

Limiters!


minmode!

T=1.0; h=0.01; dt=h/3!

Limiters!

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

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The Superbee Limiter (Roe)!

1 2 3

1

2

r

! r( )

! = 2r

! = r

superbee!

r=

fj+1

fj

fj fj1

(r) = max[0,min(2r,1),min(r,2)]

Superbee is generallyfound to be too

compressive andgenerates steps insmooth regions!

Limiters!


Superbee!

T=1.0; h=0.01; dt=h/3!

Limiters!

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


The original van Leer limiter!

1 2 3

1

2

r

! r( )

! = 2r

! = r

van Leerlimiter!

r=f

j+1 f

j

fj f

j1

(r) =

r+ r

1+ r

Smoothvariationno

abrupt switches!

Limiters!


van Leer limiter!

T=1.0; h=0.01; dt=h/3!

Limiters!

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


The MUSCL Limiter!

1 2 3

1

2

r

! r( )

!=

2r

! = r

MUSCL!

r=f

j+1 f

j

fj f

j1

(r) = max[0,min(2r,(r+1) / 2,2)]

One of the original

suggestions by vanLeer!

Limiters!


T=1.0; h=0.01; dt=h/3!

The MUSCL Limiter!

Limiters!

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

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The Sweby Limiter (Sweby)!

1 2 3

1

2

r

! r( )

! = 2r

! = r

Sweby, beta=1.5!

r=

fj+1

fj

fj fj1

(r) = max[0,min(r,1),min(r,)]; 1 2

An example of a limiterwith abrupt switching

that traverses theinterior of the allowableregion!

Limiters!


T=1.0; h=0.01; dt=h/3!

The Sweby Limiter (Sweby)!

Limiters!

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


minmode!

Superbee!

van Leer limiter!

MUSCL!

Sweby!

Limiters!

Computational Fluid DynamicsLimiters!

for m=1:nstep,m,time!hold off;plot(x,f,'linewidt',2); %axis([1 n 2.0, 4.5]); %plot solution !hold on;plot(x,fu,'r','linewidt',2);axis([0 length 0 2]); pause(0.1)!

foo=f;!for is=1:2 %two steps per timestep!fo=f;!for i=2:n-1, !bot=(fo(i)-fo(i-1));top=(fo(i+1)-fo(i)); r=top*bot/(bot^2+0.00001);!% Limiters!% psi=max([0, min([2*r,0.5*(r+1),2])]); % monotonized central (MC)!psi=max([0, min([1.5*r,1]),min([r,1.5]) ]); % Sweby !% psi=max([0, min([2*r,1]),min([r,2])]); % superbee gtext('superbee')!% psi=max([0, min([r,1])]); % minmod gtext('MINMOD')!% psi=(r+abs(r))/(r+1); % van Leer!% psi=0; % upwind!% psi=r;!fh(i)=fo(i)+0.5*psi*(fo(i)-fo(i-1));!

end;!

bot=(fo(1)-fo(n-1));top=(fo(2)-fo(1)); r=top*bot/(bot^2+0.00001);!% psi=max([0, min([2*r,0.5*(r+1),2])]); % monotonized central (MC)!psi=max([0, min([1.5*r,1]),min([r,1.5]) ]); % Sweby !% psi=max([0, min([2*r,1]),min([r,2])]); % superbee!% psi=max([0, min([r,1])]); % minmod!% psi=(r+abs(r))/(r+1); %van Leer!% psi=0; % upwind!% psi=r;!fh(1)=fo(1)+0.5*psi*(fo(1)-fo(n-1)); fh(n)=fh(1);!

for i=2:n-1,!f(i)=fo(i)-(dt/h)*(fh(i)-fh(i-1) );!

end;!

f(1)=fo(1)-(dt/h)*(fh(1)-fh(n-1) );f(n)=f(1);!end!

for i=1:n,f(i)=0.5*(f(i)+foo(i)); end % average the solution for second oder time step !

% one-dimensional linear advection using flux limiters !% This looks like the final and complete version of the program !% Feb. 21, 2011!%------------------------------------------------------------ !n=101; nstep=300; length=1.0;h=length/(n-1);!dt=0.333333333*h; time=0.0; for i=1:n, x(i)=h*(i-1);end!

f=zeros(n,1);fo=zeros(n,1);fh=zeros(n,1);foo=zeros(n,1);!fu=zeros(n,1);fuo=zeros(n,1);!for i=1:n, if (x(i)>=0.0 & x(i)=0.6 & x(i)=0.0 & x(i)=0.6 & x(i)

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fjn+1/ 2

= fjnt

2hFj+1/ 2

n Fj1/ 2

n( )

fjn+1

= fjnt

hFj+1/ 2

n+1/ 2 Fj1/ 2

n+1/ 2( )

Fj+1/ 2n+1/ 2

= F fL( )

j+1/ 2

n +1/ 2

, fR( )

j+1/ 2

n +1/ 2

( )

Predictor step!

Variables!

fj+1/ 2L

= fjn+1/ 2

+1

2

Lfj

n+1/ 2 fj1

n+1/ 2( )

fj+1/ 2

R= fj+1

n+1/21

2

R fj+1

n+1/ 2 fj

n+1/2( )

Final step!

Limiting the variables:!

=1

Find:!

Limiters!


fj

n+1/ 2= f

j

nt

2hF

j+1/ 2

n Fj1/ 2

n( )

fjn+1

= fjnt

hFj+1/ 2

n+j+1/ 2

LFj+1/ 2

n+1/ 2 Fj+1/ 2

n( )[ ](

Fj1/ 2n

+j1/ 2L

Fj1/ 2n+1/ 2

Fj1/ 2n( )[ ])

Fj+1/ 2n+1/ 2

= F fL( )

j+1/ 2

n+1/ 2

, fR( )

j+1/ 2

n+1/ 2

( )

Predictor step!

Variables!

Find:!

fj+1/ 2

L= f

j

n+1/ 2+1

2f

j

n+1/ 2 f

j1

n+1/2( )

fj+1/ 2

R= fj+1

n+1/ 2

1

2f

j+1

n+1/ 2 fj

n+1/ 2( )

Final step!

=1

Limiting the Fluxes!

Limiters!


The approach described here has, on the whole, been a

very successful one, but the difficulty of preserving sharpdiscontinuity at the same time as we accurately advectsmooth solutions continues to be a challenge and

several groups continue to advance the state of the art.!

We will discuss some of those very briefly next time andtalk about the challenges in applying these ideas to the

full nonlinear and multidimensional Euler equations!

In addition to the use of nonlinear methods to captureshocks, such methods are increasingly being used forthe advection part of Navier-Stokes solvers!


Multi-dimensional

flow


Multidimensional hyperbolic problems aregenerally done using SPLITTING, where

each coordinate direction is treatedseparately!

Multidimensional Flow!


Split versus unsplit advection!

fi, jn+1

= fi, jn+

ut

hfi1, j

n fi, j

n( ) +vt

hfi, j1

n fi, j

n( )

f

t+ u

f

x+ v

f

y= 0

Unsplit!


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Split versus unsplit advection!

ut!

vt!

(1-u) t!

(1-v)t!

fi, jn+1

= fi, jn+

uvt2

h2

fi1, j1n

fi, jn( )

+ (1 vt)ut

h2

fi1, jn

fi, jn( )+ (1 ut)

vt

h2

fi, j1n

fi, jn( )

= fi, jn+

ut

hfi1, j

n fi, j

n( )+vt

hfi, j1

n fi, j

n( )

+uvt

2

h2

fi1, j1n

fi1, jn

fi, j1n

+ fi, jn( )

f

t+ u

f

x+ v

f

y= 0

Fully unsplit!



Time splitting!

fi, j*

= fi, jn+

ut

hfi1, j

n fi, j

n( )

fi, jn+1

= fi, j*+

vt

hfi, j1

* fi, j

*( )

First!

then!

f

t+ u

f

x+ v

f

y= 0



0

510

1520

2530

0

5

10

15

20

25

30

0

0.5

1

1.5

5 1 0 1 5 20 25 3 0

5

10

15

20

25

30

5 10 15 20 25 30

5

10

15

20

25

30

5 10 15 20 25 30

5

10

15

20

25

30

Time-split! Unsplit!


Computational Fluid Dynamics% two-dimensional timesplit advection using first order upwind %------------------------------------------------------------n=32;m=32;nstep=35;length=2.0;h=length/(n-1);dt=0.3*h;f=zeros(n,m);fo=zeros(n,m);f1=zeros(n,m);time=0.0;u=1.0; v=1.0; cx=0.35;cy=0.35;

for i=2:n-1, for j=2:m-1r=sqrt((cx-h*(i-1))^2+(cy-h*(j-1))^2);if(r < 0.2) f(i,j)=1.0;end

end,endfor l=1:nstep,l,timehold off;mesh(f); axis([0 n 0 m 0 1.5]);pause;

fo=f;for i=2:n-1, for j=2:m-1

f1(i,j)=fo(i,j)-(dt*u/h)*(fo(i,j)-fo(i-1,j));end,endfor i=2:n-1, for j=2:m-1

f(i,j)=f1(i,j)-(dt*v/h)*(f1(i,j)-f1(i,j-1)); end,end time=time+dt;

end;f2=figure; contour(f,10);axis square;



Splitting thus recovers some properties of theexact advection by allowing flux from diagonal

cells. This is one reason for focusing on one-dimensional schemes, since doingmultidimensional flow by splitting is actually better

than multidimensional discretization of the originalequations!


2011-Lecture-15.pdf

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