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 xue sheng December 13, 2012 01:40

Solution of SOD Shock Tube Problem by using HARTEN Modified Flux TVD Scheme

I am solving 1D shock tube SOD problem by using Harten modified flux TVD scheme. My code is stable till CFL less than 0.6, after that it is unstable.
It should be stable till CFL less than 1.

-1=<x<0 P=1, RHO=1, V=0
0=<x<1 P=0.1, RHO=0.125, V=0

I have following question,

1. How should calculate Time Step Size (DT)? Is this formula correct?
DT = (CFL x DX)/Cmax

2. How should calculate DX? Is this formula correct?
DX=L/(200) where L = 2 (201=Grid points)

3. How should calculate Cmax? Is this formula correct?
Cmax=0
Do i=1 to 200
C=DSQRT(GAMA*Pressure(i)/Density(i))
Cmax=DMAX1(C,Cmax)
Continue

Anyone have solved this problem by using Harten modified flux TVD scheme?

 praveen December 13, 2012 04:20

You must also include fluid velocity. It might increase the wave speed

C = ABS(V(i)) + DSQRT(GAMA*Pressure(i)/Density(i))

 xue sheng December 13, 2012 05:23

Thanks for correction.
But it would further decrease CFL criterion.
HARTEN TVD scheme should be stable below CFL <1.
But in my case it is stable around CFL<0.5.

May you please point out the reason?

 praveen December 13, 2012 05:31

As I wrote your CFL condition is wrong. You must use largest wave speed to compute time step which is what I wrote in my last post.

 xue sheng December 13, 2012 06:11

Thanks.
After your recommended correction Its working well.
Now my Harten modified flux TVD scheme, ROE scheme, and LAX_WENDROFF scheme Code working well till CFL <1.

But my HARTEN-YEE TVD scheme code is stable till CFL<0.6. After that it become unstable.

May you please inform me that HARTEN modified flux TVD scheme and HARTEN-YEE TVD scheme has same CFL restriction or different?

I think both have same CFL restriction. But in my code HARTEN-YEE become unstable at CFL>0.6.
May you please point out the reason?

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