Stability of 2nd ordr one sided upwind 1st order in time
 

As mentioned in chapter 21 of Hirsch's book:
"second-order one-sided differences in space with a first-order difference in time (explicit), is unconditionally unstable"
This can be proved for linear convection equation.
I want to know that whether this is true for Euler equations or not. I get converged solutions for one dimensional nozzle flow with shock by applying the above mentioned method! However some overshoots happen at shock region.

Also, when I implement limiters (minmod) to my scheme, solution convergence trend stops at about L2norm(p(n+1)-p(n))=5e-5. I can't understand what is happening.

no answer
??

Hi Ardalan,

Maybe if you use sufficiently coarse grids, then the dissipation is sufficient to damp the enherent instability of the scheme. Try using much finer grids, I expect some troubles. I think the overshoots you observe are only the beginning of a blow up with a finer mesh.

Concerning the limiter, I am not suprised you experience convergence problems to reach steady state. This is caused by the non-differentiability of minmod function at 1. Some remedy was proposed by Venkatakrishnan to improve convergence to steady state. They use a smooth limiter function, differentiable. If you look for it on the web you will find many articles showing nice convergence graphs using smooth limiters whereas non-diff limiters such like minmod stall

Hope it helps

Thanks for your useful reply.
Now I made my code second order both in time and space.
Of course. convergence stall happens again.
The fact that makes me surprised, is that by increasing cfl number, convergence stalls at lower residuals. For example, at cfl=0.1, convergence stops at about 1e-5 but for cfl=0.5, convergence stops at about 1e-8 and for cfl=0.9, convergence stops at about 1e-12.
I'm using minmod limiter.

Is this reasonable?

Hello,

I do not know for sure about the behavior of residual as a function of CFL. My opinion is that using a lower CFL then dt is smaller and so the time truncation error. Then your scheme dissipates more with a higher truncation error and reaches convergence more easily. I really suggest you to try smooth limiters, if your scheme is well implemented it should be very easy to plug one or another limiter. On the wikipedia page you have several examples. I did not do the tests myself, but take for instance the ospre or van albada 1 limiters, it might help to reach convergence.

Cheers,
François