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I have recently learned about Discontinuous Galerkin method to solve differential equations and I was trying to implement it to solve Euler equation. For now, consider the standard Sod Shock Tube Case.

Without using any limiter I am getting an unstable scheme (even with CFL 0.01) and my solution blows up. Presently I am using 1st order polynomials in each cell using Lobatto nodes (in case someone needs to know). Also, I am calculating the interface flux using AUSM method.

Now I am not sure whether my code is wrong or does it behave like this because I haven't used any flux limiter?

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  • $\begingroup$ A first order scheme should be fairly dissipative. On the other hand you do not state how it is blowing up. There can be a few reasons why it blows up. One is because of oscillations due to the shock becoming too large but a first order scheme should dissipate that. One is using say an Euler scheme for time stepping. Another is positivity being violated. That may occur because you are using AUSM. $\endgroup$
    – Vikram
    Commented Mar 15, 2017 at 10:06
  • $\begingroup$ Yes, it is indeed because of positivity being violated. $\endgroup$
    – Manish
    Commented Mar 15, 2017 at 17:03
  • $\begingroup$ Then try a positivity preserving Riemann solver like the simple Lax Friedrichs flux. If that solves it then you know what was causing the issue. $\endgroup$
    – Vikram
    Commented Mar 15, 2017 at 17:32
  • $\begingroup$ Sure. I think that is a good idea. Will try it. $\endgroup$
    – Manish
    Commented Mar 15, 2017 at 17:46

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A finite element scheme (CG or DG) without stabilization is expected to result in a solution that violates the maximum principle, even for linear scalar problems. That is why sufficient dissipation/viscosity is needed in addition to the dissipation on the approximate Riemann solver on the interface. See the papers by Cockburn and Shu for a detailed discussion.

Various approaches exist to deal with this issue such as entropy viscosity, SUPG stabilization etc...

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  • $\begingroup$ @user814 Could you provide a link for the papers? $\endgroup$
    – ares
    Commented Oct 22, 2018 at 21:05

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