I have implemented finite-difference WENO scheme for Euler equations (with some variants - WENO-JS, WENO-Z, WENO-M, different flux splitting). It works well, but have problem with so-called carbuncles (shock instability in the areas where its front is almost aligned with the grid). This problem seems to be well-studied for finite-volume methods (quick search shows that there are some fixes for Riemann solvers), but I can't find almost any information regarding carbuncles in finite-difference methods.

Are there any known fixes for finite-difference methods?

  • 3
    $\begingroup$ How about H-correction? Some ideas: doi.org/10.1006/jcph.1998.6047 $\endgroup$ Nov 2 '17 at 18:21
  • $\begingroup$ What flux splitting are you using ? Can you give more details in your question ? $\endgroup$
    – cfdlab
    Nov 3 '17 at 3:49
  • $\begingroup$ @Praveen It seems that correct flux splitting is very important. I was using 3 versions (global and local Lax-Friedrich's and Roe-LLF) and was constructing flux splitting after weno-reconstruction of U and F(U) in local characteristic field. Now I weno-reconstruct already splitted flux. There are no more large-scaled carbuncles, but still are some weak shock front instabilities, so the question still stands. Best results so far are obtained by proper Roe-LLF splitting with Roe averaging of Jacobians at $x_{j+1/2}$. $\endgroup$
    – omican
    Nov 5 '17 at 12:05
  • $\begingroup$ Right, FD-WENO involves first doing a flux splitting and then applying weno on the split fluxes. Can you give me some reference for the Roe-LLF that you used ? $\endgroup$
    – cfdlab
    Nov 5 '17 at 12:15
  • $\begingroup$ @Praveen It's formula (2.6) from Liu, X. D., Osher, S., & Chan, T. (1994). Weighted essentially non-oscillatory schemes. Journal of computational physics, 115(1), 200-212. Here is the link: paper. In order to choose flux (f(a), f(b) or LLF-splitted), I check eigenvalues over the whole 6-point stencil. Curiously, WENO-Z is more prone to such instability type (probably, because of smaller dissipation near shocks) than WENO-JS. $\endgroup$
    – omican
    Nov 6 '17 at 12:43

After quick testing, H-correction seems to solve the problem! For future readers: see the first comment for the relevant paper link.

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    $\begingroup$ Would you please expand your answer with all the relevant information for future readers? $\endgroup$
    – nicoguaro
    Dec 30 '17 at 16:37

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