Comparison of Some Entropy Conservative Numerical Fluxes for the Euler Equations
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Entropy conservation and stability of numerical methods in gas dynamics have received much interest. Entropy conservative numerical fluxes can be used as ingredients in two kinds of schemes: firstly, as building blocks in the subcell flux differencing form of Fisher and Carpenter (Technical Report NASA/TM-2013-217971, NASA, 2013; J Comput Phys 252:518–557, 2013) and secondly (enhanced by dissipation) as numerical surface fluxes in finite volume like schemes. The purpose of this article is threefold. Firstly, the flux differencing theory is extended, guaranteeing high-order for general symmetric and consistent numerical fluxes and investigating entropy stability in a generalised framework of summation-by-parts operators applicable to multiple dimensions and simplex elements. Secondly, a general procedure to construct affordable entropy conservative fluxes is described explicitly and used to derive several new fluxes. Finally, robustness properties of entropy stable numerical fluxes are investigated and positivity preservation is proven for several entropy conservative fluxes enhanced with local Lax–Friedrichs type dissipation operators. All these theoretical investigations are supplemented with numerical experiments.
KeywordsEuler equations Summation-by-parts Flux differencing Entropy stability Positivity preservation
Mathematics Subject Classification65M70 65M60 65M06 65M12
The author would like to thank the anonymous reviewers for their helpful comments.
- 10.Fisher, T.C., Carpenter, M.H.: High-order entropy stable finite difference schemes for nonlinear conservation laws: finite domains. Technical Report NASA/TM-2013-217971, NASA, NASA Langley Research Center, Hampton (2013)Google Scholar
- 15.Jameson, A.: Formulation of kinetic energy preserving conservative schemes for gas dynamics and direct numerical simulation of one-dimensional viscous compressible flow in a shock tube using entropy and kinetic energy preserving schemes. J. Sci. Comput. 34(2), 188–208 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
- 19.Ranocha, H.: SBP operators for CPR methods. Master’s thesis, TU Braunschweig (2016)Google Scholar
- 20.Ranocha, H.: Comparison of some entropy conservative numerical fluxes for the Euler equations (2017). arXiv:1701.02264 [math.NA]
- 22.Roe, P.L.: Affordable, entropy-consistent Euler flux functions. In: Talk Presented at the Eleventh International Conference on Hyperbolic Problems: Theory, Numerics, Applications (2006). http://www2.cscamm.umd.edu/people/faculty/tadmor/references/files/Roe_Affordable_entropy_Hyp2006.pdf