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Entropy Stable Method for the Euler Equations Revisited: Central Differencing via Entropy Splitting and SBP

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Abstract

The two decades old high order central differencing via entropy splitting and summation-by-parts (SBP) difference boundary closure of Olsson and Oliger (Energy and maximum norm estimates for nonlinear conservation laws, 1994), Gerritsen and Olsson (J Comput Phys 129:245–262, 1996) and Yee et al. (J Comput Phys 162:33–81, 2000) is revisited. The objective of this paper is to prove for the first time that the entropy split methods based on physical entropies are entropy stable methods for central differencing with SBP operators for both periodic and non-periodic boundary conditions for nonlinear Euler equations. Standard high order spatial central differencing as well as high order central spatial dispersion relation preserving spatial differencing is part of the entropy stable methodology framework. The proof is to replace the spatial derivatives by SBP difference operators in the entropy split form of the equations using the physical entropy of the Euler equations. The numerical boundary closure follows directly from the SBP operator. No additional numerical boundary procedure is required. In contrast, Tadmor-type entropy conserving methods (Acta Numer 12:451–512, 2003) using mathematical entropies and more recently the methods in Winters and Gassner (J Comput Phys 304:72–108, 2016), do not naturally come with a numerical boundary closure and a generalized SBP operator has to be developed (Roanocha in Generalized summation-by-parts operators and variable coefficients, 2018. arXiv:1705.10541v2 [math.NA]). Long time integration of 2D and 3D test cases is included to show the comparison of this efficient entropy stable method with the Tadmor-type of entropy conservative methods. Studies also include the comparison among the three skew-symmetric splittings on their nonlinear stability and accuracy performance without added numerical dissipations for smooth flows. These are, namely, entropy splitting, Ducros et al. splitting and the Kennedy and Grubber splitting.

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Acknowledgements

Financial support from the NASA TTT/RCA program for the second author is gratefully acknowledged. The authors are grateful to Dr. Alan Wray of NASA Ames Research Center for the numerous invaluable discussions throughout the course of this work

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  1. Multid Analyses AB, Gothenburg, Sweden

    Björn Sjögreen

  2. NASA Ames Research Center, Mountain View, CA, USA

    H. C. Yee

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Correspondence to H. C. Yee.

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Sjögreen, B., Yee, H.C. Entropy Stable Method for the Euler Equations Revisited: Central Differencing via Entropy Splitting and SBP. J Sci Comput 81, 1359–1385 (2019). https://doi.org/10.1007/s10915-019-01013-1

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