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Journal of Scientific Computing

, Volume 81, Issue 3, pp 1359–1385 | Cite as

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

  • Björn Sjögreen
  • H. C. YeeEmail author
Article
  • 71 Downloads

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.

Keywords

High order entropy stable methods Entropy splitting of inviscid flux derivative Improve nonlinear stability Long time integration DNS and LES 

Notes

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

References

  1. 1.
    Arakawa, A.: Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. Part I. J. Comput. Phys. 1, 119–143 (1966)CrossRefGoogle Scholar
  2. 2.
    Blaisdell, G.A., Spyropoulos, E.T., Qin, J.H.: The effect of the formulation of nonlinear terms on aliasing errors in spectral methods. Appl. Num. Math. 21, 207–219 (1996)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bonito, A., Guermond, J.-L., Popov, B.: Stability analysis of explicit entropy viscosity methods for nonlinear scalar conservation equations. Math. Comput. 83(287), 1039–1062 (2014).  https://doi.org/10.1090/S0025-5718-2013-02771-8 CrossRefzbMATHGoogle Scholar
  4. 4.
    Bohm, M., Winters, A.R., Gassner, G.J., Derigs, D., Hindenlang, F., Saur, J.: An entropy stable nodal discontinuous Galerkin method for the resistive MHD equations. Part I Theoory and Numerical Verification. arXiv:1802.07341v2 [math.NA] (2018)
  5. 5.
    Ducros, F., Laporte, F., Soulères, T., Guinot, V., Moinat, P., Caruelle, B.: High-order fluxes for conservative skew-symmetric-like schemes in structured meshes: application to compressible flows. J. Comput. Phys. 161, 114–139 (2000)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fisher, T.C., Carpenter, M.H.: High-order entropy stable finite difference schemes for nonlinear conservation laws: finite domains. J. Comput. Phys. 252, 518–557 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fjordholm, U.S., Mishra, S., Tadmor, E.: Arbitrary high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws. SIAM J. Numer. Anal. 50, 544–573 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Friedrich, L., Winters, A., DelRey Fernandez, D., Gassner, G., Paarsani, M., Carpenter, M.: An entropy stable h/p non-conforming discountinuous Galerkin method with summation-by-parts property. J. Sci. Comput. 77, 689–725 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gerritsen, M., Olsson, P.: Designing an efficient solution strategy for fluid flows. I. A stable high order finite difference scheme and sharp shock resolution for the Euler equations. J. Comput. Phys. 129, 245–262 (1996)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Harten, A.: On the symmetric form of systems for conservation laws with entropy. J. Comput Phys. 49, 151 (1983)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hughes, T., Franca, L., Mallet, M.: A new finite element formulation for computational fluid dynamics: K. Symmetric forms of the compressible Euler and Navier–Stokes equations and the second law of thermodynamics. Comput. Methods Appl. Mech. Eng. 54, 223–234 (1986)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ismail, F., Roe, P.L.: Affordable, entropy-consistent Euler flux functions II: entropy production at shocks. J. Comput Phys. 228, 5410–5436 (2009)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Johansson, S.: High order summation by parts operator based on a DRP scheme applied to 2D, Technical Report 2004-050, Uppsala University, SwedenGoogle Scholar
  14. 14.
    Kennedy, C.A., Gruber, A.: Reduced aliasing formulations of the convective terms within the Navier–Stokes equations. J. Comput. Phys. 227, 1676–1700 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kotov, D.V., Yee, H.C., Wray, A.A., Sjögreen, B., Kritsuk, A.G.: Numerical disipation control in high order shock-capturing schemes for LES of low speed flows. J. Comput. Phys. 307, 189–202 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kotov, D.V., Yee, H.C., Wray, A.A., Sjögreen, B.: High order numerical methods for dynamic SGS model of turbulent flows with shocks. Commun. Comput. Phys. 19, 273–300 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    LeFloch, P.G., Mercier, J.M., Rohde, C.: Fully discrete, entropy conservative schemes of arbitrary order. SIAM J. Numer. Anal. 40, 1968–1992 (2002)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Olsson, P., Oliger, J.: Energy and maximum norm estimates for nonlinear conservation laws. RIACS Technical Report 94.01 (1994)Google Scholar
  19. 19.
    Parsani, M., Carpenter, M.H., Nielsen, E.J.: Entropy stable wall boundary conditions for the three-dimensional compressible Navier–Stokes equations. J. Comput. Phys. 292, 88–113 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Roanocha, H.: Comparison of some entropy conservative numerical fluxes for the Euler Equations. arXiv:1701.02264v2 [math.NA] (2017)
  21. 21.
    Roanocha, H.: Generalized summation-by-parts operators and variable coefficients. arXiv:1705.10541v2 [math.NA] (2018)
  22. 22.
    Sandham, N.D., Li, Q., Yee, H.C.: Entropy splitting for high-order numerical simulation of compressible turbulence. J. Comput. Phys. 23, 307–322 (2002)CrossRefGoogle Scholar
  23. 23.
    Sjögreen, B., Yee, H.C.: Multiresolution wavelet based adaptive numerical dissipation control for high order methods. J. Sci. Comput. 20, 211–255 (2004)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Sjögreen, B., Yee, H.C., Vinokur, M.: On high order finite-difference metric discretizations satisfying GCL on moving and deforming grids. J. Comput. Phys. 265, 211–220 (2014)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Sjögreen, B., Yee, H.C.: On skew-symmetric splitting and entropy conservation schemes for the Euler equations. In: Proceedings of ENUMATH09, June 29–July 2, Uppsala University, Sweden (2009)Google Scholar
  26. 26.
    Sjögreen, B., Yee, H.C.: Accuracy consideration by DRP schemes for DNS and LES of compressible flow computations. Special issue in Computers & Fluids in honor of Prof Toro’s 70th birthday, 159, 123–136 (2017)Google Scholar
  27. 27.
    Sjögreen, B., Yee, H.C.: Skew-symmetric splitting for multiscale gas dynamics and MHD turbulence flows. Extended version of Proceedings of ASTRONUM 2016, June 6–10, 2016, Monterey, CA, USA, submitted to J. Scientific Computing (2018)Google Scholar
  28. 28.
    Sjögreen, B., Yee, H.C.: High order entropy conservative central schemes for wide ranges of compressible gas dynamics and MHD flows. J. Comput. Phys. 364, 153–185 (2018)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Sjögreen, B., Yee, H.C.: Two decades old entropy stable methods for the Euler equations revisited. In: Proceedings of the ICOSAHOM-2018, July 9–13, London, UK (2018)Google Scholar
  30. 30.
    Sandham, N.D., Li, Q., Yee, H.C.: Entropy splitting for high-order numerical simulation of compressible turbulence. J. Comput. Phys. 178, 307–322 (2002)CrossRefGoogle Scholar
  31. 31.
    Svärd, M., Mishra, S.: Shock capturing artificial dissipation for high-order finite difference schemes. J. Sci. Comput. 39, 454–484 (2009)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Tadmor, E.: Numerical viscosity and the entropy condition for conservative difference schemes. Math. Comput. 43, 369–381 (1984)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Tadmor, E.: The numerical viscosity of entropy stable schemes for systems of conservation laws. I. Math. Comput. 49, 91–103 (1987)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Tadmor, E.: Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer 12, 451–512 (2003)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Taylor, G., Green, A.: Mechanism of the production of small Eddies from large ones. Proc. R. Soc. Lond. A 158, 499–521 (1937)CrossRefGoogle Scholar
  36. 36.
    Tauber, E., Sandham, N.D.: Comparison of three large-eddy simulatitons of shock-induced turbulent separation bubbles. Shock Waves 19, 469–478 (2009)CrossRefGoogle Scholar
  37. 37.
    Vinokur, M., Yee, H.C.: Extension of efficient low dissipation high-order schemes for 3D curvilinear moving grids. Front. Comput. Fluid Dyn. 129–164 (2002). Also, Proceedings of the Robert MacCormack 60th Birthday Conference, June 26–28, 2000, Half Moon Bay, CA, NASA/TM-2000-209598Google Scholar
  38. 38.
    Winters, A.R., Gassner, G.J.: Affordable, entropy conserving and entropy stable flux functions for the ideal MHD equations. J. Comput. Phys. 304, 72–108 (2016)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Yee, H.C., Sandham, N.D., Djomehri, M.J.: Low-dissipative high order shock-capturing methods using characteristtic-based filters. J. Comput. Phys. 150, 199–238 (1999)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Yee, H.C., Vinokur, M., Djomehri, M.J.: Entropy splitting and numerical dissipation. J. Comput. Phys. 162, 33–81 (2000)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Yee, H.C., Sjögreen, B.: Development of low dissipative high order filter schemes for multiscale Navier–Stokes and MHD systems. J. Comput. Phys. 225, 910–934 (2007)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Yee, H.C., Sjögreen, B.: High order filter methods for wide range of compressible flow speeds. In: Proceedings of the ICOSAHOM09, June 22–26, Trondheim, Norway (2009)Google Scholar

Copyright information

© This is a U.S. Government work and not under copyright protection in the US; foreign copyright protection may apply 2019

Authors and Affiliations

  1. 1.Multid Analyses ABGothenburgSweden
  2. 2.NASA Ames Research CenterMountain ViewUSA

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