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On High Order Entropy Conservative Numerical Flux for Multiscale Gas Dynamics and MHD Simulations

  • Björn SjögreenEmail author
  • H. C. Yee
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 119)

Abstract

The Sjögreen and Yee (On skew-symmetric splitting and entropy conservation schemes for the Euler equations, in Proceedings of ENUMATH09, June 29- July 2, Uppsala University, Sweden, 2009) high order entropy conservative numerical method for compressible gas dynamics is extended to include discontinuities and also extended to the ideal magnetohydrodynamics (MHD). The basic idea is based on Tadmor’s (Acta Numer 12:451–512, 2003) original work for the Euler gas dynamics. For the MHD four formulations of the MHD formulations are considered: (a) the conservative MHD, (b) the Godunov/Powell non-conservative form, (c) the Janhunen MHD with magnetic field source terms (Janhunen, J Comput Phys 160:649–661, 2000), and (d) a MHD with source terms by Brackbill and Barnes (J Comput Phys 35:426–430, 1980). Three forms of the high order entropy numerical fluxes in the finite difference framework are constructed. They are based on the extension of the low order form by Chandrashekar and Klingenberg (SIAM J Numer Anal 54:1313–1340, 2016), and two forms with modifications of the Winters and Gassner (J Comput Phys 304:72–108, 2016) numerical fluxes. For flows containing discontinuities and multiscale turbulence fluctuations the Yee and Sjogreen (High order filter methods for wide range of compressible flow speeds, in Proceedings of the ICOSAHOM09, Trondheim, Norway, June 22–26, 2009) and Kotov et al. (Commun Comput Phys 19:273–300, 2016; J Comput Phys 307:189–202, 2016) high order nonlinear filter approach are extended to include the high order entropy conservative numerical fluxes as the base scheme.

References

  1. 1.
    A. Arakawa, 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)CrossRefzbMATHGoogle Scholar
  2. 2.
    G.A. Blaisdell, E.T. Spyropoulos, J.H. Qin, The effect of the formulation of nonlinear terms on aliasing errors in spectral methods. Appl. Numer. Math. 21, 207–219 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    J.U. Brackbill, D.C. Barnes, The effect of nonzero ∇⋅ B on the numerical solution of the magnetohydrodynamics equations. J. Comput. Phys. 35, 426–430 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    P. Chandrashekar, C. Klingenberg, Entropy stable finite volume scheme for ideal compressible MHD on 2-D Cartesian Meshes. SIAM J. Numer. Anal. 54, 1313–1340 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    F. Ducros, F. Laporte, T. Soulères, V. Guinot, P. Moinat, B. Caruelle, High-order fluxes for conservative skew-symmetric-like schemes in structured meshes: application to compressible flows. J. Comput. Phys. 161, 114–139 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    S.K. Godunov, The symmetric form of magnetohydrodynamics equation. Num. Meth. Mech. Cont. Media 1, 26–34 (1972)Google Scholar
  7. 7.
    P. Janhunen, A positive conservative method for MHD based on HLL and Roe methods. J. Comput. Phys. 160, 649–661 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    D.V. Kotov, H.C. Yee, A.A. Wray, B. Sjögreen, A.G. Kritsuk, Numerical disipation control in high order shock-capturing schemes for LES of low speed flows. J. Comput. Phys. 307, 189–202 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    D.V. Kotov, H.C. Yee, A.A. Wray, B. Sjögreen, High order numerical methods for dynamic SGS model of turbulent flows with shocks. Commun. Comput. Phys. 19, 273–300 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    K. Linders, J. Nordström, Uniformly best wavenumber approximations by spatial central difference operators. J. Comput. Phys. 300, 695–709 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    B. Sjögreen, H.C. Yee, Multiresolution wavelet based adaptive numerical dissipation control for high order methods. J. Sci. Comput. 20, 211–255 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    B. Sjögreen, H.C. Yee, 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
  13. 13.
    B. Sjögreen, H.C. Yee, D. Kotov, Skew-symmetric splitting and stability of high order central schemes. J. Phys. 837, 012019 (2017)Google Scholar
  14. 14.
    B. Sjögreen, H.C. Yee, Construction of high order entropy conserving numerical flux for gas dynamics and MHD turbulent simulations. J. Comput. Phys. (2016, submitted)Google Scholar
  15. 15.
    E. Tadmor, Numerical viscosity and the entropy condition for conservative difference schemes. Math. Comput. 43, 369–381 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    E. Tadmor, Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer. 12, 451–512 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    A.R. Winters, G.J. Gassner, Affordable, entropy conserving and entropy stable flux functions for the ideal MHD equations. J. Comput. Phys. 304, 72–108 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    H.C. Yee, B. Sjögreen, Efficient low dissipative high order schemes for multiscale MHD flows, II: minimization of ∇⋅ B numerical error. J. Sci. Comput. 29, 115–164 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    H.C. Yee, B. Sjögreen, Development of low dissipative high order filter schemes for multiscale Navier-Stokes MHD systems. J. Comput. Phys. 225 910–934 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    H.C. Yee, B. Sjögreen, High order filter methods for wide range of compressible flow speeds, in Proceedings of the ICOSAHOM09, Trondheim, Norway, June 22–26, 2009Google Scholar
  21. 21.
    H.C. Yee, M. Vinokur, M.J. Djomehri, Entropy splitting and numerical dissipation. J. Comp. Phys. 162, 33–81 (2000)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.MultiD Analyses ABGoteborgSweden
  2. 2.NASA Ames Research CenterMountain ViewUSA

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