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.
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References
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)
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)
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)
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)
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)
S.K. Godunov, The symmetric form of magnetohydrodynamics equation. Num. Meth. Mech. Cont. Media 1, 26–34 (1972)
P. Janhunen, A positive conservative method for MHD based on HLL and Roe methods. J. Comput. Phys. 160, 649–661 (2000)
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)
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)
K. Linders, J. Nordström, Uniformly best wavenumber approximations by spatial central difference operators. J. Comput. Phys. 300, 695–709 (2015)
B. Sjögreen, H.C. Yee, Multiresolution wavelet based adaptive numerical dissipation control for high order methods. J. Sci. Comput. 20, 211–255 (2004)
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)
B. Sjögreen, H.C. Yee, D. Kotov, Skew-symmetric splitting and stability of high order central schemes. J. Phys. 837, 012019 (2017)
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)
E. Tadmor, Numerical viscosity and the entropy condition for conservative difference schemes. Math. Comput. 43, 369–381 (1984)
E. Tadmor, Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer. 12, 451–512 (2003)
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)
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)
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)
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, 2009
H.C. Yee, M. Vinokur, M.J. Djomehri, Entropy splitting and numerical dissipation. J. Comp. Phys. 162, 33–81 (2000)
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Sjögreen, B., Yee, H.C. (2017). On High Order Entropy Conservative Numerical Flux for Multiscale Gas Dynamics and MHD Simulations. In: Bittencourt, M., Dumont, N., Hesthaven, J. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016. Lecture Notes in Computational Science and Engineering, vol 119. Springer, Cham. https://doi.org/10.1007/978-3-319-65870-4_29
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