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

Abstract

The Sjogreen 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.