High order entropy conservative central schemes for wide ranges of compressible gas dynamics and MHD flows

The Sjogreen and Yee [31]high order entropy conservative numerical method for compressible gas dynamics is extended to include discontinuities and also extended to equations of ideal magnetohydrodynamics (MHD). The basic idea is based on Tadmor’s [40]original work for inviscid perfect gas flows. For the MHD four formulations of the MHD are considered: (a) the conservative MHD, (b) the Godunov [14]non-conservative form, (c) the Janhunen [19]– MHD with magnetic field source terms, and (d) a MHD with source terms by Brackbill and Barnes[5]. Three forms of the high order entropy numerical fluxes for the MHD in the finite difference framework are constructed. They are based on the extension of the low order form of Chandrashekar and Klingenberg [9], and two forms with modifications of the Winters and Gassner [49]numerical fluxes. For flows containing discontinuities and multiscale turbulence fluctuations the high order entropy conservative numerical fluxes as the new base scheme under the Yee and Sjogreen [31]and Kotov et al. [21,22]high order nonlinear filter approach is developed. The added nonlinear filter step on the high order centered entropy conservative spatial base scheme is only utilized at isolated computational regions, while maintaining high accuracy almost everywhere for long time integration of unsteady flows and DNS and LES of turbulence computations. Representative test cases for both smooth flows and problems containing discontinuities for the gas dynamics and the ideal MHD are included. The results illustrate the improved stability by using the high order entropy conservative numerical flux as the base scheme instead of the pure high order central scheme.