Skew-Symmetric Splitting and Stability of High Order Central Schemes

Skew-symmetric splittings of the inviscid flux derivative for high order central schemes are studied and developed to improve their numerical stability without added high order numerical dissipation for long time wave propagations and long time integration of compressible turbulent flows. For flows containing discontinuities and multiscale turbulence fluctuations the Yee & Sjogreen [33] and Kotov et al. [15, 14] high order nonlinear filter approach is utilized in conjunction with the skew-symmetric form of high order central schemes. Due to the incomplete hyperbolic nature of the conservative ideal magnetohydrodynamics (MHD) governing equations, not all of the skew-symmetric splittings for gas dynamics can be extended to the ideal MHD. For the MHD the Ducros et al. [6] variants are constructed. In addition, four formulations of the MHD are considered: (a) the conservative MHD, (b) the Godunov/Powell non-conservative form, (c) the Janhunen MHD with magnetic field source terms [13], and (d) a MHD with source terms of [3]. The different formulation of the equations in conjunction with the variants of Ducros et al. type skew-symmetric splitting will be shown to have a strong effect on the stability of non-dissipative approximations. Representative test cases for both smooth flows and problems containing discontinuities for the ideal MHD are included. The results illustrate the improved stability by using the skew-symmetric splitting as part of the central base scheme instead of the pure high order central scheme.