Skew-Symmetric Splitting for Multiscale Gas Dynamics and MHD Turbulence Flows

The objective of this work is to improve nonlinear stability and to minimize aliasing error by applying high order central discretizations to skew-symmetric splitting forms of the inviscid flux derivative without added high order numerical dissipation for both shock-free compressible turbulence and turbulence with weak shocks. Skew-symmetric splittings of the inviscid flux derivative for high order central schemes are studied and developed for gas dynamics and MHD systems. For problems containing discontinuities and multiscale turbulence fluctuations the Yee and Sjögreen (Proceedings of the ICOSAHOM09, Trondheim, 2016 ) high order nonlinear filter approach is utilized at isolated computational regions, while maintaining high accuracy almost everywhere for direct numerical simulation and large eddy simulations of turbulence computations. Three skew-symmetric splittings are considered for a wide range of compressible flow speeds and flow types, including a 3D forced turbulence simulation. Not all of the skew-symmetric splittings used for gas dynamics can be extended to the non-strictly hyperbolic conservation laws of ideal magnetohydrodynamics (MHD) governing equations. In this work, the Ducros et al. (J Comput Phys 161:114–139, 2000 ) splitting variants are constructed for MHD. Four formulations of the MHD equations are considered. The different formulations 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.