Development and Application of Generalized Musta Schemes

This paper is devoted to the construction of numerical fluxes for hyperbolic systems. We first present a GFORCE numerical flux, which is a weighted average of the Lax-Friedrichs and Lax-Wendroff fluxes. For the linear advection equation with constant coefficient, the new flux reduces identically to that of the Godunov first order upwind method. Then we incorporate GFORCE in the framework of the MUSTA approach, resulting in a version that we call GMUSTA. Both the GFORCE and GMUSTA fluxes are extended to multi-dimensional non-linear systems in a straightforward unsplit manner, resulting in linearly stable schemes that have the same stability regions as the straightforward multi-dimensional extension of Godunov’s method The schemes of this paper share with the family of centred methods the common properties of being simple and applicable to a large class of hyperbolic systems, but are distinctly more accurate. First-order numerical results are presented for the one-dimensional equations of nonlinear elasticity, magnetohydrodynamics. High-order results are given in the framework of the WENO methods for the two-dimensional Euler equations.