Finite Volume TVD Schemes for Magnetohydrodynamics on Unstructered Grids

Abstract

A three-dimensional (3-D) high-resolution magnetohydrodynamic (MHD) simulation scheme is developed on unstructured grid systems to solve the complexsystem problems in space science and space weather in which numerical difficulties arise from inhomogeneity due to strong background potential fields, inclusion of multispecies ions, and formations of shocks and discontinuities. The ideal MHD equations are extended to the 9-component MHD equations for multi-component ions and modified so as to avoid a direct inclusion of background potential field in dependent variables through the use of new variables. The numerical scheme adopts the finite volume method (FVM) with an upwinding numerical flux based on the linearized Riemann solver. Upwindings on unstructured grid systems are realized from the fact that the MHD equations are symmetric with respect to the rotation of the space. Despite the modifications of the equation system, the eigenvectors in the mode-synthesis matrix necessary for the evaluation of the upwinding numerical flux can still be written analytically. To get a higher order of accuracy, the upwinding flux is extended to the thirdorder total variation diminishing (TVD) numerical flux in the calculation of FVM, through the monotonic upstream scheme for conservation laws (MUSCL) approach and Van Leer’s differentiable limiter. Three numerical examples are given in order to show the efficiency of the above scheme.