A Comparative Study of Some Spatial-temporal Discretization Schemes for Nonlinear Magnetohydrodynamic Simulation of Plasmas

Abstract

A large number of numerical schemes have been developed for the integration of the hyperbolic system of partial differential equations (PDEs) arising in the magnetohydrodynamic (MHD) simulation of plasmas. These schemes can be based on either the combined space and time discretization such as the Lax-Wendroff type schemes, or one may perform first a separate space discretization leading to a semidiscretized set of ordinary differential equations (ODEs), which are then separately integrated in time. In this work, a comparative study of two schemes based on simultaneous discretization of space and time (Richtmyer two-step Lax-Wendroff scheme and MacCormack scheme) and one scheme based on centered-space semidiscretization followed by time integration by the fourth-order Runge-Kutta method, is presented. Particular attention is paid to the applicability of the linear stability criteria to the numerical integration of nonlinear MHD equations with geometry and field components of a linear theta-pinch.