Numerical Methods for Plasma Astrophysics:
From Particle Kinetics to MHD

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Central Schemes for Multi Dimensional MHD Equations

Jorge Balbas

Department of Mathematics at UCLA

Abstract:   Central schemes offer a general approach for computing the solutions of nonlinear hyperbolic conservation laws in the sense that they are not tied to the characteristic structure of the underlying PDE, this allows for the straight forward implementation of these as {\em black-box} type numerical solvers. We present a collection of central schemes for the equations of ideal magnetohydrodynamics (MHD) in one and two space dimensions. We first introduce a family of fully discrete central schemes based on the evolution of cell averages over a staggered grid. From this fully discrete approximation of cell averages, making use of the information provided by the speed of propagation of the MHD waves, we pass (in the limit $\Delta t \downarrow 0$) to a more versatile non-staggered semidiscrete formulation. The numerical solution of several MHD prototype problems presented below demonstrate the ability of central schemes to detect and resolve accurately the steep gradients that characterize the solutions of these equations.