Convergence of a Finite Difference Scheme for Two-Dimensional Incompressible Magnetohydrodynamics

Abstract

We investigate a finite difference scheme for the two-dimensional, incompressible magnetohydrodynamics equations that was introduced in [J.-G. Liu and W.-C. Wang, J. Comput. Phys., 174 (2001), pp. 12--37]. It uses central difference and averaging operators on a staggered grid and was shown not only to keep the magnetic and the velocity field divergence free but, moreover, to preserve the energy exactly. We extend Liu and Wang's result by a discrete $H^1$ bound, or more precisely, we show that the discrete solution for the velocity and the magnetic field is bounded in $L^\infty(0,\infty;H^1(\Omega))$ and $L^2(0,T;H^2(\Omega))$. This bound allows us to prove the convergence of the scheme. The convergence is strong in $L^2$ and weak in $H^1$.