Temporal error analysis of Euler semi-implicit scheme for the magnetohydrodynamics equations with variable density

Abstract

In this paper, we consider the magnetohydrodynamics (MHD) equations with variable density, which are a coupled system by the incompressible Navier-Stokes equations with variable density and the Maxwell equations. Such MHD system describes the motions of several conducting incompressible immiscible fluids without surface tension in presence of a magnetic field. A first-order Euler semi-implicit time discrete scheme is proposed to approximate the MHD system such that we only need to solve the linearized subproblems at the discrete level. Moreover, it is unconditionally stable which is a key issue for problems of multiphysical fields. A rigorous error analysis is presented and the first-order temporal convergence rate O(τ) is derived for small τ by using the discrete Lp-regularity technique, where τ is the time step. The numerical results are shown to confirm the unconditional stability and the convergence rate.