On an efficient second order backward difference Newton scheme for MHD system

Abstract

For the 2D/3D time-dependent magnetohydrodynamics (MHD) system, we propose a new second order backward difference formula Newton scheme (SBDFN) which is a combination of second order backward difference approximation of the time derivative terms and Newton treatment of the nonlinear terms. Meanwhile, the finite element method is applied as spatial discretization. Firstly, the optimal convergence of spatially semi-discrete form is deduced. Secondly, the stability and well-posedness of SBDFN scheme are provided under τ<C, where C is independent of h. Based on these results, optimal error estimates of the scheme in time are proved by negative norm technique. Finally, numerical experiments are carried out to validate the theoretical analysis.