An effective second-order scheme for the nonstationary incompressible magnetohydrodynamics equations

We are proposing the Crank-Nicolson/Adams-Bashforth (CN/AB) algorithm for the nonstationary magnetohydro-dynamic (MHD) equations. For time discretization, the time derivative terms are approximated by a first-order Euler-backward scheme, an implicit second-order Crank-Nicolson scheme acting on linear terms, and an explicit Adams-Bashforth scheme dealing with nonlinear terms. We use the finite element method for spatial discretiza-tion. Theoretical studies show that the proposed algorithm is almost unconditionally stabilized with z <= C. The optimal error estimate of the algorithm is obtained by a parabolic argument mathematical trick. Finally, the theoretical results are verified by numerical experiments.