Fully Discrete Schemes with First- and Second-Order Temporal Accuracy for the Incompressible Magnetohydrodynamic Flow Based on the Generalized Scalar Auxiliary Variable Approach

Abstract

Based on the generalized scalar auxiliary variable approach and vector penalty projection method, some fully discrete schemes with first- and second-order accuracy in time direction are constructed for solving the incompressible magnetohydrodynamic model. It is a combination of mixed finite element approximation for spatial discretization and first-order backward Euler/second-order backward differential formula for temporal discretization. The proposed schemes own several features: it decouples unknown physical variables and linearizes the nonlinear terms, then it only needs to solve some linear equations at each temporal level; although the divergence of numerical velocity is not exactly equal to zero, it can approximately meet the mass conservation when one takes small penalty parameter; while the computation of the velocity and pressure are decoupled, numerical results show that the velocity and pressure can reach second-order accuracy in time. The resulting schemes are supported by numerical analysis and simulation.