Error Estimates of Scalar Auxiliary Variable Schemes with Variable Time Steps for the Magneto-Hydrodynamic Equations

Abstract In this paper, we employ an adaptive time-stepping strategy to conduct a comprehensive theoretical analysis and numerical simulations of the magnetohydrodynamic (MHD) system. Our objective is to formulate a numerical scheme based on the variable time-steps second-order backward difference formula (VBDF2), which exhibits linear, decoupled, and semi-discrete properties. Firstly, considering the complexity of handling the nonlinear terms in the MHD equations, we will introduce a method that combines the exponential scalar auxiliary variable with the zero-energy contribution. This allows us to construct a new auxiliary variable, which is of crucial importance in dealing with the nonlinear terms. Subsequently, by reformulating the MHD system using this new auxiliary variable, we have successfully derived an equivalent system that is unconditionally stable in terms of energy. Secondly, based on the adaptive time-stepping strategy, we establish a semi-discrete MHD system to guarantee the stability of the system. It is worth noting that, under the favorable condition of being unrestricted by the time-step size, we present a rigorous error analysis. Finally, we carry out several numerical simulations to verify the theoretical results by using an effective adaptive time-step strategy.