The virtual element method for the three dimensional inductionless magnetohydrodynamic model

Abstract

This paper proposes a novel first-order and a novel second-order fully discrete virtual element schemes based on the scalar auxiliary variable method for the three dimensional inductionless magnetohydrodynamics problem. The backward Eular formula and the backward differential formula are used for the time discretization and two types conservation virtual element formulations are employed for spatial discretization. The main advantages include that the mass conservation in the velocity field and the charge conservation in the current density field are kept by taking characteristics of the virtual element method's discrete complex structures, the nonlinear term is handled explicitly by applying the scalar auxiliary variable method, the current density field is decoupled from the momentum equation, and the velocity field is decoupled from the Ohm's law. The unconditionally stable of the two fully discrete schemes are demonstrated. Finally, we present numerical experiment to verify the valid of the proposed schemes.