A virtual element method with IMEX-SAV scheme for the incompressible magnetohydrodynamics equations

TL;DR

This paper introduces a novel virtual element method combined with an IMEX-SAV scheme for solving incompressible magnetohydrodynamics equations, ensuring stability, divergence-free velocity, and decoupled magnetic field computations.

Contribution

It develops a second-order, unconditionally stable scheme that decouples magnetic field computation from velocity and pressure, with rigorous error analysis and numerical validation.

Findings

Unconditionally stable scheme demonstrated through stability analysis.

Decoupled magnetic field computation reduces complexity.

Numerical experiments confirm theoretical error estimates.

Abstract

This paper proposes a virtual element method (VEM) combined with a second-order implicit-explicit scheme based on the scalar auxiliary variable (SAV) method for the incompressible magnetohydrodynamics (MHD) equations. We employ the BDF2 scheme for time discretization and a conservative VEM for spatial discretization, in which the mass conservation in the velocity field is kept by taking advantage of the virtual element method's adaptability and its divergence-free characteristics. In our scheme, the nonlinear terms are handled explicitly using the SAV method, and the magnetic field is decoupled from the velocity and pressure. This decoupling only requires solving a sequence of linear systems with constant coefficient at each time step. The stability estimate of the fully discrete scheme is developed, demonstrating the scheme is unconditionally stable. Moreover, rigorous error estimates for the velocity and magnetic field are provided. Finally, numerical experiments are presented to verify the valid of theoretical analysis.