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

Xiaojing Dong, Yunqing Huang, Tianwen Wang

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.

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

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.

Paper Structure

This paper contains 11 sections, 11 theorems, 103 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Virtual element method
  4. Basis setting
  5. Virtual element spaces
  6. The discrete velocity and pressure spaces
  7. The discrete magnetic field space
  8. Discrete bilinear and trilinear forms and load terms approximation
  9. Fully discrete BDF2 IMEX SAV scheme
  10. Error analysis
  11. Numerical experiments

Key Result

Lemma 3.1

The discrete local bilinear forms (bilinear-eq1)-(bilinear-eq4) satisfy the following properties

Figures (3)

  • Figure 1: Polygonal meshes: $\mathcal{T}_h^1$(left), $\mathcal{T}_h^2$(middle) and $\mathcal{T}_h^3$(right).
  • Figure 2: The vector graphs of velocity (left) and magnetic (right) field at $t=1$.
  • Figure 3: The contours of velocity (upper) and magnetic (lower) field at $t=1$.

Theorems & Definitions (20)

  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Remark 4.1
  • Lemma 4.3
  • ...and 10 more