Stability and error analysis of the SAV schemes for the inductionless MHD equations

TL;DR

This paper introduces linear, decoupled, unconditionally energy stable SAV schemes for the inductionless MHD equations, providing rigorous error analysis and numerical verification of optimal convergence rates.

Contribution

It develops novel first- and second-order SAV schemes for inductionless MHD equations with proven stability and convergence, enhancing computational efficiency and accuracy.

Findings

Schemes are linear, decoupled, and unconditionally energy stable.

Optimal convergence rates are established for key variables.

Numerical examples confirm theoretical stability and accuracy.

Abstract

In this paper, we consider numerical approximations for solving the inductionless magnetohydrodynamic (MHD) equations. By utilizing the scalar auxiliary variable (SAV) approach for dealing with the convective and coupling terms, we propose some first- and second-order schemes for this system. These schemes are linear, decoupled, unconditionally energy stable, and only require solving a sequence of differential equations with constant coefficients at each time step. We further derive a rigorous error analysis for the first-order scheme, establishing optimal convergence rates for the velocity, pressure, current density and electric potential in the two-dimensional case. Numerical examples are presented to verify the theoretical findings and show the performances of the schemes.