Stability and convergence analysis of A linear, fully decoupled and unconditionally energy stable scheme for magneto-hydrodynamic equations

TL;DR

This paper introduces a linear, fully decoupled, energy-stable time-stepping scheme for magneto-hydrodynamic equations, ensuring stability and accuracy in complex simulations like Kelvin-Helmholtz instability.

Contribution

It develops a novel linear, decoupled scheme with proven unconditional energy stability and optimal error estimates for MHD systems.

Findings

Scheme is unconditionally energy stable.

Numerical experiments confirm stability and accuracy.

Effective in simulating benchmark MHD problems.

Abstract

In this paper, we consider numerical approximations for solving the nonlinear magneto-hydrodynamical system, that couples the Navier-Stokes equations and Maxwell equations together. A challenging issue to solve this model numerically is about the time marching problem, i.e., how to develop suitable temporal discretizations for the nonlinear terms in order to preserve the energy stability at the discrete level. We solve this issue in this paper by developing a linear, fully decoupled, first order time-stepping scheme, by combining the projection method and some subtle implicit-explicit treatments for nonlinear coupling terms. We further prove that the scheme is unconditional energy stable and derive the optimal error estimates rigorously. Various numerical experiments are implemented to demonstrate the stability and the accuracy in simulating some benchmark simulations, including the Kelvin-Helmholtz shear instability and the magnetic-frozen phenomenon in the lid-driven cavity.