Optimal error estimates of a Crank--Nicolson finite element projection method for magnetohydrodynamic equations

TL;DR

This paper develops and analyzes a fully discrete finite element projection method using a modified Crank--Nicolson scheme for magnetohydrodynamic equations, providing optimal error estimates and demonstrating energy stability.

Contribution

It introduces a novel decoupled finite element projection scheme with second-order accuracy and energy stability analysis for MHD equations.

Findings

Proves optimal error estimates in discrete norms.

Establishes energy stability of the proposed scheme.

Numerical examples confirm theoretical results.

Abstract

In this paper, we propose and analyze a fully discrete finite element projection method for the magnetohydrodynamic (MHD) equations. A modified Crank--Nicolson method and the Galerkin finite element method are used to discretize the model in time and space, respectively, and appropriate semi-implicit treatments are applied to the fluid convection term and two coupling terms. These semi-implicit approximations result in a linear system with variable coefficients for which the unique solvability can be proved theoretically. In addition, we use a second-order decoupling projection method of the Van Kan type \cite{vankan1986} in the Stokes solver, which computes the intermediate velocity field based on the gradient of the pressure from the previous time level, and enforces the incompressibility constraint via the Helmholtz decomposition of the intermediate velocity field. The energy stability of the scheme is theoretically proved, in which the decoupled Stokes solver needs to be analyzed in details. Error estimates are proved in the discrete $L^\infty(0,T;L^2)$ norm for the proposed decoupled finite element projection scheme. Numerical examples are provided to illustrate the theoretical results.