A Crank-Nicolson leap-frog scheme for the unsteady incompressible magnetohydrodynamics equations

TL;DR

This paper introduces a stable and convergent Crank-Nicolson leap-frog scheme for unsteady incompressible MHD equations, combining finite element spatial discretization with advanced temporal methods for improved accuracy.

Contribution

It develops a novel CNLF scheme with proven stability, convergence, and optimal error estimates for unsteady incompressible MHD equations, enhancing numerical simulation accuracy.

Findings

The scheme is stable for time steps below a certain threshold.

The method achieves second-order accuracy in the L2 norm.

Numerical results confirm theoretical convergence and effectiveness.

Abstract

This paper presents a Crank-Nicolson leap-frog (CNLF) scheme for the unsteady incompressible magnetohydrodynamics (MHD) equations. The spatial discretization adopts the Galerkin finite element method (FEM), and the temporal discretization employs the CNLF method for linear terms and the semi-implicit method for nonlinear terms. The first step uses Stokes style's scheme, the second step employs the Crank-Nicolson extrapolation scheme, and others apply the CNLF scheme. We testify that the fully discrete scheme is stable and convergent when the time step is less than or equal to a positive constant. The second order $L^{2}$ error estimates can be derived by a novel negative norm technique. The numerical results are consistent with our theoretical analysis, which indicates that the method has an optimal convergence order. Therefore, the scheme is effective for different parameters.