Solutions and Stability Analysis of Backward-Euler Method for Simplified Magnetohydrodynamics With Nonlinear Time Relaxation

Abstract

In this study, the solutions of Simplified Magnetohyrodynamics (SMHD) equations by finite element method are examined with nonlinear time relaxation term. The differential filter 𝜅(|𝑢 − 𝑢 ̅|(𝑢 − 𝑢 ̅)) term is added to SMHD equations. Also SMHD Nonlinear Time Relaxation Model (SMHDNTRM) is introduced. The model is discretized by Backward-Euler (BE) method to obtain the finite element solutions. Moreover, the stability of the method is proved. The method is found unconditionally stable. The effectiveness of the method is exemplified by several cases with comparing different methods. FreeFem++ is used for all computations.