Numerical Analysis of Backward-Euler Method for Simplified MagnetoHydroDynamics (SMHD) with Linear Time Relaxation

Abstract

In this study, the solutions of Simplified Magnetohyrodynamics (SMHD) equations by finite element method are examined with linear time relaxation term. Therefore, the differential filter $\kappa(u-\bar{u})$ term is added to SMHD equations for numerical regularization and it is introduced SMHD Linear Time Relaxation Model (SMHDLTRM). The SMHDLTRM model is discretized by Backward-Euler (BE) method to obtain finite element solutions. The stability and convergency of the method is also conducted for SMHDLTRM. The present method is unconditionally stable and convergent with small time step condition. Additionally, the effectiveness of the method has presented with some numerical examples. The BE solutions of the SMHDLTRM are compared with the BE and the Crank-Nicolson (CN) solutions of the SMHD equations. This method can be applied to some problems when necessary more time steps to get accuracy or numerical solutions blow up for classical methods (BE or CN). All computations are conducted by using FreeFem++.

Keywords

• MagnetoHydroDynamics
• backward-Euler method
• linear time relaxation
• finite element method

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