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

Authors : Gamze YÜKSEL, Mustafa Hicret YAMAN

Pages : 145-161

Doi:10.47000/tjmcs.757462

View : 10 | Download : 11

Publication Date : 2021-06-30

Article Type : Research Paper

Abstract :In this study, the solutions of Simplified Magnetohyrodynamics insert ignore into journalissuearticles values(SMHD); equations by finite element method are examined with linear time relaxation term. Therefore, the differential filter $\kappainsert ignore into journalissuearticles values(u-\bar{u});$ term is added to SMHD equations for numerical regularization and it is introduced SMHD Linear Time Relaxation Model insert ignore into journalissuearticles values(SMHDLTRM);. The SMHDLTRM model is discretized by Backward-Euler insert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(BE or CN);. All computations are conducted by using FreeFem++.
Keywords : MagnetoHydroDynamics, backward Euler method, linear time relaxation, finite element method