Numerical analysis of Crank–Nicolson method for simplified magnetohydrodynamics with linear time relaxation

Abstract

AbstractThe Crank–Nicolson (CN) finite element method is examined with a linear time relaxation term in this study. The linear differential filter term is added to simplified magnetohydrodynamics (SMHD) equations for numerical regularization and it introduced SMHD linear time relaxation model (SMHDLTRM). The SMHDLTRM model is discretized by CN method in time and the finite element method in space. The stability and convergency of the method are also conducted. The method is unconditionally stable and convergent under the small time step condition. Additionally, this study summarizes the effectiveness of four methods for SMHD and SMHDLTRM. In previous works SMHD is solved with CN and BE methods and SMHDLTRM is solved with BE method. In this study, the CN solutions of the SMHDLTRM are obtained and compared with the other solutions. All computations are conducted by using FreeFem++.