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

The Crank–Nicolson (CN) finite element method is examined with a linear time relaxation term in this study. The linear differential filter κu−u‾ term is added to simplified magnetohydrodynamics (SMHD) equations for numerical regularization and it introduced SMHD linear time relaxation model (SMHDLTR...

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Bibliographic Details
Published in:Numerical methods for partial differential equations 2022-09, Vol.38 (5), p.1232-1254
Main Authors: Yuksel, Gamze, K. Eroglu, Simge
Format: Article
Language:English
Subjects:
Convergence
Crank-Nicholson method
Crank–Nicolson method
Differential equations
Finite element method
linear time relaxation
Magnetohydrodynamics
Mathematical models
Numerical analysis
Regularization
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Summary:The Crank–Nicolson (CN) finite element method is examined with a linear time relaxation term in this study. The linear differential filter κu−u‾ 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++.
ISSN:0749-159X
1098-2426
DOI:10.1002/num.22739