A Fully Decoupled Linearized Finite Element Method with Second-Order Temporal Accuracy and Unconditional Energy Stability for Incompressible MHD Equations
For highly coupled nonlinear incompressible magnetohydrodynamic (MHD) system, a well-known numerical challenge is how to establish an unconditionally energy stable linearized numerical scheme which also has a fully decoupled structure and second-order time accuracy. This paper simultaneously reaches all of these requirements for the first time by developing an effective numerical scheme, which combines a novel decoupling technique based on the "zero-energy-contribution" feature satisfied by the coupled nonlinear terms, the second-order projection method for dealing with the fluid momentum equations, and a finite element method for spatial discretization. The implementation of the scheme is very efficient, because only a few independent linear elliptic equations with constant coefficients need to be solved by the finite element method at each time step. The unconditional energy stability and well-posedness of the scheme are proved. Various 2D and 3D numerical simulations are carried out to illustrate the developed scheme, including convergence/stability tests and some benchmark MHD problems, such as the hydromagnetic Kelvin-Helmholtz instability, and driven cavity problems.
G. D. Zhang et al., "A Fully Decoupled Linearized Finite Element Method with Second-Order Temporal Accuracy and Unconditional Energy Stability for Incompressible MHD Equations," Journal of Computational Physics, vol. 448, article no. 110752, Elsevier, Jan 2022.
The definitive version is available at https://doi.org/10.1016/j.jcp.2021.110752
Mathematics and Statistics
Keywords and Phrases
Finite Element Method; Fully-Decoupled; Linearized; MHD; Second-Order; Unconditional Energy Stability
International Standard Serial Number (ISSN)
Article - Journal
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01 Jan 2022