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.


Mathematics and Statistics


National Science Foundation, Grant DMS-1818642

Keywords and Phrases

Finite Element Method; Fully-Decoupled; Linearized; MHD; Second-Order; Unconditional Energy Stability

International Standard Serial Number (ISSN)

1090-2716; 0021-9991

Document Type

Article - Journal

Document Version


File Type





© 2022 Elsevier, All rights reserved.

Publication Date

01 Jan 2022