Fully decoupled, linearized and stabilized finite volume method for the time-dependent incompressible MHD equations

Abstract

In this paper, we consider the stability and convergence of the fully decoupled and linearized numerical scheme for the time-dependent incompressible magnetohydrodynamic equations based on the finite volume method. The lowest equal-order mixed finite element pair (P1-P1-P1) is used to approximate the velocity, pressure and magnetic fields, and the pressure projection stabilization is introduced to bypass the restriction of the discrete inf-sup condition. The semi-implicit treatment is used to linearize the nonlinear terms and the first order projection scheme is adopted to split the velocity and pressure, then a series of fully decoupled and linearized subproblems are formed. From the view of theoretical analysis, some novel stability results of the numerical solutions in both spatial semi-discrete scheme and time–space fully discrete scheme are provided, the optimal error estimates are also presented by using the energy method and choosing different test functions. From the view of computational results, the fully decoupled and linearized numerical scheme not only keeps good accuracy, but also saves a lot of computational cost.