Decoupled, Unconditionally Stable, Higher Order Discretizations for MHD Flow Simulation

Abstract

We propose, analyze, and test a new MHD discretization which decouples the system into two Oseen problems at each timestep yet maintains unconditional stability with respect to the time step size, is optimally accurate in space, and behaves like second order in time in practice. The proposed method chooses a parameter $$\theta \in [0,1]$$źź[0,1], dependent on the viscosity $$\nu $$ź and magnetic diffusivity $$\nu _m$$źm, so that the explicit treatment of certain viscous terms does not cause instabilities, and gives temporal accuracy $$O(\Delta t^2 + (1-\theta )|\nu -\nu _m|\Delta t)$$O(Δt2+(1-ź)|ź-źm|Δt). In practice, $$\nu $$ź and $$\nu _m$$źm are small, and so the method behaves like second order. When $$\theta =1$$ź=1, the method reduces to a linearized BDF2 method, but it has been proven by Li and Trenchea that such a method is stable only in the uncommon case of $$\frac{1}{2}< \frac{\nu }{\nu _m} < 2$$12<źźm<2. For the proposed method, stability and convergence are rigorously proven for appropriately chosen $$\theta $$ź, and several numerical tests are provided that confirm the theory and show the method provides excellent accuracy in cases where usual BDF2 is unstable.