High order algebraic splitting for magnetohydrodynamics simulation

This paper proposes, analyzes and tests high order algebraic splitting methods for magnetohydrodynamic (MHD) flows. The main idea is to apply, at each time step, Yosida-type algebraic splitting to a block saddle point problem that arises from a particular incremental formulation of MHD. By doing so, we dramatically reduce the complexity of the nonsymmetric block Schur complement by decoupling it into two Stokes-type Schur complements, each of which is symmetric positive definite and also is the same at each time step. We prove the splitting is O(Δt3) accurate, and if used together with (block-)pressure correction, is fourth order. A full analysis of the solver is given, both as a linear algebraic approximation, but also in a finite element context that uses the natural spatial norms. Numerical tests are given to illustrate the theory and show the effectiveness of the method.