Partitioned Conservative, Variable Step, Second-Order Method for Magneto-hydrodynamics In Els\"asser Variables

Magnetohydrodynamics (MHD) describes the interaction between electrically conducting fluids and electromagnetic fields. We propose and analyze a symplectic, second-order algorithm for the evolutionary MHD system in Els\"asser variables. We reduce the computational cost of the iterative non-linear solver, at each time step, by partitioning the coupled system into two subproblems of half size, solved in parallel. We prove that the iterations converge linearly, under a time step restriction similar to the one required in the full space-time error analysis. The variable step algorithm unconditionally conserves the energy, cross-helicity and magnetic helicity, and numerical solutions are second-order accurate in the $L^{2}$ and $H^{1}$-norms. The time adaptive mechanism, based on a local truncation error criterion, helps the variable step algorithm balance accuracy and time efficiency. Several numerical tests support the theoretical findings and verify the advantage of time adaptivity.