Implicit symmetrized streamfunction formulations of magnetohydrodynamics

We apply the finite element method to the classic tilt instability problem of two-dimensional, incompressible magnetohydrodynamics, using a streamfunction approach to enforce the divergence-free conditions on the magnetic and velocity fields. We compare two formulations of the governing equations, the standard one based on streamfunctions and a hybrid formulation with velocities and magnetic field components. We use a finite element discretization on unstructured meshes and an implicit time discretization scheme. We use the PETSc library with index sets for parallelization. To solve the nonlinear problems on each time step, we compare two nonlinear Gauss-Seidel-type methods and Newton's method with several time-step sizes. We use GMRES in PETSc with multigrid preconditioning to solve the linear subproblems within the nonlinear solvers. We also study the scalability of this simulation on a cluster.