An Implicit, Nonlinear Reduced Resistive MHD Solver

Abstract

Implicit time differencing of the resistive magnetohydrodynamic (MHD) equations can step over the limiting time scales—such as Alfvén time scales—to resolve the dynamic time scales of interest. However, nonlinearities present in these equations make an implicit implementation cumbersome. Here, viable paths for an implicit, nonlinear time integration of the MHD equations are explored using a 2D reduced viscoresistive MHD model. The implicit time integration is performed using the Newton–Raphson iterative algorithm, employing Krylov iterative techniques for the required algebraic matrix inversions, implemented Jacobian-free (i.e., without ever forming and storing the Jacobian matrix). Convergence in Krylov techniques is accelerated by preconditioning the initial problem. A “physics-based” preconditioner, based on a semi-implicit approximation to the original set of partial differential equations, is employed. The preconditioner employs low-complexity multigrid techniques to invert approximately the resulting elliptic algebraic systems. The resulting 2D reduced resistive MHD implicit algorithm is shown to be successful in dealing with large time steps (on the order of the dynamical time scale of the problem) and fine grids. The algorithm is second-order accurate in time and scalable under grid refinement. Comparison of the implicit CPU time with an explicit integration method demonstrates CPU savings even for moderate (64×64) grids, and close to an order of magnitude in fine grids (256×256).