Pseudo-transient continuation-based variable relaxation solver in nonlinear magnetohydrodynamic simulations

Efficient and robust Variable Relaxation Solver, based on pseudo-transient continuation, is developed to solve nonlinear anisotropic thermal conduction arising from fusion plasma simulations. By adding first- and/or second-order artificial time derivatives to the system, this type of method advances the resulting time-dependent nonlinear PDEs to steady state, which is the solution to be sought. In this process, only the stiffness matrix itself is involved so that the numerical complexity and errors can be greatly reduced. In fact, this work is an extension of integrating efficient linear elliptic solvers for fusion simulation on Cray X1E. Two schemes are derived in this work, first- and second-order variable relaxations. Four factors are observed to be critical for efficiency and preservation of solution's symmetric structure arising from periodic boundary condition: refining meshes in different coordinate directions, initializing nonlinear process, varying time steps in both temporal and spatial directions, and accurately generating nonlinear stiffness matrix. First finer mesh scale should be taken in strong transport direction; next the system is carefully initialized by the solution with linear conductivity; third, time step and relaxation factor are vertex-based varied and optimized at each time step; finally, the nonlinear stiffness matrix is updated by just scaling corresponding linear one with the vector generated from nonlinear thermal conductivity.