Variable relaxation solve for nonlinear thermal conduction

Efficient and robust nonlinear solvers, based on Variable Relaxation, is developed to solve nonlinear anisotropic thermal conduction arising from fusion plasma simulations. By adding first and/or second order time derivatives to the system, this type of methods advances corresponding time-dependent nonlinear systems 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 implementing efficient linear 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: mesh scales, initialization, variable time step, and nonlinear stiffness matrix computation. 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.