An efficient nonlinear solution method for non-equilibrium radiation diffusion

Abstract

A new nonlinear solution method is developed and applied to a non-equilibrium radiation diffusion problem. With this new method, Newton-like super-linear convergence is achieved in the nonlinear iteration, without the complexity of forming or inverting the Jacobian from a standard Newton method. The method is a unique combination of an outer Newton-based iteration and and inner conjugate gradient-like (Krylov) iteration. The effects of the Jacobian are probed only through approximate matrix–vector products required in the conjugate gradient-like iteration. The methodology behind the Jacobian-free Newton–Krylov method is given in detail. It is demonstrated that a simple, successive substitution, linearization produces an effective preconditioning matrix for the Krylov method. The efficiencies of different methods are compared and the benefits of converging the nonlinearities within a time step are demonstrated.