Many fundamental models in science take the form of time-dependent nonlinear reaction-diffusion differential equations. Examples include the modeling of domain walls in ferromagnetics materials, predator-prey models, the description of superconductivity in liquids, the modeling of the famous Belousov-Zhabotinsky reaction in chemical kinetics, the modeling of flame propagation, and the modeling of the spread of rabies in foxes. As well as being important for physical modeling purposes, solutions of reaction-diffusion equations can also exhibit complex and beautiful behavior arising primarily from the competition between reaction and diffusion and the nonlinear nature of the equations.
In this talk, we explore the use of Krylov-subspace techniques for the iterative solution of the linear systems that arise within the numerical solution of nonlinear reaction-diffusion equations. We first describe two suitable families of finite-element space-time discretization for such equations, the so-called continuous and discontinuous Galerkin methods. In both cases, a nonlinear system needs to be solved for each time interval, which is done by means of an inexact Newton method. We use preconditioned Krylov-subspace iterations to obtain an approximate Newton direction at each step of the Newton iteration. For higher-order Galerkin methods, the linear systems for the Newton directions are nonsymmetric, even if the spatial part of the reaction-diffusion equation is self-adjoint. Furthermore, the spectral properties of the coefficient matrices of these systems depend crucially on the choice of the basis functions used for the time discretization in the Galerkin method. We discuss the choice of basis functions that are most amenable to Krylov-subspace iterations. We then describe suitable preconditioning techniques for the linear systems to be solved at each Newton step. The matrices of these linear systems can be split into contributions from the time-derivative part, the reaction part, and the diffusion term of the reaction-diffusion equation. We discuss how to couple and balance approximations to these three parts in the construction of suitable preconditioners. Finally, we report results of numerical experiments for a variety of test problems.
This is joint work with Donald J. Estep (Georgia Institute of Technology).