Linearization methods for reaction-diffusion equations: Multidimensional problems

Abstract

Four types of linearization methods for the numerical solution of multidimensional reaction-diffusion equations are presented. The first two types are based on the discretization of the time variable, time linearization and approximate factorization. The first type also discretizes the spatial coordinates, results in block-tridiagonal matrices, and provides discrete solutions in space and time. The second type employs space linearization, yields linear, ordinary differential equations in space, and produces either piecewise continuous or piecewise differentiable solutions. The third type is based on the discretization of the spatial coordinates and time linearization, and yields continuous solutions in time. The fourth type uses time and space linearization, and results in a multidimensional, linear, elliptic equation whose solution by means of separation of variables provides continuous approximations in space and discrete in time. The fourth type also yields nine-point finite difference expressions compared with the five-point ones of the first three types.