Linearization methods for reaction-diffusion equations: 1-D problems

Abstract

Three types of linearized finite difference methods are presented. The first type is based on a θ-formulation which provides discrete solutions in both space and time. It is shown that this type is a linearly-implicit Rosenbrock's or a W-method depending on whether the full or an appropriate Jacobian is employed. The second one requires the solution of two-point, linear, ordinary differential equations and provides either piecewise continuous or piecewise differentiable solutions in space and discrete in time. The third type is based on the discretization of the spatial coordinate and provides a system of linear, ordinary differential equations in time which can be integrated analytically. Therefore, the third type of linearized finite difference methods provides continuous solutions in time and discrete in space, and may be transformed into one of the first type by discretizing the time variable.