High order parameter-robust numerical method for time dependent singularly perturbed reaction–diffusion problems

Abstract

We introduce a high order parameter-robust numerical method to solve a Dirichlet problem for one-dimensional time dependent singularly perturbed reaction-diffusion equation. A small parameter ε is multiplied with the second order spatial derivative in the equation. The parabolic boundary layers appear in the solution of the problem as the perturbation parameter ε tends to zero. To obtain the approximate solution of the problem we construct a numerical method by combining the Crank–Nicolson method on an uniform mesh in time direction, together with a hybrid scheme which is a suitable combination of a fourth order compact difference scheme and the standard central difference scheme on a generalized Shishkin mesh in spatial direction. We prove that the resulting method is parameter-robust or ε-uniform in the sense that its numerical solution converges to the exact solution uniformly well with respect to the singular perturbation parameter ε. More specifically, we prove that the numerical method is uniformly convergent of second order in time and almost fourth order in spatial variable, if the discretization parameters satisfy a non-restrictive relation. Numerical experiments are presented to validate the theoretical results and also indicate that the relation between the discretization parameters is not necessary in practice.