A robust numerical scheme for singularly perturbed parabolic reaction-diffusion problems via the method of lines

In this paper, we consider one- and two-dimensional singularly perturbed parabolic reaction-diffusion problems. We propose a parameter-uniform numerical scheme to solve these problems. The continuous problem is first discretized in the space variable using a fitted operator finite difference method. The partial differential equation is thus transformed into a system of initial value problems which are then integrated in time with the Crank–Nicolson finite difference method. A convergence analysis shows that the scheme is second-order ε-uniform convergent in space and time. Richardson extrapolation of the space variable results in a fourth order ε-uniform convergence. Numerical experiments on two test examples confirm the theoretical findings.