A higher order uniformly convergent method with Richardson extrapolation in time for singularly perturbed reaction–diffusion parabolic problems

In this paper, we are interested in solving efficiently an initial-boundary value singularly perturbed time-dependent problem of reaction–diffusion type. On a priori special mesh we construct a high order uniformly convergent finite difference scheme which combines the implicit Euler method to discretize in time, together with the Richardson extrapolation technique, and a HODIE scheme to discretize in space. The analysis of the uniform convergence splits completely the contribution to the global error of both the time and the space discretizations. We show numerical results for different test problems confirming in practice the order of uniform convergence proved.