A robust higher-order fitted mesh numerical method for solving singularly perturbed parabolic reaction-diffusion problems

In this study, a robust higher-order numerical method for solving singularly perturbed parabolic reaction-diffusion problems is presented. The Crank-Nicolson method is applied to discretize the time derivative on a uniform mesh. On a Shishkin mesh, the space derivative is discretized using a hybrid numerical method that combines the cubic spline in tension method for the boundary layer regions with the central difference method for the outer region. Theoretically, we proved that the proposed hybrid numerical method is second-order in the outer region and fourth-order in the boundary layer regions in the space direction. As a result of this, the proposed method produces an almost second-order rate of convergence in the time domain and a higher-order rate of convergence in the space domain. The newly developed method is numerically demonstrated to be uniformly convergent at higher-order, independent of the perturbation parameter. Three numerical examples are computed to support the theoretical results.