A robust fitted numerical scheme for singularly perturbed parabolic reaction–diffusion problems with a general time delay

Abstract

The present paper deals with the class of time-delayed, singularly perturbed parabolic reaction–diffusion problems. In the x−t plane, parabolic boundary layers appear on the two lateral sides of the domain when a small parameter is multiplied by the second-order space derivative. A fitted operator-based numerical method is developed to solve the considered problem, and its detailed analysis is done. To discretize the spatial domain, an exponentially fitted cubic B-spline scheme is used, and for the discretization of the time derivative, we use the implicit Euler scheme on the uniform mesh. We improve accuracy in the temporal direction using Richardson’s extrapolation method, which results in second-order parameter-uniform convergence. The stability and uniform convergence analysis of the scheme are studied. The present scheme gives a more accurate solution than existing methods in the literature. To ensure that the established numerical scheme is applicable, two test examples are carried out. The obtained numerical results support the estimated value in theory.

Keywords

• Singular perturbation
• Parabolic reaction–diffusion problem
• General time lag
• Fitted cubic B-spline method
• Error estimate