A parameter-uniform efficient numerical scheme for singularly perturbed time-delay parabolic problems with two small parameters

Abstract

In this article we study numerical approximation for two-parameter singularly perturbed parabolic partial differential equations with time delay. A priori bounds on the exact solution and its derivatives are provided to useful in the error analysis of the numerical method. The problem is discretized using an exponentially fitted scheme in the spatial direction and the Crank–Nicolson method in the time direction on a uniform mesh. The resulting scheme is shown to be second-order accurate in time and first-order accurate in space. Two test problems are used to validate the theoretical results. Calculating the maximum absolute errors and experimental orders of convergence demonstrate the method’s performance. Since the exact solutions to the test problems are unknown, the double mesh principle is used to calculate the maximum absolute errors.

Keywords

• Singular perturbation
• Two parameters
• Time-delay parabolic problem
• Exponentially fitted method
• Uniformly convergent