Fitted exact difference method for solutions of a singularly perturbed time delay parabolic PDE

Abstract

A novel numerical method has been proposed for solving singularly perturbed parabolic time delay PDE by using a fitted mesh exact finite difference method. The traditional numerical methods on uniform mesh fail to accurately approximate the solution of the considered PDE, due to the presence of a boundary layer. Therefore, developing a numerical scheme that can effectively handle the boundary layer behaviour is of great interest. The proposed scheme utilizes Crank–Nicolson method for temporal discretization and fitted mesh exact finite difference method for spatial discretization. The scheme satisfies the discrete maximum principle, stability bound, and uniform convergence with the boundary layer resolving property. Numerical test examples have been conducted to validate the scheme for different values of the perturbation parameter and mesh numbers. The results show that the proposed scheme outperforms existing methods in the literature in terms of accuracy. The proposed approach holds significant potential for solving similar PDEs that exhibit boundary layer behaviour.

Keywords

• Fitted mesh
• Exact difference method
• Singularly perturbed problem
• Time delay