Second-Order Robust Numerical Method for a Partially Singularly Perturbed Time-Dependent Reaction-Diffusion System

This article aims at the development and analysis of a numerical scheme for solving a singularly perturbed parabolic system of n reaction-diffusion equations where m of the equations (with m < n) contain a perturbation parameter while the rest do not contain it. The scheme is based on a uniform mesh in the temporal variable and a piecewise uniform Shishkin mesh in the spatial variable, together with classical finite difference approximations. Some analytical properties and error analyses are derived. Furthermore, a bound of the error is provided. Under certain assumptions, it is proved that the proposed scheme has almost second-order convergence in the space direction and almost first-order convergence in the time variable. Errors do not increase when the perturbation parameter e ? 0, proving the uniform convergence. Some numerical experiments are presented, which support the theoretical results.

Automated summary

This article develops and analyzes a numerical scheme to solve a singularly perturbed parabolic system of n reaction-diffusion equations. The scheme considers m equations with a perturbation parameter and the rest without it. It uses finite difference approximations on a uniform mesh in the temporal variable and a piecewise uniform Shishkin mesh in the spatial variable. Convergence properties and error analyses are derived, and numerical experiments are presented to support the theoretical results.