A uniformly accurate hybrid difference approximation of a system of singularly perturbed reaction-diffusion equations with delay using grid equidistribution

This paper presents a uniformly accurate difference approximation for a system of singularly perturbed reaction-diffusion equations with delay. The proposed method utilizes an appropriate combination of exponential and cubic spline difference schemes. It employs grid equidistribution to address the challenges posed by the multiscale nature of these systems, which often feature sharp gradients and boundary layers. The grid is generated based on the equidistribution of a positive monitor function, a linear combination of a constant floor and a power of the second derivative of the solution. By using adaptive mesh generation and a spline difference method, the approach enhances the accuracy of the numerical solutions while maintaining computational efficiency. Numerical experiments validate the uniform convergence and theoretical findings, demonstrating the method’s robustness irrespective of the perturbation parameter size.