A parameter uniform hybrid approach for singularly perturbed two-parameter parabolic problem with discontinuous data

TL;DR

This paper develops a hybrid numerical scheme for two-parameter singularly perturbed parabolic problems with discontinuous data, achieving uniform convergence and higher accuracy compared to existing methods.

Contribution

It introduces a novel hybrid difference scheme on a Shishkin mesh combined with Crank-Nicolson, ensuring uniform convergence for complex singularly perturbed problems.

Findings

Scheme achieves almost second-order spatial convergence

Exact second-order temporal convergence demonstrated

Numerical results outperform existing methods in accuracy and convergence

Abstract

In this article, we address singularly perturbed two-parameter parabolic problem of the reaction-convection-diffusion type in two dimensions. These problems exhibit discontinuities in the source term and convection coefficient at particular domain points, which result in the formation of interior layers. The presence of two perturbation parameters leads to the formation of boundary layers with varying widths. Our primary focus is to address these layers and develop a scheme that is uniformly convergent. So we propose a hybrid monotone difference scheme for the spatial direction, implemented on a specially designed piece-wise uniform Shishkin mesh, combined with the Crank-Nicolson method on a uniform mesh for the temporal direction. The resulting scheme is proven to be uniformly convergent, with an order of almost two in the spatial direction and exactly two in the temporal direction. Numerical experiments support the theoretically proven higher order of convergence and shows that our approach results in better accuracy and convergence compared to other existing methods in the literature.