A hybrid approach for singularly perturbed parabolic problem with discontinuous data

TL;DR

This paper introduces a hybrid numerical scheme combining central difference, upwind, and Crank-Nicolson methods on a Shishkin mesh to solve singularly perturbed parabolic equations with discontinuous data, achieving uniform second-order accuracy.

Contribution

It develops a novel hybrid difference scheme that effectively handles discontinuities and interior layers, ensuring uniform convergence for complex singularly perturbed problems.

Findings

The scheme achieves almost second-order accuracy in space.

Numerical results confirm the scheme's superior convergence and accuracy.

The method effectively manages discontinuities and interior layers.

Abstract

In this article, we study a two-dimensional singularly perturbed parabolic equation of the convection-diffusion type, characterized by discontinuities in the source term and convection coefficient at a specific point in the domain. These discontinuities lead to the development of interior layers. To address these layers and ensure uniform convergence, we propose a hybrid monotone difference scheme that combines the central difference and midpoint upwind schemes for spatial discretization, applied on a piecewise-uniform Shishkin mesh. For temporal discretization, we employ the Crank-Nicolson method on a uniform mesh. The resulting scheme is proven to be uniformly convergent, order achieving almost two in space and two in time. Numerical experiments validate the theoretical error estimates, demonstrating superior accuracy and convergence when compared to existing methods.