Robust finite difference schemes for one-dimensional parabolic singularly perturbed problems with regular layers

Sah, Kishun Kumar; Gowrisankar, Subramaniam

doi:10.22034/cmde.2025.63398.2828

Robust finite difference schemes for one-dimensional parabolic singularly perturbed problems with regular layers

Document Type : Research Paper

Authors

Department of Mathematics, National Institute of Technology Patna, India.

10.22034/cmde.2025.63398.2828

Abstract

In this article, we study the midpoint finite difference scheme for the singularly perturbed parabolic convection
diffusion problem on non-uniform meshes, which are determined by a generating function with the boundary layer
on the right side of the domain. The non-uniform meshes considered in this work are the classical Shishkin meshes,
Shishkin-Bakhvalov meshes, and Shishkin-Bakhvalov modified meshes. Uniform convergence is established with
respect to the perturbation parameter on the non-uniform meshes. The backward Euler scheme is applied in
the time direction and midpoint finite schemes in the space. Uniform convergence of up to second order in the
space and first order in time is obtained. Two numerical examples are considered to validate the numerical and
theoretical results obtained.

Keywords