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

In this article, we study the midpoint finite difference scheme for the singularly perturbed parabolic convectiondiffusion problem on non-uniform meshes, which are determined by a generating function with the boundary layeron 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 withrespect to the perturbation parameter on the non-uniform meshes. The backward Euler scheme is applied inthe time direction and midpoint finite schemes in the space. Uniform convergence of up to second order in thespace and first order in time is obtained. Two numerical examples are considered to validate the numerical andtheoretical results obtained.

کلیدواژه ها:

Singularly perturbed parabolic problem ، Regular boundary layer ، Midpoint finite difference scheme ، Shishkin mesh ، Shishkin-Bakhvalov mesh ، Shishkin-Bakhvalov modified mesh ، Generating function ، uniform convergence