# A parameter uniform essentially first order convergent numerical method for a parabolic singularly perturbed differential equation of reaction-diffusion type with initial and Robin boundary conditions

## TL;DR

This paper introduces a finite difference method on a Shishkin mesh for reaction-diffusion parabolic equations with boundary layers, achieving uniform first-order convergence in space and time regardless of perturbation parameters.

## Contribution

The paper proposes a novel numerical scheme that attains uniform essentially first-order convergence for a class of singularly perturbed parabolic equations with boundary layers.

## Key findings

- Method is uniformly convergent in maximum norm.
- Achieves first-order accuracy in time and essentially first-order in space.
- Effective for reaction-diffusion equations with boundary layers.

## Abstract

In this paper, a class of linear parabolic singularly perturbed second order differential equations of reaction-diffusion type with initial and Robin boundary conditions is considered. The solution u of this equation is smooth, whereas the first derivative in the space variable exhibits parabolic boundary layers. A numerical method composed of a classical finite difference scheme on a piecewise uniform Shishkin mesh is suggested. This method is proved to be first order convergent in time and essentially first order convergent in the space variable in the maximum norm uniformly in the perturbation parameters.