# Robust numerical schemes for singularly perturbed delay parabolic convection diffusion problems with degenerate coefficient

## TL;DR

This paper develops and analyzes a robust numerical scheme combining backward Euler and specialized spatial discretization on a modified Shishkin mesh to effectively solve singularly perturbed delay convection diffusion problems with degenerate coefficients.

## Contribution

It introduces a new numerical method with stability and convergence analysis for challenging delay convection diffusion equations with degenerate coefficients.

## Key findings

- The proposed scheme is stable and convergent as predicted theoretically.
- Numerical results demonstrate the scheme's effectiveness over traditional methods.
- Comparison shows improved accuracy on Shishkin mesh.

## Abstract

This article studies a dirichlet boundary value problem for singularly perturbed time delay convection diffusion equation with degenerate coefficient. A priori explicit bounds are established on the solution and its derivatives. For asymptotic analysis of the spatial derivatives the solution is decomposed into regular and singular parts. To approximate the solution a numerical method is considered which consists of backward Euler scheme for time discretization on uniform mesh and a combination of midpoint upwind and central difference scheme for the spatial discretization on modified Shishkin mesh. Stability analysis is carried out, numerical results are presented and comparison is done with upwind scheme on uniform mesh as well as upwind scheme on Shishkin mesh to demonstrate the effectiveness of the proposed method. The convergence obtained in practical satisfies the theoretical predictions.