# A higher order numerical scheme for singularly perturbed parabolic turning point problems exhibiting twin boundary layers

## TL;DR

This paper introduces a higher order, parameter-uniform numerical scheme for solving singularly perturbed parabolic turning point problems with twin boundary layers, demonstrating improved accuracy and convergence on specialized meshes.

## Contribution

It develops a novel hybrid discretization scheme on a generalized Shishkin mesh that achieves higher order uniform convergence for complex boundary layer problems.

## Key findings

- Scheme is $$-uniformly convergent of order one in time and order two in space.
- Numerical results confirm the scheme's higher accuracy over traditional methods.
- Comparison shows superior performance on generalized Shishkin mesh.

## Abstract

In this article, a parameter-uniform numerical method is presented to solve one-dimensional singularly perturbed parabolic convection-diffusion turning point problem exhibiting two exponential boundary layers. We study the asymptotic behaviour of the solution and its partial derivatives. The problem is discretized using the implicit Euler method for time discretization on a uniform mesh and a hybrid scheme for spatial discretization on a generalized Shishkin mesh. The scheme is shown to be $\varepsilon$-uniformly convergent of order one in time direction and order two in spatial direction upto a logarithmic factor. Numerical experiments are conducted to validate the theoretical results. Comparison is done with upwind scheme on uniform mesh as well as on standard Shishkin mesh to demonstrate the higher order accuracy of the proposed scheme on a generalized Shishkin mesh. \end{abstract} \begin{keyword} Singular perturbation, parabolic convection-diffusion equations, turning point, hybrid scheme, twin boundary layers, Shishkin mesh