A singularly perturbed convection-diffusion problem posed on an annulus

TL;DR

This paper develops a finite difference method using polar coordinates, upwind schemes, and a Shishkin mesh to solve a singularly perturbed convection-diffusion problem on an annulus, with proven uniform error bounds.

Contribution

It introduces a novel combination of polar coordinates, upwind differences, and a Shishkin mesh for annular domains, ensuring uniform convergence and avoiding interior layers.

Findings

The method achieves parameter-uniform error bounds.

Numerical results confirm the theoretical error estimates.

The approach effectively handles characteristic points near the annulus.

Abstract

A finite difference method is constructed for a singularly perturbed convection diffusion problem posed on an annulus. The method involves combining polar coordinates, an upwind finite difference operator and a piecewise-uniform Shishkin mesh in the radial direction. Compatibility constraints are imposed on the data in the vicinity of certain characteristic points to ensure that interior layers do not form within the annulus. A theoretical parameter-uniform error bound is established and numerical results are presented to illustrate the performance of the numerical method applied to two particular test problems.