Parameter-uniform numerical methods for singularly perturbed linear transport problems

TL;DR

The paper develops and analyzes parameter-uniform numerical methods using Shishkin meshes for singularly perturbed linear transport problems, demonstrating their effectiveness through theoretical analysis and numerical experiments.

Contribution

It introduces new pointwise accurate, parameter-uniform numerical methods for first-order transport problems using Shishkin meshes, with proven convergence and practical validation.

Findings

Methods achieve parameter-uniform convergence in maximum norm.

Numerical results confirm theoretical error bounds.

Test problem demonstrates practical applicability of the methods.

Abstract

Pointwise accurate numerical methods are constructed and analysed for three classes of singularly perturbed first order transport problems. The methods involve piecewise-uniform Shishkin meshes and the numerical approximations are shown to be parameter-uniformly convergent in the maximum norm. A transport problem from the modelling of fluid-particle interaction is formulated and used as a test problem for these numerical methods. Numerical results are presented to illustrate the performance of the numerical methods and to confirm the theoretical error bounds established in the paper.