Parameter-uniform approximations for a singularly perturbed convection-diffusion problem with a discontinuous initial condition

TL;DR

This paper develops a parameter-uniform numerical method for a singularly perturbed convection-diffusion problem with a discontinuous initial condition, using a special analytical function and a layer-adapted mesh to ensure accuracy.

Contribution

It introduces a new analytical function matching the initial discontinuity and demonstrates a parameter-uniform numerical method with proven error bounds for such problems.

Findings

The numerical method is proven to be parameter-uniform.

Numerical results confirm the theoretical error bounds.

The approach effectively handles discontinuities in initial conditions.

Abstract

A singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. A particular complimentary error function is identified which matches the discontinuity in the initial condition. The difference between this analytical function and the solution of the parabolic problem is approximated numerically. A co-ordinate transformation is used so that a layer-adapted mesh can be aligned to the interior layer present in the solution. Numerical analysis is presented for the associated numerical method, which establishes that the numerical method is a parameter-uniform numerical method. Numerical results are presented to illustrate the pointwise error bounds established in the paper.