Numerical approximations to a singularly perturbed convection-diffusion problem with a discontinuous initial condition

TL;DR

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

Contribution

It introduces a parameter-uniform numerical approach combining an analytical matching function with layer-adapted meshes for singularly perturbed problems.

Findings

Numerical results confirm the theoretical error bounds.

The method effectively handles discontinuities in initial conditions.

The approach achieves uniform convergence across parameter ranges.

Abstract

A singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. An analytic function is identified which matches the discontinuity in the initial condition and also satisfies the homogenous parabolic differential equation associated with the problem. The difference between this analytical function and the solution of the parabolic problem is approximated numerically, using an upwind finite difference operator combined with an appropriate layer-adapted mesh. The numerical method is shown to be parameter-uniform. Numerical results are presented to illustrate the theoretical error bounds established in the paper.