A parameter uniform method for two-parameter singularly perturbed boundary value problems with discontinuous data

TL;DR

This paper introduces a parameter-uniform numerical method using a Shishkin-Bakhvalov mesh to accurately solve one-dimensional two-parameter singularly perturbed boundary value problems with discontinuous data, achieving first-order convergence.

Contribution

It proposes a novel parameter-uniform scheme on a Shishkin-Bakhvalov mesh for problems with interior and boundary layers caused by data discontinuities, ensuring first-order convergence.

Findings

The method achieves first-order parameter-uniform convergence.

Numerical results confirm the theoretical error estimates.

The Shishkin-Bakhvalov mesh improves convergence over the Shishkin mesh.

Abstract

A two-parameter singularly perturbed problem with discontinuous source and convection coefficient is considered in one dimension. Both convection coefficient and source term are discontinuous at a point in the domain. The presence of perturbation parameters results in boundary layers at the boundaries. Also, an interior layer occurs due to the discontinuity of data at an interior point. An upwind scheme on an appropriately defined Shishkin-Bakhvalov mesh is used to resolve the boundary layers and interior layers. A three-point formula is used at the point of discontinuity. The proposed method has first-order parameter uniform convergence. Theoretical error estimates derived are verified using the numerical method on some test problems. Numerical results authenticate the claims made. The use of the Shiskin-Bakhvalov mesh helps achieve the first-order convergence, unlike the Shishkin mesh, where the order of convergence deteriorates due to a logarithmic term.