Piecewise equidistant meshes for quasilinear turning point problems: Technical report

TL;DR

This paper introduces a specialized finite-difference scheme on piecewise equidistant meshes for solving quasilinear singularly perturbed boundary value problems with turning points, achieving uniform accuracy and decreasing errors as perturbation diminishes.

Contribution

It develops a new mesh design and a combined scheme that improve accuracy and uniformity in solving complex turning point problems.

Findings

Scheme is practically second-order accurate.

L1 errors decrease as perturbation parameter tends to zero.

Meshes generalize the Shishkin mesh for better accuracy.

Abstract

A class of quasilinear singularly perturbed boundary value problems with a turning point of attractive type is considered. The problems are solved numerically by a finite-difference scheme on a special discretization mesh which is dense near the turning point. The scheme is a combination of the standard central and midpoint schemes and is practically second-order accurate. Pointwise accuracy is uniform in the perturbation parameter and, moreover, L1 errors decrease when the perturbation parameter tends to 0. This is achieved by the use of meshes which generalize the piecewise equidistant Shishkin mesh. Two particular types of meshes are considered and compared.