Higher-order finite difference schemes on layer-adapted meshes for singularly perturbed problems in 2D

Abstract

In this paper, a midpoint upwind scheme and a new hybrid difference scheme on tensor-product layer-adapted meshes are originally proposed for general two-dimensional singularly perturbed boundary value problems with exponential boundary layers. The higher-order truncation errors are obtained. Two numerical examples demonstrate that the midpoint upwind scheme obtains second-order convergence outside the boundary layers and almost first-order convergence in the boundary layers on the Shishkin mesh, and second-order convergence outside the boundary layers and first-order convergence in the boundary layers on the Bakhvalov-Shishkin mesh, and the new hybrid difference scheme attains second-order convergence outside the boundary layers and almost second-order convergence in the boundary layers on the Shishkin mesh.