High-Order Compact Finite-Difference Scheme for Singularly-Perturbed Reaction-Diffusion Problems on a New Mesh of Shishkin Type

Abstract

In this work, a high-order compact finite-difference (HOCFD) scheme has been proposed to solve 1-dimensional (1D) and 2-dimensional (2D) elliptic and parabolic singularly-perturbed reaction-diffusion problems. A new kind of piecewise uniform mesh of Shishkin type (Miller et al. in Fitted Numerical Methods for Singular Perturbation Problems, 1996) has also been proposed and using this mesh the HOCFD scheme gives better results as compared to the results using the Shishkin mesh. Moreover, the stated method gives e-uniform convergence and improved orders of convergence which have also been provided in the results for some test problems.