Higher Order Non-Uniform Grids for Singularly Perturbed Convection-Diffusion-Reaction Problems

Abstract

In this paper, a higher order nonuniform grid strategy is developed for solving singularly perturbed convection-diffusion-reaction problems with boundary layers. A new nonuniform grid finite difference method (FDM) based on a coordinate transformation is adopted to establish higher order accuracy. To achieve this, we study and make use of the truncation error of the discretized system to obtain a fourth-order nonuniform grid transformation. Considering a three-point central finite-difference scheme, we create not only fourth-order but even sixth-order approximations (which is the maximum order that can be obtained) by a suitable choice of the underlying nonuniform grids. Further, an adaptive nonuniform grid method based on equidistribution principle is used to demonstrate the sixth-order of convergence. Unlike several other adaptive numerical methods, our strategy uses no pre-knowledge of the location and the width of the layers. Numerical experiments for various test problems are presented to verify the theoretical aspects. We also show that other, slightly different, choices of the grid distributions already lead to a substantial degradation of the accuracy. The numerical results illustrate the effectiveness of the proposed higher order numerical strategy for nonlinear convection dominated singularly perturbed boundary value problems.