The Novel Rational Spectral Collocation Method for a Singularly Perturbed Reaction-Diffusion Problem with Nonsmooth Data

Lufeng Yang,Fotios Georgiades

doi:10.1155/2021/8844163

Lufeng Yang, Fotios Georgiades

Open Access

https://doi.org/10.1155/2021/8844163

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Abstract

A singularly perturbed reaction-diffusion problem with a discontinuous source term is considered. A novel rational spectral collocation combined with a singularity-separated method for this problem is presented. The solution is expressed as u = w + v , where w is the solution of corresponding auxiliary boundary value problem and v is a singular correction with direct expressions. The rational spectral collocation method combined with a sinh transformation is applied to solve the weakened singularly boundary value problem. According to the asymptotic analysis, the sinh transformation parameters can be determined by the width and position of the boundary layers. The parameters in the singular correction can be determined by the boundary conditions of the original problem. Numerical experiment supports theoretical results and shows that compared with previous research results, the novel method has the advantages of a high computational accuracy in singularly perturbed reaction-diffusion problems with nonsmooth data.

Highlights

Singular perturbation problems arise in mathematical modeling of physical and engineering problems, such as the boundary layer of fluid mechanics, the turning point of quantum mechanics, and the flow of large Reynolds numbers
Detailed theories and analysis of the singular perturbation problem can be found in the literature [1, 2]. ese problems have steep gradients in narrow layers, which is a serious obstacle in calculating classical numerical methods
Many studies of singularly perturbed problems with nonsmooth data have been researched. e first-order Schwarz method with uniform parameters on Shishkin’s mesh was proposed for a singularly perturbed reaction-diffusion problem with a nonsmooth source term [3].Chandru and Shanthi applied a boundary value technique and hybrid difference scheme for singularly perturbed boundary value problem of reactiondiffusion type with discontinuous source term in [4, 5], respectively. e second-order finite element method was presented on a Bakhvalov–Shishkin’s mesh for singularly perturbed problems with the interior layer [6]

Summary

Introduction

Singular perturbation problems arise in mathematical modeling of physical and engineering problems, such as the boundary layer of fluid mechanics, the turning point of quantum mechanics, and the flow of large Reynolds numbers. A hybrid difference scheme on a Shishkin’s mesh was introduced for a singularly perturbed reaction-diffusion problem with one and two parameters [9] with interior layers. E second-order singularly perturbed reaction-diffusion boundary value problem with discontinuous source term is considered: Mathematical Problems in Engineering. A novel rational spectral collocation method is presented combined with the singularity-separated technique for a system of singularly perturbed boundary value problems [14]. A novel numerical method based on the rational spectral collocation in a barycentric form with a singularly-separated method (RSC-SSM) is proposed for solving the secondorder singularly perturbed reaction-diffusion problem with nonsmooth source term.

Preliminaries

Numerical Experiments

Conclusions