文章基本信息

标题：A high-order finite difference scheme for a singularly perturbed reaction-diffusion problem with an interior layer
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作者：Zhongdi Cen ; Anbo Le ; Aimin Xu 等
期刊名称：Advances in Difference Equations
印刷版ISSN：1687-1839
电子版ISSN：1687-1847
出版年度：2017
卷号：2017
期号：1
页码：202
DOI：10.1186/s13662-017-1268-1
语种：English
出版社：Hindawi Publishing Corporation
摘要：In this paper, we consider a singularly perturbed reaction-diffusion problem with a discontinuous source term. Boundary and interior layers appear in the solution. The problem is discretized by using a hybrid finite difference scheme on a Shishkin-type mesh. A nonequidistant generalization of the Numerov scheme is used on the Shishkin-type mesh except for the point of discontinuity, whereas a second-order difference scheme with an additional refined mesh is used for the point of discontinuity. Although the difference scheme does not satisfy the discrete maximum principle, the maximum norm stability of the scheme is established. The maximum error in the mesh points is shown to be uniformly bounded by ( N − 1 ln N ) 4 $( N^{-1}\ln N ) ^{4}$ with a constant independent of the perturbation parameter. Numerical results supporting the theory are presented.
关键词：singular perturbation ; reaction-diffusion equation ; Shishkin-type mesh ; finite difference scheme ; uniform convergence