A high-order finite difference scheme for a singularly perturbed reaction-diffusion problem with an interior layer
Abstract 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...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2017-07-01
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| Series: | Advances in Difference Equations |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13662-017-1268-1 |
| _version_ | 1828307402091397120 |
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| author | Zhongdi Cen Anbo Le Aimin Xu |
| author_facet | Zhongdi Cen Anbo Le Aimin Xu |
| author_sort | Zhongdi Cen |
| collection | DOAJ |
| description | Abstract 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. |
| first_indexed | 2024-04-13T14:56:43Z |
| format | Article |
| id | doaj.art-eee7c9fc722e49a0a0aed2a28e1f0c25 |
| institution | Directory Open Access Journal |
| issn | 1687-1847 |
| language | English |
| last_indexed | 2024-04-13T14:56:43Z |
| publishDate | 2017-07-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Advances in Difference Equations |
| spelling | doaj.art-eee7c9fc722e49a0a0aed2a28e1f0c252022-12-22T02:42:25ZengSpringerOpenAdvances in Difference Equations1687-18472017-07-012017111410.1186/s13662-017-1268-1A high-order finite difference scheme for a singularly perturbed reaction-diffusion problem with an interior layerZhongdi Cen0Anbo Le1Aimin Xu2College of Finance and Trade, Ningbo Dahongying UniversityInstitute of Mathematics, Zhejiang Wanli UniversityInstitute of Mathematics, Zhejiang Wanli UniversityAbstract 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.http://link.springer.com/article/10.1186/s13662-017-1268-1singular perturbationreaction-diffusion equationShishkin-type meshfinite difference schemeuniform convergence |
| spellingShingle | Zhongdi Cen Anbo Le Aimin Xu A high-order finite difference scheme for a singularly perturbed reaction-diffusion problem with an interior layer Advances in Difference Equations singular perturbation reaction-diffusion equation Shishkin-type mesh finite difference scheme uniform convergence |
| title | A high-order finite difference scheme for a singularly perturbed reaction-diffusion problem with an interior layer |
| title_full | A high-order finite difference scheme for a singularly perturbed reaction-diffusion problem with an interior layer |
| title_fullStr | A high-order finite difference scheme for a singularly perturbed reaction-diffusion problem with an interior layer |
| title_full_unstemmed | A high-order finite difference scheme for a singularly perturbed reaction-diffusion problem with an interior layer |
| title_short | A high-order finite difference scheme for a singularly perturbed reaction-diffusion problem with an interior layer |
| title_sort | high order finite difference scheme for a singularly perturbed reaction diffusion problem with an interior layer |
| topic | singular perturbation reaction-diffusion equation Shishkin-type mesh finite difference scheme uniform convergence |
| url | http://link.springer.com/article/10.1186/s13662-017-1268-1 |
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