Document Zbl 1004.65079

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I.</a></div> <h2 class="title"> <strong>A numerical method for a system of singularly perturbed reaction-diffusion equations.</strong> <i>(English)</i> <a class="label nowrap" href="/1004.65079">Zbl 1004.65079</a> </h2> <div class="source"> <a href="/serials/482" title="Journal Profile">J. Comput. Appl. Math.</a> <a href="/?q=in%3A86354" title="Articles in this Issue">145, No. 1, 151-166 (2002)</a>. </div> <div class="abstract">It is well-known that standard numerical methods are not appropriate for singularly perturbed problems. The authors concentrate on the case of a Dirichlet problem for a system of two singularly perturbed reaction-diffusion problems, with diffusion coefficients $\varepsilon_1$ and $\varepsilon_2$ such that $\varepsilon_1=\varepsilon$ and $\varepsilon_2=1$. A numerical parameter-uniform method whose solutions converge pointwise at all points of the domain independently of the singular perturbation parameters is constructed and analysed. The authors establish essentially second-order parameter-uniform convergence. Numerical results are presented, which illustrate the theoretical results. <br class="zbmathjax-paragraph">This paper extends the results for piecewise-uniform meshes applied to a single one-dimensional singularly perturbed reaction-diffusion problem [<span class="zbmathjax-textit">J. J. H. Miller, E. O’Riordan</span> and <span class="zbmathjax-textit">G. I. Shishkin</span>, Fitted mesh methods for the singular perturbation reaction diffusion problem, in: Proceedings of the Fifth International Colloquium on Numerical Analysis, 1996, Plovdiv, Bulgaria (Academic Publications, 1997), pp. 99-105] to a system of two coupled reaction-diffusion problems.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A10061">Oscar Lopez-Pouso (Santiago de Compostela)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A1787680">Cited in <strong>62</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65L10" title="MSC2020">65L10</a> </td> <td class="space"> Numerical solution of boundary value problems involving ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34B15" title="MSC2020">34B15</a> </td> <td class="space"> Nonlinear boundary value problems for ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34E15" title="MSC2020">34E15</a> </td> <td class="space"> Singular perturbations for ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65L20" title="MSC2020">65L20</a> </td> <td class="space"> Stability and convergence of numerical methods for ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65L12" title="MSC2020">65L12</a> </td> <td class="space"> Finite difference and finite volume methods for ordinary differential equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Areaction-diffusion+equation">reaction-diffusion equation</a>; <a href="/?q=ut%3Asingular+perturbation">singular perturbation</a>; <a href="/?q=ut%3ADirichlet+problem">Dirichlet problem</a>; <a href="/?q=ut%3Aparameter-uniform+method">parameter-uniform method</a>; <a href="/?q=ut%3Aconvergence">convergence</a>; <a href="/?q=ut%3Anumerical+results">numerical results</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1004.65079" data-ciurl="/ci/01787680" data-biburl="/bibtex/01787680.bib" data-amsurl="/amsrefs/01787680.bib" data-xmlurl="/xml/01787680.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/01787680.pdf" title="Zbl 1004.65079 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/S0377-0427(01)00541-6" aria-label="DOI for “A numerical method for a system of singularly perturbed reaction-diffusion equations”" title="10.1016/S0377-0427(01)00541-6">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Farrell, P. A.; Hegarty, A. F.; Miller, J. J.H.; O’Riordan, E.; Shishkin, G. I., Robust Computational Techniques for Boundary Layers (2000), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton · <a href="/0964.65083" class="nowrap">Zbl 0964.65083</a></td> </tr><tr> <td>[2]</td> <td class="space">Kan-On, Y.; Mimura, M., Singular perturbation approach to a 3-component reaction-diffusion system arising in population dynamics, SIAM J. Math Anal., 29, 6, 1519-1536 (1998) · <a href="/0920.35015" class="nowrap">Zbl 0920.35015</a></td> </tr><tr> <td>[3]</td> <td class="space">N. Madden, Numerical methods for wave-current interactions, Ph.D., National University of Ireland, Cork, 2000.; N. Madden, Numerical methods for wave-current interactions, Ph.D., National University of Ireland, Cork, 2000.</td> </tr><tr> <td>[4]</td> <td class="space">S. Matthews, Parameter robust methods for a system of coupled singularly perturbed ordinary differential reaction diffusion equations, M.Sc., School of Mathematical Sciences, Dublin City University, 2000.; S. Matthews, Parameter robust methods for a system of coupled singularly perturbed ordinary differential reaction diffusion equations, M.Sc., School of Mathematical Sciences, Dublin City University, 2000.</td> </tr><tr> <td>[5]</td> <td class="space">S. Matthews, J.J.H. Miller, E. O’Riordan, G.I. Shishkin, Parameter-robust numerical methods for a system of reaction-diffusion problems with boundary layers, in: J.J.H. Miller, G.I. Shishkin, L. Vulkov (Eds.), Analytical and Numerical Methods for Convection-Dominated and Singularly Perturbed Problems, Nova Science Publishers, Inc., New York, USA, 2000, pp. 219-224.; S. Matthews, J.J.H. Miller, E. O’Riordan, G.I. Shishkin, Parameter-robust numerical methods for a system of reaction-diffusion problems with boundary layers, in: J.J.H. Miller, G.I. Shishkin, L. Vulkov (Eds.), Analytical and Numerical Methods for Convection-Dominated and Singularly Perturbed Problems, Nova Science Publishers, Inc., New York, USA, 2000, pp. 219-224.</td> </tr><tr> <td>[6]</td> <td class="space">Miller, J. J.H.; O’Riordan, E.; Shishkin, G. I., Fitted numerical methods for singular perturbation problems (1996), World Scientific: World Scientific Singapore · <a href="/0945.65521" class="nowrap">Zbl 0945.65521</a></td> </tr><tr> <td>[7]</td> <td class="space">J.J.H. Miller, E. O’Riordan, G.I. Shishkin, Fitted mesh methods for the singular perturbation reaction diffusion problem, in: Proceedings of the Fifth International Colloquium on Numerical Analysis, 1996, Plovdiv, Bulgaria, (Academic Publications, 1997), pp. 99-105.; J.J.H. Miller, E. O’Riordan, G.I. Shishkin, Fitted mesh methods for the singular perturbation reaction diffusion problem, in: Proceedings of the Fifth International Colloquium on Numerical Analysis, 1996, Plovdiv, Bulgaria, (Academic Publications, 1997), pp. 99-105.</td> </tr><tr> <td>[8]</td> <td class="space">Miller, J. J.H.; O’Riordan, E.; Shishkin, G. I.; Shishkina, L. P., Fitted mesh methods for problems with parabolic boundary layers, Math. Proc. Roy. Irish Acad., 98A, 2, 173-190 (1998) · <a href="/0934.65091" class="nowrap">Zbl 0934.65091</a></td> </tr><tr> <td>[9]</td> <td class="space">O’Riordan, E.; Stynes, M., A uniformly accurate finite-element method for a singularly perturbed one-dimensional reaction-diffusion problem, Math. Comput., 47, 176, 555-570 (1986) · <a href="/0625.65073" class="nowrap">Zbl 0625.65073</a></td> </tr><tr> <td>[10]</td> <td class="space">Roos, H.-G.; Stynes, M.; Tobiska, L., Numerical Methods for Singularly Perturbed Differential Equations, Convection-Diffusion and Flow Problems (1996), Springer-Verlag: Springer-Verlag New York · <a href="/0844.65075" class="nowrap">Zbl 0844.65075</a></td> </tr><tr> <td>[11]</td> <td class="space">Shishkin, G. I., Mesh approximation of singularly perturbed boundary-value problems for systems of elliptic and parabolic equations, Comput. Maths. Math. Phys., 35, 4, 429-446 (1995) · <a href="/0852.65071" class="nowrap">Zbl 0852.65071</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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