Document Zbl 1205.65221

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Math. Appl.</a> <a href="/?q=in%3A268505" title="Articles in this Issue">60, No. 7, 1924-1939 (2010)</a>. </div> <div class="abstract">Summary: We present the analysis of an upwind scheme for obtaining the global solution and the normalized flux for a convection-diffusion two-point boundary value problem. The solution of the upwind scheme is obtained on a suitable nonuniform mesh which is formed by equidistributing the arc-length monitor function. It is shown that the discrete solution obtained by the upwind scheme and the global solution obtained via interpolation converges uniformly with respect to the perturbation parameter. In addition, we prove the uniform first-order convergence of the weighted derivative of the numerical solution on this nonuniform mesh and the uniform convergence of the global normalized flux on the whole domain. Numerical results are presented that demonstrate the sharpness of our results.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5851153">Cited in <strong>15</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65L11" title="MSC2020">65L11</a> </td> <td class="space"> Numerical solution of singularly perturbed problems involving 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%3Asingularly+perturbed+convection-diffusion+problems">singularly perturbed convection-diffusion problems</a>; <a href="/?q=ut%3Aupwind+scheme">upwind scheme</a>; <a href="/?q=ut%3Aadaptive+mesh">adaptive mesh</a>; <a href="/?q=ut%3Anormalized+flux">normalized flux</a>; <a href="/?q=ut%3Auniform+convergence">uniform convergence</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 1205.65221" data-ciurl="/ci/05851153" data-biburl="/bibtex/05851153.bib" data-amsurl="/amsrefs/05851153.bib" data-xmlurl="/xml/05851153.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05851153.pdf" title="Zbl 1205.65221 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/j.camwa.2010.07.026" aria-label="DOI for “Parameter-uniform numerical method for global solution and global normalized flux of singularly perturbed boundary value problems using grid equidistribution”" title="10.1016/j.camwa.2010.07.026">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">Beckett, M. G.; Mackenzie, J. A., Convergence analysis of finite differences approximations on equidistributed grids to a singularly perturbed boundary value problem, Appl. Numer. Math., 35, 166, 87-109 (2000) · <a href="/0963.65086" class="nowrap">Zbl 0963.65086</a></td> </tr><tr> <td>[2]</td> <td class="space">Kellogg, R. B.; Tsan, A., Analysis of some difference approximations for a singular perturbation problem without turning point, Math. Comp., 32, 144, 1025-1039 (1978) · <a href="/0418.65040" class="nowrap">Zbl 0418.65040</a></td> </tr><tr> <td>[3]</td> <td class="space">Mohapatra, J.; Natesan, S., Uniform convergence analysis of finite difference scheme for singularly perturbed delay differential equation on an adaptively generated grid, Numer. Math. Theory Methods Appl., 3, 1, 1-22 (2010) · <a href="/1224.65186" class="nowrap">Zbl 1224.65186</a></td> </tr><tr> <td>[4]</td> <td class="space">Qiu, Y.; Sloan, D. M.; Tang, T., Numerical solution of a singularly perturbed two point boundary value problem using equidistribution: analysis of convergence, J. Comput. Appl. Math., 116, 121-143 (2000) · <a href="/0977.65069" class="nowrap">Zbl 0977.65069</a></td> </tr><tr> <td>[5]</td> <td class="space">Roos, H. G.; Stynes, M.; Tobiska, L., Numerical Methods for Singularly Perturbed Differential Equations (1996), Springer: Springer Berlin · <a href="/0844.65075" class="nowrap">Zbl 0844.65075</a></td> </tr><tr> <td>[6]</td> <td class="space">Bawa, R. K.; Clavero, C., Higher order global solution and normalized flux for singularly perturbed reaction-diffusion problems, Appl. Math. Comput., 216, 7, 2058-2068 (2010) · <a href="/1211.65098" class="nowrap">Zbl 1211.65098</a></td> </tr><tr> <td>[7]</td> <td class="space">Clavero, C.; Bawa, R. K.; Natesan, S., A robust second-order numerical method for global solution and global normalized flux for singularly perturbed self-adjoint boundary value problems, Int. J. Comput. Math., 86, 10-11, 1731-1745 (2009) · <a href="/1180.65089" class="nowrap">Zbl 1180.65089</a></td> </tr><tr> <td>[8]</td> <td class="space">Ervin, V.; Layton, W., On the approximation of the derivatives of singularly perturbed boundary value prblems, SIAM J. Sci. Comput., 8, 3, 265-277 (1987) · <a href="/0619.65071" class="nowrap">Zbl 0619.65071</a></td> </tr><tr> <td>[9]</td> <td class="space">Kopteva, N.; Stynes, M., Approximation of derivatives in a convection-diffusion two-point boundary value problem, Appl. Numer. Math., 39, 47-60 (2001) · <a href="/0987.65069" class="nowrap">Zbl 0987.65069</a></td> </tr><tr> <td>[10]</td> <td class="space">Natesan, S.; Bawa, R. K.; Clavero, C., Uniformly convergent compact numerical scheme for the normalized flux of singularly perturbed reaction-diffusion problems, Int. J. Inf. Syst. Sci., 3, 2, 207-221 (2007) · <a href="/1130.65081" class="nowrap">Zbl 1130.65081</a></td> </tr><tr> <td>[11]</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 Press: Chapman & Hall/CRC Press Boca Raton, FL · <a href="/0964.65083" class="nowrap">Zbl 0964.65083</a></td> </tr><tr> <td>[12]</td> <td class="space">Mackenzie, J., Uniform convergence analysis of an upwind finite difference approximation of a convection-diffusion boundary value problem on an adaptive grid, IMA J. Numer. Anal., 19, 233-249 (1999) · <a href="/0929.65047" class="nowrap">Zbl 0929.65047</a></td> </tr><tr> <td>[13]</td> <td class="space">Chen, Y., Uniform convergence analysis of finite difference approximations for singularly perturbation problems on an adapted grid, Adv. Comput. Math., 24, 197-212 (2006) · <a href="/1095.65065" class="nowrap">Zbl 1095.65065</a></td> </tr><tr> <td>[14]</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></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. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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