Document Zbl 1202.65139

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Numer. Anal.</a> <a href="/?q=in%3A270089" title="Articles in this Issue">2010, Article ID 352174, 17 p. (2010)</a>. </div> <div class="abstract">Summary: We derive a family of sixth-order compact finite-difference schemes for the three-dimensional Poisson’s equation. As opposed to other research regarding higher-order compact difference schemes, our approach includes consideration of the discretization of the source function on a compact finite-difference stencil. The schemes derived approximate the solution to Poisson’s equation on a compact stencil, and thus the schemes can be easily implemented and resulting linear systems are solved in a high-performance computing environment. The resulting discretization is a one-parameter family of finite-difference schemes which may be further optimized for accuracy and stability. Computational experiments are implemented which illustrate the theoretically demonstrated truncation errors.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5816753">Cited in <strong>3</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65N06" title="MSC2020">65N06</a> </td> <td class="space"> Finite difference methods for boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35J05" title="MSC2020">35J05</a> </td> <td class="space"> Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Anumerical+examples">numerical examples</a>; <a href="/?q=ut%3Asixth-order+compact+finite-difference+schemes">sixth-order compact finite-difference schemes</a>; <a href="/?q=ut%3Athree-dimensional+Poisson%27s+equation">three-dimensional Poisson’s equation</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 1202.65139" data-ciurl="/ci/05816753" data-biburl="/bibtex/05816753.bib" data-amsurl="/amsrefs/05816753.bib" data-xmlurl="/xml/05816753.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05816753.pdf" title="Zbl 1202.65139 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/10.1155/2010/352174" aria-label="DOI for “A family of sixth-order compact finite-difference schemes for the three-dimensional Poisson equation”" title="10.1155/2010/352174">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://eudml.org/doc/229327" title="Open Access at EuDML">EuDML</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, Wadsworth & Brooks-Cole Advanced Books & Software, Pacific Grove, Calif, USA, 1989. · <a href="/1071.65118" class="nowrap">Zbl 1071.65118</a></td> </tr><tr> <td>[2]</td> <td class="space">L. Ge and J. Zhang, “Symbolic computation of high order compact difference schemes for three dimensional linear elliptic partial differential equations with variable coefficients,” Journal of Computational and Applied Mathematics, vol. 143, no. 1, pp. 9-27, 2002. · <a href="/1001.65112" class="nowrap">Zbl 1001.65112</a> · <a href="https://doi.org/10.1016/S0377-0427(01)00504-0" class="nowrap">doi:10.1016/S0377-0427(01)00504-0</a></td> </tr><tr> <td>[3]</td> <td class="space">M. Piller and E. 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Carlson, “A fourth-order compact difference scheme on face centered cubic grids with multigrid method for solving 2D convection diffusion equation,” Mathematics and Computers in Simulation, vol. 63, no. 6, pp. 651-661, 2003. · <a href="/1036.65082" class="nowrap">Zbl 1036.65082</a> · <a href="https://doi.org/10.1016/S0378-4754(03)00095-8" class="nowrap">doi:10.1016/S0378-4754(03)00095-8</a></td> </tr><tr> <td>[6]</td> <td class="space">S. K. Lele, “Compact finite difference schemes with spectral-like resolution,” Journal of Computational Physics, vol. 103, no. 1, pp. 16-42, 1992. · <a href="/0759.65006" class="nowrap">Zbl 0759.65006</a> · <a href="https://doi.org/10.1016/0021-9991(92)90324-R" class="nowrap">doi:10.1016/0021-9991(92)90324-R</a></td> </tr><tr> <td>[7]</td> <td class="space">K. Ito, Z. Li, and Y. 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O’Leary, “Efficient iterative solution of the three-dimensional Helmholtz equation,” Journal of Computational Physics, vol. 142, no. 1, pp. 163-181, 1998. · <a href="/0929.65089" class="nowrap">Zbl 0929.65089</a> · <a href="https://doi.org/10.1006/jcph.1998.5939" class="nowrap">doi:10.1006/jcph.1998.5939</a></td> </tr><tr> <td>[10]</td> <td class="space">Y. A. Erlangga, C. Vuik, and C. W. Oosterlee, “On a class of preconditioners for solving the Helmholtz equation,” Applied Numerical Mathematics, vol. 50, no. 3-4, pp. 409-425, 2004. · <a href="/1051.65101" class="nowrap">Zbl 1051.65101</a> · <a href="https://doi.org/10.1016/j.apnum.2004.01.009" class="nowrap">doi:10.1016/j.apnum.2004.01.009</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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