Document Zbl 1499.65556

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Math., Theory Methods Appl.</a> <a href="/?q=in%3A482841" title="Articles in this Issue">15, No. 2, 279-303 (2022)</a>. </div> <div class="abstract">Summary: In this paper we propose and analyze a backward differentiation formula (BDF) type numerical scheme for the Cahn-Hilliard equation with third order temporal accuracy. The Fourier pseudo-spectral method is used to discretize space. The surface diffusion and the nonlinear chemical potential terms are treated implicitly, while the expansive term is approximated by a third order explicit extrapolation formula for the sake of solvability. In addition, a third order accurate Douglas-Dupont regularization term, in the form of $- A_0\Delta t^2\Delta_N (\phi^{n+1}-\phi^n),$ is added in the numerical scheme. In particular, the energy stability is carefully derived in a modified version, so that a uniform bound for the original energy functional is available, and a theoretical justification of the coefficient $A$ becomes available. As a result of this energy stability analysis, a uniform-in-time $L^6_N$ bound of the numerical solution is obtained. And also, the optimal rate convergence analysis and error estimate are provided, in the $L^\infty_{ \Delta t} (0, T ;L^2_N) \cap L^2_{ \Delta t} (0,T; H^2_h)$ norm, with the help of the $L^6_N$ bound for the numerical solution. A few numerical simulation results are presented to demonstrate the efficiency of the numerical scheme and the third order convergence.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7538551">Cited in <strong>18</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65M70" title="MSC2020">65M70</a> </td> <td class="space"> Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65M12" title="MSC2020">65M12</a> </td> <td class="space"> Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65M15" title="MSC2020">65M15</a> </td> <td class="space"> Error bounds for initial value and initial-boundary value problems involving PDEs </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3ACahn-Hilliard+equation">Cahn-Hilliard equation</a>; <a href="/?q=ut%3Athird+order+backward+differentiation+formula">third order backward differentiation formula</a>; <a href="/?q=ut%3Aunique+solvability">unique solvability</a>; <a href="/?q=ut%3Aenergy+stability">energy stability</a>; <a href="/?q=ut%3Adiscrete+%5C%28L%5E6_N%5C%29+estimate">discrete $L^6_N$ estimate</a>; <a href="/?q=ut%3Aoptimal+rate+convergence+analysis">optimal rate convergence analysis</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 1499.65556" data-ciurl="/ci/07538551" data-biburl="/bibtex/07538551.bib" data-amsurl="/amsrefs/07538551.bib" data-xmlurl="/xml/07538551.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07538551.pdf" title="Zbl 1499.65556 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/10.4208/nmtma.OA-2021-0165" aria-label="DOI for “A third order accurate in time, BDF-type energy stable scheme for the Cahn-Hilliard equation”" title="10.4208/nmtma.OA-2021-0165">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">S. 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