Document Zbl 1094.65085

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B.</a></div> <h2 class="title"> <strong>Weighted average finite difference methods for fractional diffusion equations.</strong> <i>(English)</i> <a class="label nowrap" href="/1094.65085">Zbl 1094.65085</a> </h2> <div class="source"> <a href="/serials/127" title="Journal Profile">J. Comput. Phys.</a> <a href="/?q=in%3A126217" title="Articles in this Issue">216, No. 1, 264-274 (2006)</a>. </div> <div class="abstract">Summary: A class of finite difference methods for solving fractional diffusion equations is considered. These methods are an extension of the weighted average methods for ordinary (non-fractional) diffusion equations. Their accuracy is of order $(\Delta x)^{2}$ and $\Delta t$, except for the fractional version of the Crank-Nicolson method, where the accuracy with respect to the timestep is of order $(\Delta t)^{2}$ if a second-order approximation to the fractional time-derivative is used. Their stability is analyzed by means of a recently proposed procedure akin to the standard von Neumann stability analysis [<span class="zbmathjax-textit">S. B. Yuste</span> and <span class="zbmathjax-textit">L. Acedo</span>, SIAM J. Numer. Anal. 42, No. 5, 1862–1874 (2005; <a href="/1119.65379">Zbl 1119.65379</a>)]. A simple and accurate stability criterion valid for different discretization schemes of the fractional derivative, arbitrary weight factor, and arbitrary order of the fractional derivative, is found and checked numerically. Some examples are provided in which the new methods’ numerical solutions are obtained and compared against exact solutions.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5037689">Cited in <strong>207</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65M06" title="MSC2020">65M06</a> </td> <td class="space"> Finite difference methods for initial value and initial-boundary value problems involving PDEs </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Afractional+diffusion+equation">fractional diffusion equation</a>; <a href="/?q=ut%3Avon+Neumann+stability+analysis">von Neumann stability analysis</a>; <a href="/?q=ut%3Afinite+difference+methods">finite difference methods</a>; <a href="/?q=ut%3Aanomalous+diffusion">anomalous diffusion</a>; <a href="/?q=ut%3ACrank-Nicolson+method">Crank-Nicolson method</a>; <a href="/?q=ut%3Anumerical+examples">numerical examples</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1119.65379">Zbl 1119.65379</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 1094.65085" data-ciurl="/ci/05037689" data-biburl="/bibtex/05037689.bib" data-amsurl="/amsrefs/05037689.bib" data-xmlurl="/xml/05037689.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05037689.pdf" title="Zbl 1094.65085 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.jcp.2005.12.006" aria-label="DOI for “Weighted average finite difference methods for fractional diffusion equations”" title="10.1016/j.jcp.2005.12.006">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/cs/0408053"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">(Hilfer, R., Applications of Fractional Calculus in Physics (2000), World Scientific: World Scientific Singapore) · <a href="/0998.26002" class="nowrap">Zbl 0998.26002</a></td> </tr><tr> <td>[2]</td> <td class="space">Sokolov, I. 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