Document Zbl 1443.65145

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Math. Appl.</a> <a href="/?q=in%3A453938" title="Articles in this Issue">71, No. 5, 1114-1128 (2016)</a>. </div> <div class="abstract">Summary: In this paper, we propose a conservative and effective difference scheme for solving the two dimensional nonlinear space-fractional Schrödinger equation with the Riesz fractional derivative. The scheme is constructed by combining the split-step method for handling the nonlinearity with the alternating direction implicit (ADI) method for resolving the multi-dimensions difficulty. The Riesz space-fractional derivative is approximated by the second order accurate fractional centered difference. Based on matrix analysis, we show that in the discrete sense the scheme conserves the mass and energy for linear problems and conserves the mass for nonlinear problems. The unconditional convergence is proved rigorously in the linear case. Numerical tests are performed to support our theoretical results and show the efficiency of the proposed scheme.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7258795">Cited in <strong>33</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><tr> <td> <a class="mono" href="/classification/?q=cc%3A35R11" title="MSC2020">35R11</a> </td> <td class="space"> Fractional partial differential equations </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></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Afractional+Schr%C3%B6dinger+equation">fractional Schrödinger equation</a>; <a href="/?q=ut%3ARiesz+fractional+derivative">Riesz fractional derivative</a>; <a href="/?q=ut%3Asplit-step">split-step</a>; <a href="/?q=ut%3AADI">ADI</a>; <a href="/?q=ut%3Aconservation">conservation</a>; <a href="/?q=ut%3Aconvergence">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 1443.65145" data-ciurl="/ci/07258795" data-biburl="/bibtex/07258795.bib" data-amsurl="/amsrefs/07258795.bib" data-xmlurl="/xml/07258795.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07258795.pdf" title="Zbl 1443.65145 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.2016.01.022" aria-label="DOI for “Split-step alternating direction implicit difference scheme for the fractional Schrödinger equation in two dimensions”" title="10.1016/j.camwa.2016.01.022">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">Laskin, N., Fractional quantum mechanics, Phys. 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