Document Zbl 1297.65095

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Comput. Phys.</a> <a href="/?q=in%3A335951" title="Articles in this Issue">242, 715-725 (2013)</a>. </div> <div class="abstract">Summary: The implicit finite difference scheme with the shifted Grünwald formula, which is unconditionally stable, is employed to discretize fractional diffusion equations. The resulting systems are Toeplitz-like and then the fast Fourier transform can be used to reduce the computational cost of the matrix-vector multiplication. The preconditioned conjugate gradient normal residual method with a circulant preconditioner is proposed to solve the discretized linear systems. The spectrum of the preconditioned matrix is proven to be clustered around 1 if diffusion coefficients are constant; hence the convergence rate of the proposed iterative algorithm is superlinear. Numerical experiments are carried out to demonstrate that our circulant preconditioner works very well, even though for cases of variable diffusion coefficients.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6334682">Cited in <strong>196</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%3A35K05" title="MSC2020">35K05</a> </td> <td class="space"> Heat equation </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><tr> <td> <a class="mono" href="/classification/?q=cc%3A65F08" title="MSC2020">65F08</a> </td> <td class="space"> Preconditioners for iterative methods </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65T50" title="MSC2020">65T50</a> </td> <td class="space"> Numerical methods for discrete and fast Fourier transforms </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Afractional+diffusion+equations">fractional diffusion equations</a>; <a href="/?q=ut%3Ashifted+Gr%C3%BCnwald+discretization">shifted Grünwald discretization</a>; <a href="/?q=ut%3AToeplitz">Toeplitz</a>; <a href="/?q=ut%3Acirculant+preconditioner">circulant preconditioner</a>; <a href="/?q=ut%3Afast+Fourier+transform">fast Fourier transform</a>; <a href="/?q=ut%3Astability">stability</a>; <a href="/?q=ut%3Afinite+difference+scheme">finite difference scheme</a>; <a href="/?q=ut%3Apreconditioned+conjugate+gradient+normal+residual+method">preconditioned conjugate gradient normal residual method</a>; <a href="/?q=ut%3Anumerical+experiment">numerical experiment</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 1297.65095" data-ciurl="/ci/06334682" data-biburl="/bibtex/06334682.bib" data-amsurl="/amsrefs/06334682.bib" data-xmlurl="/xml/06334682.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06334682.pdf" title="Zbl 1297.65095 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.2013.02.025" aria-label="DOI for “A circulant preconditioner for fractional diffusion equations”" title="10.1016/j.jcp.2013.02.025">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">Bai, J.; Feng, X., Fractional-order anisotropic diffusion for image denoising, IEEE Trans. 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