Document Zbl 07996403

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<a href="/authors/cheng.juan" title="Author Profile">Cheng, Juan</a><a class="btn btn-default btn-xs matrix" type="button" href="https://donald.zentralblatt-math.org/lethe/recruit/?first_name=Juan&last_name=Cheng&document_id=7996403"> <span class="text-danger">Recruit</span> </a>; <a href="/authors/shu.chi-wang" title="Author Profile">Shu, Chi-Wang</a><a class="btn btn-default btn-xs matrix" type="button" href="https://donald.zentralblatt-math.org/lethe/recruit/?first_name=Chi-Wang&last_name=Shu&document_id=7996403"> <span class="text-danger">Recruit</span> </a></div> <h2 class="title"> <strong>The high-order variable-coefficient explicit-implicit-null method for diffusion and dispersion equations.</strong> <i>(English)</i> <a class="label nowrap" href="/7996403">Zbl 07996403</a> </h2> <div class="source"> <a href="/serials/9600" title="Journal Profile">Commun. Appl. Math. Comput.</a> <a href="/?q=in%3A524797" title="Articles in this Issue">7, No. 1, 115-150 (2025)</a>. </div> <div class="abstract"><div class="pre">Summary: For the high-order diffusion and dispersion equations, the general practice of the explicit-implicit-null (EIN) method is to add and subtract an appropriately large linear highest derivative term with a constant coefficient at one side of the equation, and then apply the standard implicit-explicit method to the equivalent equation. We call this approach the constant-coefficient EIN method in this paper and hereafter denote it by “CC-EIN”. To reduce the error in the CC-EIN method, the variable-coefficient explicit-implicit-null (VC-EIN) method, which is obtained by adding and subtracting a linear highest derivative term with a variable coefficient, is proposed and studied in this paper. Coupled with the local discontinuous Galerkin (LDG) spatial discretization, the VC-EIN method is shown to be unconditionally stable and can achieve high order of accuracy for both one-dimensional and two-dimensional quasi-linear and nonlinear equations. In addition, although the computational cost slightly increases, the VC-EIN method can obtain more accurate results than the CC-EIN method, if the diffusion coefficient or the dispersion coefficient has a few high and narrow bumps and the bumps only account for a small part of the whole computational domain.</div></div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65M60" title="MSC2020">65M60</a> </td> <td class="space"> Finite element, Rayleigh-Ritz and Galerkin 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></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Adiffusion+equation">diffusion equation</a>; <a href="/?q=ut%3Adispersion+equation">dispersion equation</a>; <a href="/?q=ut%3Astability">stability</a>; <a href="/?q=ut%3Aexplicit-implicit-null+%28EIN%29+time+discretization">explicit-implicit-null (EIN) time discretization</a>; <a href="/?q=ut%3Alocal+discontinuous+Galerkin+%28LDG%29+method">local discontinuous Galerkin (LDG) method</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 07996403" data-ciurl="/ci/07996403" data-biburl="/bibtex/07996403.bib" data-amsurl="/amsrefs/07996403.bib" data-xmlurl="/xml/07996403.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07996403.pdf" title="Zbl 07996403 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/10.1007/s42967-023-00359-6" aria-label="DOI for “The high-order variable-coefficient explicit-implicit-null method for diffusion and dispersion equations”" title="10.1007/s42967-023-00359-6">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">Ascher, UM; Ruuth, SJ; Spiteri, RJ, Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Appl. 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