Document Zbl 1513.65401

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Appl. Math.</a> <a href="/?q=in%3A489465" title="Articles in this Issue">41, No. 8, Paper No. 387, 27 p. (2022)</a>. </div> <div class="abstract">Summary: This paper deals with the convergence and stability of the spectral collocation method for a hyperbolic partial differential equation with piecewise continuous arguments. Firstly, the convergence of continuous-time and discrete-time collocation methods is analyzed by means of equivalent schemes with $L^2$-norm rigorously. We obtain the order of convergence of the continuous-time collocation method is $O(h^4)$ and the discrete-time collocation method is $O(h^4+p)$, where $h$ and $p$ are spatial step and temporal step, respectively. Secondly, the stability of two numerical schemes is analyzed by Fourier analysis method. It is proven that the continuous-time collocation method is unconditionally stable. The stability conditions for the discrete-time collocation method are derived under which the analytic solution is asymptotically stable. Finally, some numerical experiments are carried out to demonstrate our theoretical results.</div> <div class="clear"></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%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%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%3A65N35" title="MSC2020">65N35</a> </td> <td class="space"> Spectral, collocation and related methods for 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%3A35B35" title="MSC2020">35B35</a> </td> <td class="space"> Stability in context of PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35B40" title="MSC2020">35B40</a> </td> <td class="space"> Asymptotic behavior of solutions to PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35L10" title="MSC2020">35L10</a> </td> <td class="space"> Second-order hyperbolic equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Ahyperbolic+partial+differential+equation">hyperbolic partial differential equation</a>; <a href="/?q=ut%3Apiecewise+continuous+arguments">piecewise continuous arguments</a>; <a href="/?q=ut%3Aspectral+collocation+method">spectral collocation method</a>; <a href="/?q=ut%3Aconvergence">convergence</a>; <a href="/?q=ut%3Astability">stability</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 1513.65401" data-ciurl="/ci/07622780" data-biburl="/bibtex/07622780.bib" data-amsurl="/amsrefs/07622780.bib" data-xmlurl="/xml/07622780.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07622780.pdf" title="Zbl 1513.65401 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/s40314-022-02106-8" aria-label="DOI for “Convergence and stability of spectral collocation method for hyperbolic partial differential equation with piecewise continuous arguments”" title="10.1007/s40314-022-02106-8">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">Aftabizadeh, AR; 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