Document Zbl 0797.65067

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A.</a></div> <h2 class="title"> <strong>Method of lines and direct discretization: A comparison for linear advection.</strong> <i>(English)</i> <a class="label nowrap" href="/0797.65067">Zbl 0797.65067</a> </h2> <div class="source"> <a href="/serials/1033" title="Journal Profile">Appl. Numer. Math.</a> <a href="/?q=in%3A18782" title="Articles in this Issue">13, No. 6, 469-490 (1994)</a>. </div> <div class="abstract">Two related schemes are compared for two-dimensional linear advection: a scheme based on dimension splitting starting from fully discrete one- dimensional discretization, and a scheme based on the method of lines. Both schemes use the same four-point upwind biased stencil in each direction and both are considered with flux limiting to avoid oscillations and negative values of the solution.<br class="zbmathjax-paragraph">Theoretical aspects of the application of the methods for 2-D models involving error analysis and computational cost are discussed. Numerical tests are run on uniform grids with local uniform grid refinement to compare their accuracy and computational efficiency.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A5348">R.Anguelov (Welkom)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A550352">Cited in <strong>8</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65M20" title="MSC2020">65M20</a> </td> <td class="space"> Method of lines 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%3A65M15" title="MSC2020">65M15</a> </td> <td class="space"> Error bounds for initial value and initial-boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65M50" title="MSC2020">65M50</a> </td> <td class="space"> Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35L15" title="MSC2020">35L15</a> </td> <td class="space"> Initial value problems for second-order hyperbolic equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65Y20" title="MSC2020">65Y20</a> </td> <td class="space"> Complexity and performance of numerical algorithms </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Anumerical+tests">numerical tests</a>; <a href="/?q=ut%3Alinear+advection">linear advection</a>; <a href="/?q=ut%3Adimension+splitting">dimension splitting</a>; <a href="/?q=ut%3Amethod+of+lines">method of lines</a>; <a href="/?q=ut%3Afour-point+upwind+biased+stencil">four-point upwind biased stencil</a>; <a href="/?q=ut%3Aerror+analysis">error analysis</a>; <a href="/?q=ut%3Agrid+refinement">grid refinement</a>; <a href="/?q=ut%3Acomputational+efficiency">computational efficiency</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 0797.65067" data-ciurl="/ci/00550352" data-biburl="/bibtex/00550352.bib" data-amsurl="/amsrefs/00550352.bib" data-xmlurl="/xml/00550352.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/00550352.pdf" title="Zbl 0797.65067 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/10.1016/0168-9274(94)90009-4" aria-label="DOI for “Method of lines and direct discretization: A comparison for linear advection”" title="10.1016/0168-9274(94)90009-4">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://ir.cwi.nl/pub/1464" title="Full Text Link">Link</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Bott, A., Monotone flux limitation in the area-preserving flux-form advection algorithm, Monthly Weather Rev., 120, 2595-2602 (1992)</td> </tr><tr> <td>[2]</td> <td class="space">Colella, P.; Woodward, P., The piecewise parabolic method (PPM) for gas-dynamical simulations, J. Comput. Phys., 54, 174-201 (1984) · <a href="/0531.76082" class="nowrap">Zbl 0531.76082</a></td> </tr><tr> <td>[3]</td> <td class="space">Hov, Ø.; Zlatev, Z.; Berkowicz, R.; Eliassen, A.; Prahm, L. P., Comparison of numerical techniques for use in air pollution models with non-linear chemical reactions, Atmos. Envir., 23, 967-983 (1989)</td> </tr><tr> <td>[4]</td> <td class="space">Hundsdorfer, W.; Koren, B.; Loon, M.van; Verwer, J. G., A positive finite-difference advection scheme applied on locally refined grids, (Report NM-R9309 (1993), CWI: CWI Amsterdam) · <a href="/0860.65073" class="nowrap">Zbl 0860.65073</a></td> </tr><tr> <td>[5]</td> <td class="space">Koren, B.; Vreugdenhil, C. B.; Koren, B., A robust upwind discretization for advection, diffusion and source terms, Numerical Methods of Advection-Diffusion Problems, 45 (1993), Vieweg: Vieweg Braunschweig, Notes on Numerical Fluid Mechanics · <a href="/0788.00033" class="nowrap">Zbl 0788.00033</a></td> </tr><tr> <td>[6]</td> <td class="space">Leonard, B. P.; Minkowycz, W. J.; Sparrow, E. M.; Schneider, G. E.; Pletcher, R. H., Elliptic systems: finite difference method IV, Handbook of Numerical Heat Transfer (1988), Wiley: Wiley New York</td> </tr><tr> <td>[7]</td> <td class="space">Leonard, B. P., Simple high accuracy resolution program for convective modeling of discontinuities, Internat. J. Numer. Methods Fluids, 8, 1291-1318 (1988) · <a href="/0667.76125" class="nowrap">Zbl 0667.76125</a></td> </tr><tr> <td>[8]</td> <td class="space">LeVeque, R. J., Time-split methods for partial differential equations, Ph.D. Thesis (1982), Department of Computer Science, Stanford University: Department of Computer Science, Stanford University Stanford, CA</td> </tr><tr> <td>[9]</td> <td class="space">LeVeque, R. J., Numerical Methods for Conservation Laws, Lecture Notes in Mathematics (1992), Birkhäuser: Birkhäuser Basel, ETH Zürich · <a href="/0847.65053" class="nowrap">Zbl 0847.65053</a></td> </tr><tr> <td>[10]</td> <td class="space">McRea, G. J.; Goodin, W. R.; Seinfeld, J. H., Numerical solution of atmospheric diffusion for chemically reacting flows, J. Comput. Phys., 77, 1-42 (1982) · <a href="/0502.76098" class="nowrap">Zbl 0502.76098</a></td> </tr><tr> <td>[11]</td> <td class="space">Roe, P. L., Some contributions to the modeling of discontinuous flows, (Lectures in Applied Mathematics, 22 (1985), AMS: AMS Providence, RI), 163-193 · <a href="/0665.76072" class="nowrap">Zbl 0665.76072</a></td> </tr><tr> <td>[12]</td> <td class="space">Shu, C. W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77, 439-471 (1988) · <a href="/0653.65072" class="nowrap">Zbl 0653.65072</a></td> </tr><tr> <td>[13]</td> <td class="space">Strang, G., Trigonometric polynomials and difference methods of maximal accuracy, J. Math. Phys., 41, 147-154 (1962) · <a href="/0111.31601" class="nowrap">Zbl 0111.31601</a></td> </tr><tr> <td>[14]</td> <td class="space">Sweby, P. K., High resolution schemes using flux-limiters for hyperbolic conservation laws, SIAM J. Numer. Anal., 21, 995-1011 (1984) · <a href="/0565.65048" class="nowrap">Zbl 0565.65048</a></td> </tr><tr> <td>[15]</td> <td class="space">Tamamidis, P.; Assanis, D. N., Evaluation of various high order accuracy schemes with and without flux limiters, Internat. J. Numer. Methods Fluids, 16, 931-948 (1993) · <a href="/0781.76068" class="nowrap">Zbl 0781.76068</a></td> </tr><tr> <td>[16]</td> <td class="space">Trompert, R. A.; Verwer, J. G., A static regridding method for two-dimensional parabolic partial differential equations, Appl. Numer. Math., 8, 65-90 (1991) · <a href="/0745.65057" class="nowrap">Zbl 0745.65057</a></td> </tr><tr> <td>[17]</td> <td class="space">Leer, B.van, Towards the ultimate conservative difference scheme IV: a new approach to numerical convection, J. Comput. Phys., 23, 276-299 (1977) · <a href="/0339.76056" class="nowrap">Zbl 0339.76056</a></td> </tr><tr> <td>[18]</td> <td class="space">Leer, B.van, Upwind-difference methods for aerodynamic problems governed by the Euler equations, (Engquist, B. E.; Osher, S.; Sommerville, R. C.J., Large-scale Computations in Fluid Mechanics (1985), American Mathematical Society: American Mathematical Society Providence, RI), 327-336 · <a href="/0582.76065" class="nowrap">Zbl 0582.76065</a></td> </tr><tr> <td>[19]</td> <td class="space">Vreugdenhil, C. B.; Koren, B., Numerical Method for Advection-Diffusion Problems, Notes on Numerical Fluid Mechanics, 45 (1993), Vieweg: Vieweg Braunschweig · <a href="/0806.76071" class="nowrap">Zbl 0806.76071</a></td> </tr><tr> <td>[20]</td> <td class="space">Zalesak, S. T., A preliminary comparison of modern shock-capturing schemes: linear advection, (Vichnevetsky, R.; Stepleman, R. S., Advances in Computer Methods for Partial Differential Equations IV (1987), IMACS: IMACS New Brunswick, NJ), 15-22</td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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