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Comput. Phys.</a> <a href="/?q=in%3A253468" title="Articles in this Issue">228, No. 20, 7565-7595 (2009)</a>. </div> <div class="abstract">Summary: The projection method is a widely used fractional-step algorithm for solving the incompressible Navier-Stokes equations. Despite numerous improvements to the methodology, however, imposing physical boundary conditions with projection-based fluid solvers remains difficult, and obtaining high-order accuracy may not be possible for some choices of boundary conditions. In this work, we present an unsplit, linearly-implicit discretization of the incompressible Navier-Stokes equations on a staggered grid along with an efficient solution method for the resulting system of linear equations. Since our scheme is not a fractional-step algorithm, it is straightforward to specify general physical boundary conditions accurately; however, this capability comes at the price of having to solve the time-dependent incompressible Stokes equations at each timestep. To solve this linear system efficiently, we employ a Krylov subspace method preconditioned by the projection method. In our implementation, the subdomain solvers required by the projection preconditioner employ the conjugate gradient method with geometric multigrid preconditioning. The accuracy of the scheme is demonstrated for several problems, including forced and unforced analytic test cases and lid-driven cavity flows. These tests consider a variety of physical boundary conditions with Reynolds numbers ranging from 1 to 30000. The effectiveness of the projection preconditioner is compared to an alternative preconditioning strategy based on an approximation to the Schur complement for the time-dependent incompressible Stokes operator. The projection method is found to be a more efficient preconditioner in most cases considered in the present work.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A5615504">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A5615504">Cited in <strong>55</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A76M20" title="MSC2020">76M20</a> </td> <td class="space"> Finite difference methods applied to problems in fluid mechanics </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%3A76D05" title="MSC2020">76D05</a> </td> <td class="space"> Navier-Stokes equations for incompressible viscous fluids </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aincompressible+flow">incompressible flow</a>; <a href="/?q=ut%3ANavier-Stokes+equations">Navier-Stokes equations</a>; <a href="/?q=ut%3Apreconditioner">preconditioner</a>; <a href="/?q=ut%3Aprojection+method">projection method</a>; <a href="/?q=ut%3Ablock+factorization">block factorization</a>; <a href="/?q=ut%3Aapproximate+Schur+complement">approximate Schur complement</a>; <a href="/?q=ut%3Aphysical+boundary+conditions">physical boundary conditions</a>; <a href="/?q=ut%3Amultigrid">multigrid</a></div> <div class="software"> <h3>Software:</h3><a href="/software/4012">PETSc</a>; <a href="/software/426">hypre</a>; <a href="/software/341">Gerris</a></div> <!-- Modal used to show zbmath metadata in different output formats--> <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;"> <!-- alert for connection errors etc --> </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"> <!-- Button trigger metadata modal --> <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1391.76474" data-ciurl="/ci/05615504" data-biburl="/bibtex/05615504.bib" data-amsurl="/amsrefs/05615504.bib" data-xmlurl="/xml/05615504.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05615504.pdf" title="Zbl 1391.76474 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.2009.07.001" aria-label="DOI for “An accurate and efficient method for the incompressible Navier-Stokes equations using the projection method as a preconditioner”" title="10.1016/j.jcp.2009.07.001">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">Chorin, A. 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