A staggered semi-implicit discontinuous Galerkin method for the two dimensional incompressible Navier–Stokes equations
In this paper we propose a new spatially high order accurate semi-implicit discontinuous Galerkin (DG) method for the solution of the two dimensional incompressible Navier–Stokes equations on staggered unstructured curved meshes. While the discrete pressure is defined on the primal grid, the discret...
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Veröffentlicht in: | Applied mathematics and computation 2014-12, Vol.248, p.70-92 |
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description | In this paper we propose a new spatially high order accurate semi-implicit discontinuous Galerkin (DG) method for the solution of the two dimensional incompressible Navier–Stokes equations on staggered unstructured curved meshes. While the discrete pressure is defined on the primal grid, the discrete velocity vector field is defined on an edge-based dual grid. The flexibility of high order DG methods on curved unstructured meshes allows to discretize even complex physical domains on rather coarse grids.
Formal substitution of the discrete momentum equation into the discrete continuity equation yields one sparse linear equation system with four non-zero blocks per element for only one scalar unknown, namely the pressure. The method is computationally efficient, since the resulting system is not only very sparse but also symmetric and positive definite for appropriate boundary conditions. Furthermore, all the volume and surface integrals needed by the scheme presented in this paper depend only on the geometry and the polynomial degree of the basis and test functions and can therefore be precomputed and stored in a preprocessor stage, which leads to savings in terms of computational effort for the time evolution part. In this way also the extension to a fully curved isoparametric approach becomes natural and affects only the preprocessing step. The method is validated for polynomial degrees up to p=3 by solving some typical numerical test problems and comparing the numerical results with available analytical solutions or other numerical and experimental reference data. |
doi_str_mv | 10.1016/j.amc.2014.09.089 |
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Formal substitution of the discrete momentum equation into the discrete continuity equation yields one sparse linear equation system with four non-zero blocks per element for only one scalar unknown, namely the pressure. The method is computationally efficient, since the resulting system is not only very sparse but also symmetric and positive definite for appropriate boundary conditions. Furthermore, all the volume and surface integrals needed by the scheme presented in this paper depend only on the geometry and the polynomial degree of the basis and test functions and can therefore be precomputed and stored in a preprocessor stage, which leads to savings in terms of computational effort for the time evolution part. In this way also the extension to a fully curved isoparametric approach becomes natural and affects only the preprocessing step. The method is validated for polynomial degrees up to p=3 by solving some typical numerical test problems and comparing the numerical results with available analytical solutions or other numerical and experimental reference data.</description><identifier>ISSN: 0096-3003</identifier><identifier>EISSN: 1873-5649</identifier><identifier>DOI: 10.1016/j.amc.2014.09.089</identifier><language>eng</language><publisher>Elsevier Inc</publisher><subject>Computation ; Curved ; Curved isoparametric elements ; Galerkin methods ; High order staggered finite element schemes ; Incompressible Navier–Stokes equations ; Mathematical analysis ; Mathematical models ; Navier-Stokes equations ; Polynomials ; Semi-implicit discontinuous Galerkin schemes ; Staggered unstructured triangular meshes ; Two dimensional</subject><ispartof>Applied mathematics and computation, 2014-12, Vol.248, p.70-92</ispartof><rights>2014 Elsevier Inc.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c373t-6ea4004fe9e4f54af17af0b8436602bde8378975cb78d6d3fbabefb683c0ad8b3</citedby><cites>FETCH-LOGICAL-c373t-6ea4004fe9e4f54af17af0b8436602bde8378975cb78d6d3fbabefb683c0ad8b3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://dx.doi.org/10.1016/j.amc.2014.09.089$$EHTML$$P50$$Gelsevier$$H</linktohtml><link.rule.ids>314,780,784,3550,27924,27925,45995</link.rule.ids></links><search><creatorcontrib>Tavelli, Maurizio</creatorcontrib><creatorcontrib>Dumbser, Michael</creatorcontrib><title>A staggered semi-implicit discontinuous Galerkin method for the two dimensional incompressible Navier–Stokes equations</title><title>Applied mathematics and computation</title><description>In this paper we propose a new spatially high order accurate semi-implicit discontinuous Galerkin (DG) method for the solution of the two dimensional incompressible Navier–Stokes equations on staggered unstructured curved meshes. While the discrete pressure is defined on the primal grid, the discrete velocity vector field is defined on an edge-based dual grid. The flexibility of high order DG methods on curved unstructured meshes allows to discretize even complex physical domains on rather coarse grids.
Formal substitution of the discrete momentum equation into the discrete continuity equation yields one sparse linear equation system with four non-zero blocks per element for only one scalar unknown, namely the pressure. The method is computationally efficient, since the resulting system is not only very sparse but also symmetric and positive definite for appropriate boundary conditions. Furthermore, all the volume and surface integrals needed by the scheme presented in this paper depend only on the geometry and the polynomial degree of the basis and test functions and can therefore be precomputed and stored in a preprocessor stage, which leads to savings in terms of computational effort for the time evolution part. In this way also the extension to a fully curved isoparametric approach becomes natural and affects only the preprocessing step. The method is validated for polynomial degrees up to p=3 by solving some typical numerical test problems and comparing the numerical results with available analytical solutions or other numerical and experimental reference data.</description><subject>Computation</subject><subject>Curved</subject><subject>Curved isoparametric elements</subject><subject>Galerkin methods</subject><subject>High order staggered finite element schemes</subject><subject>Incompressible Navier–Stokes equations</subject><subject>Mathematical analysis</subject><subject>Mathematical models</subject><subject>Navier-Stokes equations</subject><subject>Polynomials</subject><subject>Semi-implicit discontinuous Galerkin schemes</subject><subject>Staggered unstructured triangular meshes</subject><subject>Two dimensional</subject><issn>0096-3003</issn><issn>1873-5649</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2014</creationdate><recordtype>article</recordtype><recordid>eNp9kLFu2zAURYkiBeqk-YBsHLNIfTRlikImw0icAkE7pJ0JinxMaEuiTdJpsvUf-of9ktJw5k5vOfcB5xByxaBmwMSXTa1HU8-BNTV0NcjuA5kx2fJqIZrujMwAOlFxAP6JnKe0AYBWsGZGXpc0Zf30hBEtTTj6yo-7wRufqfXJhCn76RAOia71gHHrJzpifg6WuhBpfkaaf4VCjjglHyY9UD-ZMO4ipuT7Aek3_eIx_v395zGHLSaK-4POhUyfyUenh4SX7_eC_Ly7_bG6rx6-r7-ulg-V4S3PlUDdADQOO2zcotGOtdpBLxsuBMx7i5K3smsXpm-lFZa7XvfoeiG5AW1lzy_I9envLob9AVNWY_HCYdATFi_FxIJx2bF2XlB2Qk0MKUV0ahf9qOObYqCOldVGlcrqWFlBp0rlsrk5bbA4HFVVMh4ng9ZHNFnZ4P-z_gf8son_</recordid><startdate>20141201</startdate><enddate>20141201</enddate><creator>Tavelli, Maurizio</creator><creator>Dumbser, Michael</creator><general>Elsevier Inc</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>20141201</creationdate><title>A staggered semi-implicit discontinuous Galerkin method for the two dimensional incompressible Navier–Stokes equations</title><author>Tavelli, Maurizio ; Dumbser, Michael</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c373t-6ea4004fe9e4f54af17af0b8436602bde8378975cb78d6d3fbabefb683c0ad8b3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2014</creationdate><topic>Computation</topic><topic>Curved</topic><topic>Curved isoparametric elements</topic><topic>Galerkin methods</topic><topic>High order staggered finite element schemes</topic><topic>Incompressible Navier–Stokes equations</topic><topic>Mathematical analysis</topic><topic>Mathematical models</topic><topic>Navier-Stokes equations</topic><topic>Polynomials</topic><topic>Semi-implicit discontinuous Galerkin schemes</topic><topic>Staggered unstructured triangular meshes</topic><topic>Two dimensional</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Tavelli, Maurizio</creatorcontrib><creatorcontrib>Dumbser, Michael</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Applied mathematics and computation</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Tavelli, Maurizio</au><au>Dumbser, Michael</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A staggered semi-implicit discontinuous Galerkin method for the two dimensional incompressible Navier–Stokes equations</atitle><jtitle>Applied mathematics and computation</jtitle><date>2014-12-01</date><risdate>2014</risdate><volume>248</volume><spage>70</spage><epage>92</epage><pages>70-92</pages><issn>0096-3003</issn><eissn>1873-5649</eissn><abstract>In this paper we propose a new spatially high order accurate semi-implicit discontinuous Galerkin (DG) method for the solution of the two dimensional incompressible Navier–Stokes equations on staggered unstructured curved meshes. While the discrete pressure is defined on the primal grid, the discrete velocity vector field is defined on an edge-based dual grid. The flexibility of high order DG methods on curved unstructured meshes allows to discretize even complex physical domains on rather coarse grids.
Formal substitution of the discrete momentum equation into the discrete continuity equation yields one sparse linear equation system with four non-zero blocks per element for only one scalar unknown, namely the pressure. The method is computationally efficient, since the resulting system is not only very sparse but also symmetric and positive definite for appropriate boundary conditions. Furthermore, all the volume and surface integrals needed by the scheme presented in this paper depend only on the geometry and the polynomial degree of the basis and test functions and can therefore be precomputed and stored in a preprocessor stage, which leads to savings in terms of computational effort for the time evolution part. In this way also the extension to a fully curved isoparametric approach becomes natural and affects only the preprocessing step. The method is validated for polynomial degrees up to p=3 by solving some typical numerical test problems and comparing the numerical results with available analytical solutions or other numerical and experimental reference data.</abstract><pub>Elsevier Inc</pub><doi>10.1016/j.amc.2014.09.089</doi><tpages>23</tpages><oa>free_for_read</oa></addata></record> |
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subjects | Computation Curved Curved isoparametric elements Galerkin methods High order staggered finite element schemes Incompressible Navier–Stokes equations Mathematical analysis Mathematical models Navier-Stokes equations Polynomials Semi-implicit discontinuous Galerkin schemes Staggered unstructured triangular meshes Two dimensional |
title | A staggered semi-implicit discontinuous Galerkin method for the two dimensional incompressible Navier–Stokes equations |
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