Convergence of the Finite Volume Method for Multidimensional Conservation Laws
We establish the convergence of the finite volume method applied to multidimensional hyperbolic conservation laws and based on monotone numerical flux-functions. Our technique applies with a fairly unrestrictive assumption on the triangulations ("flat elements" are allowed) and to Lipschit...
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| Veröffentlicht in: | SIAM journal on numerical analysis 1995-06, Vol.32 (3), p.687-705 |
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| creator | Cockburn, B. Coquel, F. Lefloch, P. G. |
| description | We establish the convergence of the finite volume method applied to multidimensional hyperbolic conservation laws and based on monotone numerical flux-functions. Our technique applies with a fairly unrestrictive assumption on the triangulations ("flat elements" are allowed) and to Lipschitz continuous flux-functions. We treat the initial and boundary value problem and obtain the strong convergence of the scheme to the unique entropy discontinuous solution in the sense of Kruzkov. The proof of convergence is based on a convergence framework [Coquel and LeFloch, Math. Comp., 57 (1991), pp. 169-210 and J. Numer. Anal., 30 (1993), pp. 675-700]. From a convex decomposition of the scheme, we derive a new estimate for the rate of entropy dissipation and a new formulation of the discrete entropy inequalities. These estimates are shown to be sufficient for the passage to the limit in the discrete equation. Convergence follows from DiPerna's uniqueness result in the class of entropy measure-valued solutions. |
| doi_str_mv | 10.1137/0732032 |
| format | Article |
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G.</creatorcontrib><title>Convergence of the Finite Volume Method for Multidimensional Conservation Laws</title><title>SIAM journal on numerical analysis</title><description>We establish the convergence of the finite volume method applied to multidimensional hyperbolic conservation laws and based on monotone numerical flux-functions. Our technique applies with a fairly unrestrictive assumption on the triangulations ("flat elements" are allowed) and to Lipschitz continuous flux-functions. We treat the initial and boundary value problem and obtain the strong convergence of the scheme to the unique entropy discontinuous solution in the sense of Kruzkov. The proof of convergence is based on a convergence framework [Coquel and LeFloch, Math. Comp., 57 (1991), pp. 169-210 and J. Numer. Anal., 30 (1993), pp. 675-700]. From a convex decomposition of the scheme, we derive a new estimate for the rate of entropy dissipation and a new formulation of the discrete entropy inequalities. 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Scientific computation</subject><subject>Partial differential equations</subject><subject>Partial differential equations, initial value problems and time-dependant initial-boundary value problems</subject><subject>Polyhedrons</subject><subject>Scalars</subject><subject>Sciences and techniques of general use</subject><subject>Triangulation</subject><issn>0036-1429</issn><issn>1095-7170</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>1995</creationdate><recordtype>article</recordtype><sourceid>8G5</sourceid><sourceid>ABUWG</sourceid><sourceid>AFKRA</sourceid><sourceid>AZQEC</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><sourceid>GNUQQ</sourceid><sourceid>GUQSH</sourceid><sourceid>M2O</sourceid><recordid>eNo9kE9LAzEQxYMoWKv4BTwEETyt5n-yRylWhVYv6nVJ04ndst3UJK347U1p8TTMzG_ePB5Cl5TcUcr1PdGcEc6O0ICSWlaaanKMBoRwVVHB6lN0ltKSlN5QPkCvo9BvIX5B7wAHj_MC8Ljt2wz4M3SbFeAp5EWYYx8inm663M7bFfSpDb3tcDlOELc2lxZP7E86RyfedgkuDnWIPsaP76PnavL29DJ6mFSOcZYrZ_1cSTrTHgwzwtDaErAzqQQzyglqam_LEDgY7pXlxBW3hM0INUoJpfkQXe911zF8byDlZhk2sVhKTc2YNpJyUaDbPeRiSCmCb9axXdn421DS7LJqDlkV8uYgZ5OznY-2d236x7mUsta7r1d7bJlyiP9rRqURQvA_y1pv1g</recordid><startdate>19950601</startdate><enddate>19950601</enddate><creator>Cockburn, B.</creator><creator>Coquel, F.</creator><creator>Lefloch, P. 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G.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Convergence of the Finite Volume Method for Multidimensional Conservation Laws</atitle><jtitle>SIAM journal on numerical analysis</jtitle><date>1995-06-01</date><risdate>1995</risdate><volume>32</volume><issue>3</issue><spage>687</spage><epage>705</epage><pages>687-705</pages><issn>0036-1429</issn><eissn>1095-7170</eissn><coden>SJNAEQ</coden><abstract>We establish the convergence of the finite volume method applied to multidimensional hyperbolic conservation laws and based on monotone numerical flux-functions. Our technique applies with a fairly unrestrictive assumption on the triangulations ("flat elements" are allowed) and to Lipschitz continuous flux-functions. We treat the initial and boundary value problem and obtain the strong convergence of the scheme to the unique entropy discontinuous solution in the sense of Kruzkov. The proof of convergence is based on a convergence framework [Coquel and LeFloch, Math. Comp., 57 (1991), pp. 169-210 and J. Numer. Anal., 30 (1993), pp. 675-700]. From a convex decomposition of the scheme, we derive a new estimate for the rate of entropy dissipation and a new formulation of the discrete entropy inequalities. These estimates are shown to be sufficient for the passage to the limit in the discrete equation. Convergence follows from DiPerna's uniqueness result in the class of entropy measure-valued solutions.</abstract><cop>Philadelphia, PA</cop><pub>Society for Industrial and Applied Mathematics</pub><doi>10.1137/0732032</doi><tpages>19</tpages></addata></record> |
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| ispartof | SIAM journal on numerical analysis, 1995-06, Vol.32 (3), p.687-705 |
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| source | SIAM Journals Online; JSTOR Mathematics & Statistics; JSTOR Archive Collection A-Z Listing |
| subjects | A priori knowledge Applied mathematics Approximation Boundary conditions Boundary value problems Conservation laws Entropy Exact sciences and technology Finite volume method Mathematical analysis Mathematics Numerical analysis Numerical analysis. Scientific computation Partial differential equations Partial differential equations, initial value problems and time-dependant initial-boundary value problems Polyhedrons Scalars Sciences and techniques of general use Triangulation |
| title | Convergence of the Finite Volume Method for Multidimensional Conservation Laws |
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