An approximation scheme for a Hamilton–Jacobi equation defined on a network
In this paper we study approximation of Hamilton–Jacobi equations defined on a network. We introduce an appropriate notion of viscosity solution on networks which satisfies existence, uniqueness and stability properties. Then we define an approximation scheme of semi-Lagrangian type by discretizing...
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| Veröffentlicht in: | Applied numerical mathematics 2013-11, Vol.73, p.33-47 |
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| container_title | Applied numerical mathematics |
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| creator | Camilli, Fabio Festa, Adriano Schieborn, Dirk |
| description | In this paper we study approximation of Hamilton–Jacobi equations defined on a network. We introduce an appropriate notion of viscosity solution on networks which satisfies existence, uniqueness and stability properties. Then we define an approximation scheme of semi-Lagrangian type by discretizing in time the representation formula for the solution of Hamilton–Jacobi equations and we prove that the discrete problem admits a unique solution. Moreover we prove that the solution of the approximation scheme converges to the solution of the continuous problem uniformly on the network.
In the second part of the paper we study a fully discrete scheme obtained via a finite elements discretization of the semi-discrete problem. Also for fully discrete scheme we prove the well posedness and the convergence to the viscosity solution of the Hamilton–Jacobi equation. We also discuss some issues concerning the implementation of the algorithm and we present some numerical examples. |
| doi_str_mv | 10.1016/j.apnum.2013.05.003 |
| format | Article |
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In the second part of the paper we study a fully discrete scheme obtained via a finite elements discretization of the semi-discrete problem. Also for fully discrete scheme we prove the well posedness and the convergence to the viscosity solution of the Hamilton–Jacobi equation. We also discuss some issues concerning the implementation of the algorithm and we present some numerical examples.</description><identifier>ISSN: 0168-9274</identifier><identifier>EISSN: 1873-5460</identifier><identifier>DOI: 10.1016/j.apnum.2013.05.003</identifier><language>eng</language><publisher>Elsevier B.V</publisher><subject>Analysis of PDEs ; Approximation ; Convergence ; Discretization ; Hamilton-Jacobi equation ; Mathematical analysis ; Mathematical models ; Mathematics ; Network ; Networks ; Numerical Analysis ; Viscosity ; Viscosity solution</subject><ispartof>Applied numerical mathematics, 2013-11, Vol.73, p.33-47</ispartof><rights>2013 IMACS</rights><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c481t-cf49527d25fcd44e2a1d7dcc099f64a31dd89e74a6394bf4b439d3d9303d654a3</citedby><cites>FETCH-LOGICAL-c481t-cf49527d25fcd44e2a1d7dcc099f64a31dd89e74a6394bf4b439d3d9303d654a3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://www.sciencedirect.com/science/article/pii/S0168927413000792$$EHTML$$P50$$Gelsevier$$H</linktohtml><link.rule.ids>230,314,776,780,881,3537,27901,27902,65306</link.rule.ids><backlink>$$Uhttps://inria.hal.science/hal-00724768$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Camilli, Fabio</creatorcontrib><creatorcontrib>Festa, Adriano</creatorcontrib><creatorcontrib>Schieborn, Dirk</creatorcontrib><title>An approximation scheme for a Hamilton–Jacobi equation defined on a network</title><title>Applied numerical mathematics</title><description>In this paper we study approximation of Hamilton–Jacobi equations defined on a network. We introduce an appropriate notion of viscosity solution on networks which satisfies existence, uniqueness and stability properties. Then we define an approximation scheme of semi-Lagrangian type by discretizing in time the representation formula for the solution of Hamilton–Jacobi equations and we prove that the discrete problem admits a unique solution. Moreover we prove that the solution of the approximation scheme converges to the solution of the continuous problem uniformly on the network.
In the second part of the paper we study a fully discrete scheme obtained via a finite elements discretization of the semi-discrete problem. Also for fully discrete scheme we prove the well posedness and the convergence to the viscosity solution of the Hamilton–Jacobi equation. We also discuss some issues concerning the implementation of the algorithm and we present some numerical examples.</description><subject>Analysis of PDEs</subject><subject>Approximation</subject><subject>Convergence</subject><subject>Discretization</subject><subject>Hamilton-Jacobi equation</subject><subject>Mathematical analysis</subject><subject>Mathematical models</subject><subject>Mathematics</subject><subject>Network</subject><subject>Networks</subject><subject>Numerical Analysis</subject><subject>Viscosity</subject><subject>Viscosity solution</subject><issn>0168-9274</issn><issn>1873-5460</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2013</creationdate><recordtype>article</recordtype><recordid>eNp9kLFu2zAQhokiBeq4eYIsGpNBylEkJXHIYBhJncJFl3YmaPIE05FEm5STdOs79A37JKWjImOmOxy-_-7-n5BLCgUFWt3sCr0fjn1RAmUFiAKAfSAz2tQsF7yCMzJLVJPLsuafyHmMOwAQgsOMfFsMmd7vg39xvR6dH7Jotthj1vqQ6Wyle9eNfvj7-89XbfzGZXg4TpzF1g1os9TqbMDx2YfHz-Rjq7uIF__rnPy8v_uxXOXr718elot1bnhDx9y0XIqytqVojeUcS01tbY0BKduKa0atbSTWXFdM8k3LN5xJy6xkwGwlEjAn19Pere7UPqTXwy_ltVOrxVqdZgB1yeuqeaKJvZrYZPJwxDiq3kWDXacH9MeoaFVTIaAEmVA2oSb4GAO2b7spqFPQaqdeg1anoBWIdIcl1e2kwuT4yWFQ0TgcDFoX0IzKeveu_h92iIfs</recordid><startdate>20131101</startdate><enddate>20131101</enddate><creator>Camilli, Fabio</creator><creator>Festa, Adriano</creator><creator>Schieborn, Dirk</creator><general>Elsevier B.V</general><general>Elsevier</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>1XC</scope></search><sort><creationdate>20131101</creationdate><title>An approximation scheme for a Hamilton–Jacobi equation defined on a network</title><author>Camilli, Fabio ; Festa, Adriano ; Schieborn, Dirk</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c481t-cf49527d25fcd44e2a1d7dcc099f64a31dd89e74a6394bf4b439d3d9303d654a3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2013</creationdate><topic>Analysis of PDEs</topic><topic>Approximation</topic><topic>Convergence</topic><topic>Discretization</topic><topic>Hamilton-Jacobi equation</topic><topic>Mathematical analysis</topic><topic>Mathematical models</topic><topic>Mathematics</topic><topic>Network</topic><topic>Networks</topic><topic>Numerical Analysis</topic><topic>Viscosity</topic><topic>Viscosity solution</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Camilli, Fabio</creatorcontrib><creatorcontrib>Festa, Adriano</creatorcontrib><creatorcontrib>Schieborn, Dirk</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><collection>Hyper Article en Ligne (HAL)</collection><jtitle>Applied numerical mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Camilli, Fabio</au><au>Festa, Adriano</au><au>Schieborn, Dirk</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>An approximation scheme for a Hamilton–Jacobi equation defined on a network</atitle><jtitle>Applied numerical mathematics</jtitle><date>2013-11-01</date><risdate>2013</risdate><volume>73</volume><spage>33</spage><epage>47</epage><pages>33-47</pages><issn>0168-9274</issn><eissn>1873-5460</eissn><abstract>In this paper we study approximation of Hamilton–Jacobi equations defined on a network. We introduce an appropriate notion of viscosity solution on networks which satisfies existence, uniqueness and stability properties. Then we define an approximation scheme of semi-Lagrangian type by discretizing in time the representation formula for the solution of Hamilton–Jacobi equations and we prove that the discrete problem admits a unique solution. Moreover we prove that the solution of the approximation scheme converges to the solution of the continuous problem uniformly on the network.
In the second part of the paper we study a fully discrete scheme obtained via a finite elements discretization of the semi-discrete problem. Also for fully discrete scheme we prove the well posedness and the convergence to the viscosity solution of the Hamilton–Jacobi equation. We also discuss some issues concerning the implementation of the algorithm and we present some numerical examples.</abstract><pub>Elsevier B.V</pub><doi>10.1016/j.apnum.2013.05.003</doi><tpages>15</tpages><oa>free_for_read</oa></addata></record> |
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| ispartof | Applied numerical mathematics, 2013-11, Vol.73, p.33-47 |
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| source | Elsevier ScienceDirect Journals |
| subjects | Analysis of PDEs Approximation Convergence Discretization Hamilton-Jacobi equation Mathematical analysis Mathematical models Mathematics Network Networks Numerical Analysis Viscosity Viscosity solution |
| title | An approximation scheme for a Hamilton–Jacobi equation defined on a network |
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