Structure Preserving Discretizations of the Liouville Equation and their Numerical Tests
The main purpose of this article is to show how symmetry structures in partial differential equations can be preserved in a discrete world and reflected in difference schemes. Three different structure preserving discretizations of the Liouville equation are presented and then used to solve specific...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2015 |
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Інститут математики НАН України
2015
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/147153 |
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| Zitieren: | Structure Preserving Discretizations of the Liouville Equation and their Numerical Tests / D. Levi, L. Martina, P. Winternitz // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 42 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862730926467842048 |
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| author | Levi, D. Martina, L. Winternitz, P. |
| author_facet | Levi, D. Martina, L. Winternitz, P. |
| citation_txt | Structure Preserving Discretizations of the Liouville Equation and their Numerical Tests / D. Levi, L. Martina, P. Winternitz // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 42 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The main purpose of this article is to show how symmetry structures in partial differential equations can be preserved in a discrete world and reflected in difference schemes. Three different structure preserving discretizations of the Liouville equation are presented and then used to solve specific boundary value problems. The results are compared with exact solutions satisfying the same boundary conditions. All three discretizations are on four point lattices. One preserves linearizability of the equation, another the infinite-dimensional symmetry group as higher symmetries, the third one preserves the maximal finite-dimensional subgroup of the symmetry group as point symmetries. A 9-point invariant scheme that gives a better approximation of the equation, but significantly worse numerical results for solutions is presented and discussed.
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| first_indexed | 2025-12-07T19:23:25Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-147153 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T19:23:25Z |
| publishDate | 2015 |
| publisher | Інститут математики НАН України |
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| spelling | Levi, D. Martina, L. Winternitz, P. 2019-02-13T17:45:44Z 2019-02-13T17:45:44Z 2015 Structure Preserving Discretizations of the Liouville Equation and their Numerical Tests / D. Levi, L. Martina, P. Winternitz // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 42 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B80; 22E60; 39A14; 65Mxx DOI:10.3842/SIGMA.2015.080 https://nasplib.isofts.kiev.ua/handle/123456789/147153 The main purpose of this article is to show how symmetry structures in partial differential equations can be preserved in a discrete world and reflected in difference schemes. Three different structure preserving discretizations of the Liouville equation are presented and then used to solve specific boundary value problems. The results are compared with exact solutions satisfying the same boundary conditions. All three discretizations are on four point lattices. One preserves linearizability of the equation, another the infinite-dimensional symmetry group as higher symmetries, the third one preserves the maximal finite-dimensional subgroup of the symmetry group as point symmetries. A 9-point invariant scheme that gives a better approximation of the equation, but significantly worse numerical results for solutions is presented and discussed. This paper is a contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of
 Luc Vinet. The full collection is available at http://www.emis.de/journals/SIGMA/ESSA2014.html.
 DL has been partly supported by the Italian Ministry of Education and Research, 2010 PRIN
 Continuous and discrete nonlinear integrable evolutions: from water waves to symplectic maps.
 LM has been partly supported by the Italian Ministry of Education and Research, 2011 PRIN
 Teorie geometriche e analitiche dei sistemi Hamiltoniani in dimensioni f inite e inf inite. DL
 and LM are supported also by INFN IS-CSN4 Mathematical Methods of Nonlinear Physics.
 The research of PW is partially supported by a research grant from NSERC of Canada.
 PW thanks the European Union Research Executive Agency for the award of a Marie Curie International Incoming Research Fellowship making his stay at University Roma Tre possible.
 He thanks the Department of Mathematics and Physics of Roma Tre for hospitality.
 We thank the referees for many valuable comments which allowed us to greatly improve the
 article. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Structure Preserving Discretizations of the Liouville Equation and their Numerical Tests Article published earlier |
| spellingShingle | Structure Preserving Discretizations of the Liouville Equation and their Numerical Tests Levi, D. Martina, L. Winternitz, P. |
| title | Structure Preserving Discretizations of the Liouville Equation and their Numerical Tests |
| title_full | Structure Preserving Discretizations of the Liouville Equation and their Numerical Tests |
| title_fullStr | Structure Preserving Discretizations of the Liouville Equation and their Numerical Tests |
| title_full_unstemmed | Structure Preserving Discretizations of the Liouville Equation and their Numerical Tests |
| title_short | Structure Preserving Discretizations of the Liouville Equation and their Numerical Tests |
| title_sort | structure preserving discretizations of the liouville equation and their numerical tests |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147153 |
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