Gröbner Bases and Generation of Difference Schemes for Partial Differential Equations
In this paper we present an algorithmic approach to the generation of fully conservative difference schemes for linear partial differential equations. The approach is based on enlargement of the equations in their integral conservation law form by extra integral relations between unknown functions a...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2006 |
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Інститут математики НАН України
2006
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/146172 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Gröbner Bases and Generation of Difference Schemes for Partial Differential Equations / V.P. Gerdt, Y.A. Blinkov, V.V. Mozzhilkin// Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 50 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862566172656926720 |
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| author | Gerdt, V.P. Blinkov, Y.A. Mozzhilkin, V.V. |
| author_facet | Gerdt, V.P. Blinkov, Y.A. Mozzhilkin, V.V. |
| citation_txt | Gröbner Bases and Generation of Difference Schemes for Partial Differential Equations / V.P. Gerdt, Y.A. Blinkov, V.V. Mozzhilkin// Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 50 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In this paper we present an algorithmic approach to the generation of fully conservative difference schemes for linear partial differential equations. The approach is based on enlargement of the equations in their integral conservation law form by extra integral relations between unknown functions and their derivatives, and on discretization of the obtained system. The structure of the discrete system depends on numerical approximation methods for the integrals occurring in the enlarged system. As a result of the discretization, a system of linear polynomial difference equations is derived for the unknown functions and their partial derivatives. A difference scheme is constructed by elimination of all the partial derivatives. The elimination can be achieved by selecting a proper elimination ranking and by computing a Gröbner basis of the linear difference ideal generated by the polynomials in the discrete system. For these purposes we use the difference form of Janet-like Gröbner bases and their implementation in Maple. As illustration of the described methods and algorithms, we construct a number of difference schemes for Burgers and Falkowich-Karman equations and discuss their numerical properties.
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| first_indexed | 2025-11-26T00:08:35Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-146172 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-26T00:08:35Z |
| publishDate | 2006 |
| publisher | Інститут математики НАН України |
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| spelling | Gerdt, V.P. Blinkov, Y.A. Mozzhilkin, V.V. 2019-02-07T20:10:52Z 2019-02-07T20:10:52Z 2006 Gröbner Bases and Generation of Difference Schemes for Partial Differential Equations / V.P. Gerdt, Y.A. Blinkov, V.V. Mozzhilkin// Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 50 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 68W30; 65M06; 13P10; 39A05; 65Q05 https://nasplib.isofts.kiev.ua/handle/123456789/146172 In this paper we present an algorithmic approach to the generation of fully conservative difference schemes for linear partial differential equations. The approach is based on enlargement of the equations in their integral conservation law form by extra integral relations between unknown functions and their derivatives, and on discretization of the obtained system. The structure of the discrete system depends on numerical approximation methods for the integrals occurring in the enlarged system. As a result of the discretization, a system of linear polynomial difference equations is derived for the unknown functions and their partial derivatives. A difference scheme is constructed by elimination of all the partial derivatives. The elimination can be achieved by selecting a proper elimination ranking and by computing a Gröbner basis of the linear difference ideal generated by the polynomials in the discrete system. For these purposes we use the difference form of Janet-like Gröbner bases and their implementation in Maple. As illustration of the described methods and algorithms, we construct a number of difference schemes for Burgers and Falkowich-Karman equations and discuss their numerical properties. We would like to thank the referees for their important remarks that allowed us to correct the manuscript. We are also grateful to Daniel Robertz and Viktor Levandovskyy for useful discussions and comments. The contribution of two authors (V.P.G. and Yu.A.B.) was partially supported by grants 04-01-00784 and 05-02-17645 from the Russian Foundation for Basic Research and by grant 2339.2003.2 from the Ministry of Education and Science of the Russian Federation. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Gröbner Bases and Generation of Difference Schemes for Partial Differential Equations Article published earlier |
| spellingShingle | Gröbner Bases and Generation of Difference Schemes for Partial Differential Equations Gerdt, V.P. Blinkov, Y.A. Mozzhilkin, V.V. |
| title | Gröbner Bases and Generation of Difference Schemes for Partial Differential Equations |
| title_full | Gröbner Bases and Generation of Difference Schemes for Partial Differential Equations |
| title_fullStr | Gröbner Bases and Generation of Difference Schemes for Partial Differential Equations |
| title_full_unstemmed | Gröbner Bases and Generation of Difference Schemes for Partial Differential Equations |
| title_short | Gröbner Bases and Generation of Difference Schemes for Partial Differential Equations |
| title_sort | gröbner bases and generation of difference schemes for partial differential equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146172 |
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