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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| Дата: | 2006 |
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| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2006
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/146172 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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irk-123456789-1461722019-02-08T01:23:02Z Gröbner Bases and Generation of Difference Schemes for Partial Differential Equations Gerdt, V.P. Blinkov, Y.A. Mozzhilkin, V.V. 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. 2006 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/146172 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| 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. |
| format |
Article |
| author |
Gerdt, V.P. Blinkov, Y.A. Mozzhilkin, V.V. |
| spellingShingle |
Gerdt, V.P. Blinkov, Y.A. Mozzhilkin, V.V. Gröbner Bases and Generation of Difference Schemes for Partial Differential Equations Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Gerdt, V.P. Blinkov, Y.A. Mozzhilkin, V.V. |
| author_sort |
Gerdt, V.P. |
| title |
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_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_sort |
gröbner bases and generation of difference schemes for partial differential equations |
| publisher |
Інститут математики НАН України |
| publishDate |
2006 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/146172 |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
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2025-07-10T23:20:48Z |
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2025-07-10T23:20:48Z |
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