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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2003
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/math/0302299 |
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| _version_ | 1866911217489018880 |
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| author | Su, Hongling Qin, Mengzhao |
| author_facet | Su, Hongling Qin, Mengzhao |
| contents | The multi-symplectic form for Hamiltonian PDEs leads to a general framework for geometric numerical schemes that preserve a discrete version of the conservation of symplecticity. The cases for systems or PDEs with dissipation terms has never been extended. In this paper, we suggest a new extension for generalizing the multi-symplectic form for Hamiltonian systems to systems with dissipation which never have remarkable energy and momentum conservation properties. The central idea is that the PDEs is of a first-order type that has a symplectic structure depended explicitly on time variable, and decomposed into distinct components representing space and time directions. This suggest a natural definition of multi-symplectic Birkhoff's equation as a multi-symplectic structure from that a multi-symplectic dissipation law is constructed. We show that this definition leads to deeper understanding relationship between functional principle and PDEs. The concept of multi-symplectic integrator is also discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_0302299 |
| institution | arXiv |
| publishDate | 2003 |
| record_format | arxiv |
| spellingShingle | Multi-symplectic Birkhoffian Structure for PDEs with Dissipation Terms Su, Hongling Qin, Mengzhao Numerical Analysis The multi-symplectic form for Hamiltonian PDEs leads to a general framework for geometric numerical schemes that preserve a discrete version of the conservation of symplecticity. The cases for systems or PDEs with dissipation terms has never been extended. In this paper, we suggest a new extension for generalizing the multi-symplectic form for Hamiltonian systems to systems with dissipation which never have remarkable energy and momentum conservation properties. The central idea is that the PDEs is of a first-order type that has a symplectic structure depended explicitly on time variable, and decomposed into distinct components representing space and time directions. This suggest a natural definition of multi-symplectic Birkhoff's equation as a multi-symplectic structure from that a multi-symplectic dissipation law is constructed. We show that this definition leads to deeper understanding relationship between functional principle and PDEs. The concept of multi-symplectic integrator is also discussed. |
| title | Multi-symplectic Birkhoffian Structure for PDEs with Dissipation Terms |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/math/0302299 |