Multi-symplectic Birkhoffian Structure for PDEs with Dissipation Terms

Details Multi-symplectic Birkhoffian Structure for PDEs with Dissipation Terms Multi-symplectic Birkhoffian Structure for PDEs with Dissipation Terms

2003-02-25

arxiv.org/abs/math/0302299v1

Mathematics

Numerical Analysis

7 pages

Scientific paper

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.

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