Multi-symplectic Birkhoffian Structure for PDEs with Dissipation Terms
Hongling Su, Mengzhao Qin
TL;DR
This work extends the multi-symplectic framework to dissipative PDEs by introducing a time-dependent, covariant multi-symplectic Birkhoffian formulation and a first-order representation $M(z,x,t)z_t+K(z,x,t)z_x=\nabla_z B(z,x,t)+\partial_t F+\partial_x G$, linking dissipation to a variational perspective. It develops the theory of self-adjointness for the covariant Birkhoffian system and derives continuous and discrete dissipation laws for symplecticity and related quantities, enabling structure-preserving numerics via multi-symplectic integrators. The paper provides a concrete damped-string example to illustrate the construction and demonstrates a discrete scheme that preserves the dissipation law, highlighting potential for stable, physically consistent simulations of nonconservative PDEs. Collectively, this framework offers a principled approach to modeling, analysis, and numerics of dissipative Hamiltonian PDEs with explicit time dependence in their geometric structure.
Abstract
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.