Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity

The symplectic numerical integration of finite-dimensional Hamiltonian syst ems is a well established subject and has led to a deeper understanding of existing methods as well as to the development of new very efficient and ac curate schemes, e.g., for rigid body, constrained. and molecular dynamics. The numerical integration of infinite-dimensional Hamiltonian systems or Ha miltonian PDEs is much less explored. In this Letter, we suggest a new theo retical framework for generalizing symplectic numerical integrators for ODE s to Hamiltonian PDEs in R-2: time plus one space dimension. The central id ea is that symplecticity for Hamiltonian PDEs is directional: the symplecti c structure of the PDE is decomposed into distinct components representing space and time independently. In this setting PDE integrators can be constr ucted by concatenating uni-directional ODE symplectic integrators. This sug gests a natural definition of multi-symplectic integrator as a discretizati on that conserves a discrete version of the conservation of symplecticity f or Hamiltonian PDEs. We show that this approach leads to a general framewor k for geometric numerical schemes for Hamiltonian PDEs. which have remarkab le energy and momentum conservation properties. Generalizations, including development of higher-order methods, application to the Euler equations in fluid mechanics. application to perturbed systems, and extension to more th an one space dimension are also discussed. (C) 2001 Elsevier Science B.V. A ll rights reserved.