Geometric integrators for hamiltonian pdes

Abstract

We consider methods for systematic construction of algorithms for a class of time-dependent PDEs with Hamiltonian structure. These systems possess phase space geometry and constants of the motion that need to be preserved by the integration algorithm to reflect the qualitative features of the system. We exploit the structure of Hamiltonian systems, in particular their variational formulation based on a Lagrangian, and the dual covariant formulation, to expose the geometric features of the system that have natural analogs when discretized. We emphasize the local space-time approach to the constructions, making them amenable to parallelization and preconditioning using domain decomposition methods, and enabling treatment of complex spatial geometries and boundary conditions. These methods are applied to the two representative problems: the Nonlinear Schrodinger equation (NLS) and the Heisenberg magnet model (HM), both of which possess highly nontrivial geometric structures. We treat the “1+1” case of one spatial and one temporal dimension with periodic boundary conditions, but both systems have higher-dimensional “m+1” generalizations and the analyses extend accordingly. In addition, NLS has an integrable semidiscretization due to Ablowitz and Ladik (AL), which, as an ODE, possesses a highly nonlinear symplectic structure, For this system we consider a generating function approach to the nonstandard symplectic form, and derive novel symplectic integrators of arbitrary order exploiting the particular structure of AL where the standard techniques fail. To facilitate the construction of scalable, parallelizeable implementations, we have developed an extension class library for PETSc [8], the Dynamical Systems Toolkit (DST), that implements parallel mesh manipulation and discrete grid function and operator routines, which serve as the basic building blocks for efficient geometric integrators. While a piece of research code, it also facilitates rapid development and testing of algorithms obtained using the methods developed above and can serve as a foundation for further software development in this direction, In addition, the question of the best object framework for expression of natural PDE operations deserves investigation in its own right and we include its discussion.