Multisymplecticity of Hybridizable Discontinuous Galerkin Methods

In this paper, we prove necessary and sufficient conditions for a hybridizable discontinuous Galerkin method to satisfy a multisymplectic conservation law, when applied to a canonical Hamiltonian system of partial differential equations. We show that these conditions are satisfied by the “hybridized” versions of several of the most commonly used finite element methods, including mixed, nonconforming, and discontinuous Galerkin methods. (Interestingly, for the continuous Galerkin method in dimension greater than one, we show that multisymplecticity only holds in a weaker sense.) Consequently, these general-purpose finite element methods may be used for structure-preserving discretization (or semidiscretization) of canonical Hamiltonian systems of ODEs or PDEs. This establishes multisymplecticity for a large class of arbitrarily high-order methods on unstructured meshes.