# Multisymplecticity of hybridizable discontinuous Galerkin methods

## TL;DR

This paper establishes necessary and sufficient conditions for hybridizable discontinuous Galerkin methods to satisfy multisymplectic conservation laws, enabling structure-preserving discretizations of Hamiltonian PDEs with high-order accuracy.

## Contribution

It provides a comprehensive characterization of multisymplecticity conditions for HDG methods, including common finite element variants, extending structure-preserving discretizations to high-order methods.

## Key findings

- HDG methods satisfy multisymplectic conservation laws under specific conditions
- Common finite element methods like mixed and nonconforming are multisymplectic when hybridized
- Multisymplecticity holds for high-order methods on unstructured meshes

## Abstract

In this paper, we prove necessary and sufficient conditions for a hybridizable discontinuous Galerkin (HDG) 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.