Multisymplecticity of hybridizable discontinuous Galerkin methods
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2017
- DOI:
- 10.48550/arXiv.1705.08609
- arXiv:
- arXiv:1705.08609
- Bibcode:
- 2017arXiv170508609M
- Keywords:
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- Mathematics - Numerical Analysis;
- 65N30;
- 37K05
- E-Print:
- 34 pages