Discontinuous Galerkin Finite Element Methods for Linear Port-Hamiltonian Dynamical Systems

Authors

Xiaoyu Cheng, J. J. W. van der Vegt, Yan Xu, H. J. Zwart

Abstract

In this paper, we present discontinuous Galerkin (DG) finite element discretizations for a class of linear hyperbolic port-Hamiltonian dynamical systems. The key point in constructing a port-Hamiltonian system is a Stokes-Dirac structure. Instead of following the traditional approach of defining the strong form of the Dirac structure, we define a Dirac structure in weak form, specifically in the input-state-output form. This is implemented within broken Sobolev spaces on a tessellation with polyhedral elements. After that, we state the weak port-Hamiltonian formulation and prove that it relates to a Poisson bracket. In our work, a crucial aspect of constructing the above-mentioned Dirac structure is that we provide a conservative relation between the boundary ports. Next, we state DG discretizations of the port-Hamiltonian system by using the weak form of the Dirac structure and broken polynomial spaces of differential forms, and we provide a priori error estimates for the structure-preserving port-Hamiltonian discontinuous Galerkin (PHDG) discretizations. The accuracy and capability of the methods developed in this paper are demonstrated by presenting several numerical experiments.

Keywords

Port-Hamiltonian systems; Dirac structure; Discontinuous Galerkin methods; Exterior calculus

Citation

• Journal: Journal of Scientific Computing
• Year: 2025
• Volume: 104
• Issue: 1
• Pages:
• Publisher: Springer Science and Business Media LLC
• DOI: 10.1007/s10915-025-02926-w

BibTeX

@article{Cheng_2025,
  title={{Discontinuous Galerkin Finite Element Methods for Linear Port-Hamiltonian Dynamical Systems}},
  volume={104},
  ISSN={1573-7691},
  DOI={10.1007/s10915-025-02926-w},
  number={1},
  journal={Journal of Scientific Computing},
  publisher={Springer Science and Business Media LLC},
  author={Cheng, Xiaoyu and van der Vegt, J. J. W. and Xu, Yan and Zwart, H. J.},
  year={2025}
}

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