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

• Published in: Journal of scientific computing Vol. 104; no. 1
• Main Authors: Cheng, Xiaoyu, van der Vegt, J. J. W., Xu, Yan, Zwart, H. J.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.07.2025
• Subjects: Algorithms; Computational Mathematics and Numerical Analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Theoretical; Exterior calculus; Discontinuous Galerkin methods; Dirac structure; Port-Hamiltonian systems
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-025-02926-w

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.