Energy-consistent Petrov–Galerkin time discretization of port-Hamiltonian systems

Authors

Jan Giesselmann, Attila Karsai, Tabea Tscherpel

Abstract

For a general class of nonlinear port-Hamiltonian systems we develop a high-order time discretization scheme with certain structure preservation properties. The finite or infinite-dimensional system under consideration possesses a Hamiltonian function, which represents an energy in the system and is conserved or dissipated along solutions. For infinite-dimensional systems this structure is preserved under suitable Galerkin discretization in space. The numerical scheme is energy-consistent in the sense that the Hamiltonian of the approximate solutions at time grid points behaves accordingly. This structure preservation property is achieved by specific design of a continuous Petrov–Galerkin (cPG) method in time. It coincides with standard cPG methods in special cases, in which the latter are energy-consistent. Examples of port-Hamiltonian ODEs and PDEs are presented to visualize the framework. In numerical experiments the energy consistency is verified and the convergence behavior is investigated.

Citation

• Journal: The SMAI Journal of computational mathematics
• Year: 2025
• Volume: 11
• Issue:
• Pages: 335–367
• Publisher: MathDoc/Centre Mersenne
• DOI: 10.5802/smai-jcm.127

BibTeX

@article{Giesselmann_2025,
  title={{Energy-consistent Petrov–Galerkin time discretization of port-Hamiltonian systems}},
  volume={11},
  ISSN={2426-8399},
  DOI={10.5802/smai-jcm.127},
  journal={The SMAI Journal of computational mathematics},
  publisher={MathDoc/Centre Mersenne},
  author={Giesselmann, Jan and Karsai, Attila and Tscherpel, Tabea},
  year={2025},
  pages={335--367}
}

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