Authors

Paul Kotyczka, Bernhard Maschke, Laurent Lefèvre

Abstract

We present the mixed Galerkin discretization of distributed parameter port-Hamiltonian systems. On the prototypical example of hyperbolic systems of two conservation laws in arbitrary spatial dimension, we derive the main contributions: (i) A weak formulation of the underlying geometric (Stokes–Dirac) structure with a segmented boundary according to the causality of the boundary ports. (ii) The geometric approximation of the Stokes–Dirac structure by a finite-dimensional Dirac structure is realized using a mixed Galerkin approach and power-preserving linear maps, which define minimal discrete power variables. (iii) With a consistent approximation of the Hamiltonian, we obtain finite-dimensional port-Hamiltonian state space models. By the degrees of freedom in the power-preserving maps, the resulting family of structure-preserving schemes allows for trade-offs between centered approximations and upwinding. We illustrate the method on the example of Whitney finite elements on a 2D simplicial triangulation and compare the eigenvalue approximation in 1D with a related approach.

Keywords

Systems of conservation laws with boundary energy flows; Port-Hamiltonian systems; Mixed Galerkin methods; Geometric spatial discretization; Structure-preserving discretization

Citation

  • Journal: Journal of Computational Physics
  • Year: 2018
  • Volume: 361
  • Issue:
  • Pages: 442–476
  • Publisher: Elsevier BV
  • DOI: 10.1016/j.jcp.2018.02.006

BibTeX

@article{Kotyczka_2018,
  title={{Weak form of Stokes–Dirac structures and geometric discretization of port-Hamiltonian systems}},
  volume={361},
  ISSN={0021-9991},
  DOI={10.1016/j.jcp.2018.02.006},
  journal={Journal of Computational Physics},
  publisher={Elsevier BV},
  author={Kotyczka, Paul and Maschke, Bernhard and Lefèvre, Laurent},
  year={2018},
  pages={442--476}
}

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References