Port-Hamiltonian Discontinuous Galerkin Finite Element Methods

N. Kumar; J. J. W. van der Vegt; H. J. Zwart

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arXiv · 2212.07041

Port-Hamiltonian Discontinuous Galerkin Finite Element Methods

N. Kumar·J. J. W. van der Vegt·H. J. Zwart

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Abstract

A port-Hamiltonian (pH) system formulation is a geometrical notion used to formulate conservation laws for various physical systems. The distributed parameter port-Hamiltonian formulation models infinite dimensional Hamiltonian dynamical systems that have a non-zero energy flow through the boundaries. In this paper we propose a novel framework for discontinuous Galerkin (DG) discretizations of pH-systems. Linking DG methods with pH-systems gives rise to compatible structure preserving finite element discretizations along with flexibility in terms of geometry and function spaces of the variables involved. Moreover, the port-Hamiltonian formulation makes boundary ports explicit, which makes the choice of structure and power preserving numerical fluxes easier. We state the Discontinuous Finite Element Stokes-Dirac structure with a power preserving coupling between elements, which provides the mathematical framework for a large class of pH discontinuous Galerkin discretizations. We also provide an a priori error analysis for the port-Hamiltonian discontinuous Galerkin Finite Element Method (pH-DGFEM). The port-Hamiltonian discontinuous Galerkin finite element method is demonstrated for the scalar wave equation showing optimal rates of convergence.

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BibTeX RIS

N. Kumar, J. J. W. van der Vegt, H. J. Zwart. 2022-12-14. Port-Hamiltonian Discontinuous Galerkin Finite Element Methods. https://arxiv.org/abs/2212.07041

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