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Energy-consistent Petrov–Galerkin time discretization of port-Hamiltonian systems
Jan Giesselmann1; Attila Karsai2; Tabea Tscherpel1
1 Department of Mathematics, Technische Universität Darmstadt, Dolivostr. 15, 64293 Darmstadt,Germany
2 Institute of Mathematics, Technische Universität Berlin, Str. des 17. Juni 136, 10623 Berlin,Germany
The SMAI Journal of computational mathematics, Volume 11 (2025), pp. 335-367.

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.

Published online:
DOI: 10.5802/smai-jcm.127
Classification: 35K55, 37L65, 37K58, 65J08, 65J15, 65L60, 65M60, 65P10
Keywords: port-Hamiltonian system, Hamiltonian system, gradient system, energy conserving, structure preservation, Petrov-Galerkin

Jan Giesselmann 1; Attila Karsai 2; Tabea Tscherpel 1

1 Department of Mathematics, Technische Universität Darmstadt, Dolivostr. 15, 64293 Darmstadt,Germany
2 Institute of Mathematics, Technische Universität Berlin, Str. des 17. Juni 136, 10623 Berlin,Germany 
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     title = {Energy-consistent {Petrov{\textendash}Galerkin} time discretization of {port-Hamiltonian} systems},
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     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
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Jan Giesselmann; Attila Karsai; Tabea Tscherpel. Energy-consistent Petrov–Galerkin time discretization of port-Hamiltonian systems. The SMAI Journal of computational mathematics, Volume 11 (2025), pp. 335-367. doi : 10.5802/smai-jcm.127. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.127/

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