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at - Automatisierungstechnik

Methoden und Anwendungen der Steuerungs-, Regelungs- und Informationstechnik

[AT - Automation Technology: Methods and Applications of Control, Regulation, and Information Technology
]

Editor-in-Chief: Jumar, Ulrich


IMPACT FACTOR 2017: 0.503

CiteScore 2017: 0.47

SCImago Journal Rank (SJR) 2017: 0.212
Source Normalized Impact per Paper (SNIP) 2017: 0.546

Online
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2196-677X
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Volume 65, Issue 5

Issues

Discrete port-Hamiltonian formulation and numerical approximation for systems of two conservation laws

Diskrete Port-Hamiltonsche Darstellung und numerische Approximation für Systeme zweier Erhaltungsgleichungen

Paul Kotyczka
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  • Univ Lyon, Université Claude Bernard Lyon 1, CNRS, LAGEP UMR 5007, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex France
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/ Bernhard Maschke
  • Univ Lyon, Université Claude Bernard Lyon 1, CNRS, LAGEP UMR 5007, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex France
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Published Online: 2017-05-03 | DOI: https://doi.org/10.1515/auto-2016-0098

Abstract

We discuss the discrete formulation of systems of conservation laws in port-Hamiltonian form on dual chain complexes. Based on integral balance equations and topological information, this representation is exact and qualifies as a control model. The finite-dimensional approximation requires an energy discretization that yields discrete constitutive equations. We give (i) a brief overview of discrete modeling of conservation laws on n-complexes and (ii) extend existing results by allowing for mixed physical types of boundary inputs. This requires the construction of a primal and a dual complex based on the underlying staggered grids and the localization of the inputs on the system boundary. Finally, (iii) we discuss the properties of the resulting structure-preserving discretization scheme based on a consistency analysis for the 2D nonlinear shallow water equations.

Zusammenfassung

Wir diskutieren die diskrete Beschreibung von Systemen von Erhaltungsgleichungen auf dualen Kettenkomplexen in Port-Hamiltonscher Form. Die Darstellung beruht auf den integralen Bilanzgleichungen und der Topologie der Diskretisierungsgitter. Sie ist exakt und eignet sich als Regelungsmodell. Die endlich-dimensionale Näherung erfordert eine Diskretisierung des Energiefunktionals, welche auf diskrete Konstitutivgleichungen führt. Wir geben (i) einen kurzen Überblick über die diskrete Modellierung von Erhaltungsgleichungen auf n-Komplexen. Die systematische Konstruktion der dualen Komplexe auf Grundlage versetzter Gitter und der Verortung der Randeingriffe erweitert (ii) bestehende Ergebnisse um die Abbildung verschiedenartiger physikalischer Randeingänge. Schließlich diskutieren wir (iii) am Beispiel der nichtlinearen 2D-Flachwassergleichungen und auf Basis einer Konsistenzanalyse die Eigenschaften des resultierenden strukturerhaltenden Diskretisierungsverfahrens.

Keywords: Conservation laws; port-Hamiltonian systems; discrete formulation; structure-preserving discretization; finite volumes

Schlagwörter: Erhaltungsgleichungen; Port-Hamiltonsche Systeme; diskrete Formulierung; strukturerhaltende Diskretisierung; Finite Volumen

About the article

Paul Kotyczka

Paul Kotyczka is lecturer (Akademischer Rat) at the Chair of Automatic Control, Technical University of Munich. He is on a two-year research stay at University Claude Bernard Lyon 1, France, where he joined the Laboratory of Automatic Control and Chemical Engineering (LAGEP) in September 2015 as a Marie Skłodowska-Curie Fellow. Currently, he works as a postdoctoral researcher within the DFG-ANR cofunded project INFIDHEM. His research interests concern physical modelling, numerical methods for distributed parameter port-Hamiltonian systems and nonlinear control.

Univ Lyon, Université Claude Bernard Lyon 1, CNRS, LAGEP UMR 5007, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France

Bernhard Maschke

Bernhard Maschke is Professor of Automatic Control at the Laboratory of Automatic Control and Chemical Engineering (LAGEP UMR CNRS 5007) of the University Claude Bernard of Lyon, Villeurbanne, France. The main streamline of research that he developed, concerns the generalizations of Hamiltonian systems for the modelling, simulation and control of complex physical systems derived from network theory and thermodynamic theory known as port-Hamiltonian systems. His current research interests concern the control of port-Hamiltonian systems defined on contact manifolds, of Hamiltonian systems of conservation laws with boundary control and systems defined on graphs and kcomplexes.

Univ Lyon, Université Claude Bernard Lyon 1, CNRS, LAGEP UMR 5007, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France


Revised: 2017-02-23

Accepted: 2017-03-23

Received: 2016-08-07

Published Online: 2017-05-03

Published in Print: 2017-05-29


Citation Information: at - Automatisierungstechnik, Volume 65, Issue 5, Pages 308–322, ISSN (Online) 2196-677X, ISSN (Print) 0178-2312, DOI: https://doi.org/10.1515/auto-2016-0098.

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[2]
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Journal of Computational Physics, 2018, Volume 373, Page 673
[4]
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Journal of Computational Physics, 2018, Volume 361, Page 442

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