On Port-Hamiltonian Approximation of a Nonlinear Flow Problem on Networks

Authors

Björn Liljegren-Sailer, Nicole Marheineke

Abstract

This paper deals with the systematic development of structure-preserving approximations for a class of nonlinear partial differential equations on networks. The class includes, for example, gas pipe network systems described by barotropic Euler equations. Our approach is guided throughout by energy-based modeling concepts (port-Hamiltonian formalism, theory of Legendre transformation), which provide a convenient and general line of reasoning. Under mild assumptions on the approximation, local conservation of mass, an energy bound, and the inheritance of the port-Hamiltonian structure can be shown. Our approach is not limited to conventional space discretization but also covers complexity reduction of the nonlinearities by inexact integration. Thus, it can serve as a basis for structure-preserving model reduction. Combined with an energy-stable time integration, we demonstrate the applicability and good stability properties using the example of the Euler equations on networks.

Citation

• Journal: SIAM Journal on Scientific Computing
• Year: 2022
• Volume: 44
• Issue: 3
• Pages: B834–B859
• Publisher: Society for Industrial & Applied Mathematics (SIAM)
• DOI: 10.1137/21m1443480

BibTeX

@article{Liljegren_Sailer_2022,
  title={{On Port-Hamiltonian Approximation of a Nonlinear Flow Problem on Networks}},
  volume={44},
  ISSN={1095-7197},
  DOI={10.1137/21m1443480},
  number={3},
  journal={SIAM Journal on Scientific Computing},
  publisher={Society for Industrial & Applied Mathematics (SIAM)},
  author={Liljegren-Sailer, Björn and Marheineke, Nicole},
  year={2022},
  pages={B834--B859}
}

Download the bib file

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