On port-Hamiltonian approximation of a nonlinear flow problem on networks

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-...

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Bibliographic Details
Published in:arXiv.org 2022-03
Main Authors: Liljegren-Sailer, Björn, Marheineke, Nicole
Format: Article
Language:English
Subjects:
Approximation
Euler-Lagrange equation
Gas pipes
Mathematical analysis
Mathematical models
Model reduction
Nonlinear differential equations
Nonlinear equations
Partial differential equations
Well posed problems
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Summary: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 numerically demonstrate the applicability and good stability properties of the approach using the Euler equations as an example.
ISSN:2331-8422