This paper is on the systematic development of robust and online-efficient approximations for a class of nonlinear partial differential equations on networks. The class includes, e.g. , gas pipe network systems described by one-dimensional barotropic Euler equations. All steps necessary in nonlinear model reduction are covered by our analysis. These are the space discretization by conventional methods, the projection-based model order reduction and the complexity reduction of nonlinearities. Special attention is paid to the structure-preservation on all levels. The proposed reduced models are shown to be locally massconservative, to fulfill energy bounds and to inherit port-Hamiltonian structure. The main ingredients of our analysis are energy-based modeling concepts like the port-Hamiltonian framework and the theory on the Legendre transform, which allow a convenient and general line of argumentation. Moreover, the case of the barotropic Euler equations is examined in more detail and a well-posedness result is proven for their approximation in our framework .
This paper deals with the systematic development of structure-preserving and robust 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 .
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All steps necessary in nonlinear model reduction are covered by our analysis. These are the space discretization by conventional methods, the projection-based model order reduction and the complexity reduction of nonlinearities. Special attention is paid to the structure-preservation on all levels. The proposed reduced models are shown to be locally massconservative, to fulfill energy bounds and to inherit port-Hamiltonian structure. The main ingredients of our analysis are
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framework and the theory on the Legendre transform, which allow a convenient and general line of argumentation. Moreover,
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
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case of the barotropic Euler equations is examined in more detail and a well-posedness result is proven for their approximation in our framework
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