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Port-Hamiltonian approximation of a nonlinear flow problem. Part I: Space approximation ansatz

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 mass conservative, 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.