ON PORT-HAMILTONIAN APPROXIMATION OF A NONLINEAR FLOW PROBLEM ON NETWORKS

This paper deals with the systematic development of structure-preserving approx-imations 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 assump-tions 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.

Automated summary

This paper deals with the systematic development of structure-preserving approximations for a class of nonlinear partial differential equations on networks. The approach is guided by energy-based modeling concepts, such as the port-Hamiltonian formalism and the theory of Legendre transformation. The results show that under mild assumptions, local conservation of mass, an energy bound, and the inheritance of the port-Hamiltonian structure can be achieved. The method is not limited to conventional space discretization and can also handle complexity reduction of the nonlinearities by inexact integration.