  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.
