  This paper presents a structure-preserving spatial discretization method for
distributed parameter port-Hamiltonian systems. The class of considered systems
are hyperbolic systems of two conservation laws in arbitrary spatial dimension
and geometries. For these systems, a partioned finite element method is
derived, based on the integration by parts of one of the two conservation laws
written in weak form. The nonlinear 1D Shallow Water Equation (SWE) is first
considered as a motivation example. Then the method is investigated on the
example of the nonlinear 2D SWE. Complete derivation of the reduced
finite-dimensional port-Hamiltonian system is provided and numerical
experiments are performed. Extensions to curvilinear (polar) coordinate
systems, space-varying coefficients and higher-order port-Hamiltonian systems
(Euler-Bernoulli beam equation) are provided.
