Port-Hamiltonian formulation of shallow water equations with coriolis force and topography∗

Port based network modeling of complex lumped parameter physical systems naturally leads to a generalized Hamiltonian formulation of its dynamics. The resulting class of open dynamical systems are called “Port-Hamiltonian systems” [12] which are defined using a Dirac structure, the Hamiltonian and dissipative elements. This formulation has been successfully extended to classes of distributed parameter systems by introducing an infinite-dimensional Dirac structure based on Stokes’ Theorem [11]. The port-Hamiltonian formulation is a non-trivial extension of the Hamiltonian formulation. The key extension is the non-zero energy flow through the boundary of the spatial domain by using Dirac structures as opposed to the Poisson structures in the standard Hamiltonian formulation which assume zero energy flow through the boundary. Port-Hamiltonian formulation can be applied to infinite-dimensional fluid dynamical systems through which governing equations emerge from basic conservation laws and follow additional conservation laws. For example, port-Hamiltonian formulation of shallow water flows gives rise to partial differential equations arising from mass and momentum conservation laws and naturally (because of the Dirac structure) derive additional conservation laws of energy, potential vorticity and enstrophy. Many numerical methods such as finite difference, finite volume and (dis)continuous Galerkin finite element methods have been developed to approximate these infinite-dimensional systems into finitedimensional systems. Such finite-dimensional systems may satisfy basic conservation laws of the original system to the discrete level but they do not satisfy accurately the additional conservation laws. Such numerical methods for shallow water equations are already developed (see [2, 3, 10, 1]. In this paper, we therefore aim to develop a novel numerical approach by approximating the shallow water equations based on port-Hamiltonian formulation such that all conservation laws are satisfied to the discrete level. Finite-dimensional port-Hamiltonian approximation of one-dimensional shallow water equations is presented. We first consider the infinite-dimensional port-Hamiltonian formulation of shallow water equations with respect to a non-constant Dirac structure. The discretization essentially consists of two parts: First we discretize the interconnection structure and next we discretize the constitutive relations of the energy storage. In the current work we consider linear shallow water equations, i.e. linearize the system around a given height and zero velocities. Given the port-Hamiltonian formulation of distributed parameter systems it is natural to use different finite-elements for the ap-