A Finite Dimensional Approximation of the shallow water Equations: The port-Hamiltonian Approach

We look into the problem of approximating a distributed parameter port-Hamiltonian system which is represented by a non-constant Stokes-Dirac structure. We here employ the idea where we use different finite elements for the approximation of geometric variables (forms) describing a infinite-dimensional system, to spatially discretize the system and obtain a finite-dimensional port-Hamiltonian system. In particular we take the example of a special case of the shallow water equations.

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

We look into the problem of approximating the shallow water equations with Coriolis forces and topography. We model the system as an infinite-dimensional port-Hamiltonian system which is represented by a non-constant Stokes-Dirac structure. We here employ the idea of using different finite elements for the approximation of geometric variables (forms) describing a distributed parameter system, to spatially discretize the system and obtain a lumped parameter port-Hamiltonian system. The discretized model then captures the physical laws of its infinite-dimensional couterpart such as conservation of energy. We present some preliminary numerical results to justify our claims

Explicit Simplicial Discretization of Distributed-Parameter Port-Hamiltonian Systems

Simplicial Dirac structures as finite analogues of the canonical Stokes-Dirac structure, capturing the topological laws of the system, are defined on simplicial manifolds in terms of primal and dual cochains related by the coboundary operators. These finite-dimensional Dirac structures offer a framework for the formulation of standard input-output finite-dimensional port-Hamiltonian systems that emulate the behavior of distributed-parameter port-Hamiltonian systems. This paper elaborates on the matrix representations of simplicial Dirac structures and the resulting port-Hamiltonian systems on simplicial manifolds. Employing these representations, we consider the existence of structural invariants and demonstrate how they pertain to the energy shaping of port-Hamiltonian systems on simplicial manifolds