Port-Hamiltonian systems and their discontinuous Galerkin discretization

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This dissertation discusses port-Hamiltonian formulations and their numerical discretization for several classes of hyperbolic partial differential equations. The thesis focuses on three key topics: Hamiltonian formulations of the incompressible Euler equations with a free surface, port-Hamiltonian formulations of the incompressible Euler equations with a free surface, and port-Hamiltonian discontinuous Galerkin discretizations for a class of linear hyperbolic partial differential equations. In Chapter 2, based on the classical formulations, we derive generalized Hamiltonian formulations of the incompressible Euler equations with a free surface using the language of dierential forms. Three sets of variables, including velocity, solenoidal velocity, potential, vorticity, and free surface, are used to represent the incompressible Euler equations with a free surface. Additionally, we derive the corresponding Poisson bracket for these sets of variables and express the Hamiltonian systems using these Poisson brackets. Next, we extend the generalized Hamiltonian formulations of the incompressible Euler equations with a free surface to include conditions that permit energy exchange at the boundary of the spatial domain. We derive the corresponding Dirac structure and port-Hamiltonian formulations of the incompressible Euler equations with a domain boundary, consisting of a free surface and a fixed surface with inhomogeneous boundary conditions. In Chapter 3, we first obtain the weak form of the Dirac structure for a class of linear hyperbolic partial differential equations defined in broken Sobolev spaces. Next, by approximating all variables using piecewise polynomial spaces of differential forms, we derive port-Hamiltonian discontinuous Galerkin (PHDG) discretizations and demonstrate their power conservation properties. We also obtain the corresponding pseudo-Poisson brackets and prove they are also Poisson brackets. Finally, we present several numerical experiments to verify the accuracy and capabilities of PHDG methods.

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