Abstract
This dissertation discusses portHamiltonian 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, portHamiltonian formulations of the incompressible Euler equations with a free surface, and portHamiltonian 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 portHamiltonian 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 portHamiltonian discontinuous Galerkin (PHDG) discretizations and demonstrate their power conservation properties. We also obtain the corresponding pseudoPoisson brackets and prove they are also Poisson brackets. Finally, we present several numerical experiments to verify the accuracy and capabilities of PHDG methods.
equations. The thesis focuses on three key topics: Hamiltonian formulations of the incompressible Euler equations with a free surface, portHamiltonian formulations of the incompressible Euler equations with a free surface, and portHamiltonian 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 portHamiltonian 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 portHamiltonian discontinuous Galerkin (PHDG) discretizations and demonstrate their power conservation properties. We also obtain the corresponding pseudoPoisson brackets and prove they are also Poisson brackets. Finally, we present several numerical experiments to verify the accuracy and capabilities of PHDG methods.
Original language  English 

Qualification  Doctor of Philosophy 
Awarding Institution 

Supervisors/Advisors 

Award date  7 Jun 2024 
Place of Publication  Enschede 
Publisher  
Print ISBNs  9789036561266 
Electronic ISBNs  9789036561273 
DOIs  
Publication status  Published  Jun 2024 