Port-Hamiltonian Systems and their Discontinuous Galerkin Discretization
Xiaoyu Cheng is a PhD student in the department Mathematics of Computational Science. Promotors are prof.dr.ir.J.J.W. van der Vegt and prof.dr. H.J. Zwart from the faculty EEMCS and prof.dr. Y. Xu from the University of Science & Technology of China.
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.
Firstly, based on the classical formulations, we derive generalized Hamiltonian formulations of the incompressible Euler equations with a free surface using the language of differential 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.
Secondly, 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.Our main results are the construction and proof of Dirac structures in suitable Sobolev spaces of differential forms for each variable set, which provides the core of any port-Hamiltonian formulation. Finally, it is proven that the state dependent Dirac structures are related to Poisson brackets that are linear, skew-symmetric and satisfy the Jacobi identity. Consequently, we derive the 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.
Finally, we obtain discontinuous Galerkin (DG) finite element discretizations of a class for linear hyperbolic port-Hamiltonian dynamical systems. The key point in constructing a port-Hamiltonian system is a Stokes-Dirac structure. Instead of following the traditional approach of defining the strong form of the Dirac structure, we define a Dirac structure in weak form, specifically in the input-state-output form. This is implemented within broken Sobolev spaces on a tessellation with polyhedral elements. After that, we state the weak port-Hamiltonian formulation and prove that it relates to a Poisson bracket. In our work, a crucial aspect of constructing the above-mentioned Dirac structure is that we provide a conservative relation between the boundary ports. Next, we state DG discretizations of the port-Hamiltonian system by using the weak form of the Dirac structure and broken polynomial spaces of differential forms, and we provide a priori error estimates for the structure-preserving port-Hamiltonian discontinuous Galerkin (PHDG) discretizations. The accuracy and capabilities of the methods developed in this chapter are demonstrated by presenting several numerical experiments.
