Yan Xu, Professor, University of Science and Technology of China

Xiaoying Dai, Professor

Port-Hamiltonian systems of hyperbolic partial differential equations and their discretizations

9:00-10:00 March 29 (Friday), N933

In this talk, we present port-Hamiltonian formulations and their numerical discretization for several classes of hyperbolic partial differential equations. We present the port-Hamiltonian formulations of incompressible Euler equations in three sets of variables: velocity, solenoidal velocity and vorticity. The novelty of our work is to consider the nonhomogeneous surface that is composed of free surface and permeable fixed surface. The key point of constructing a port-Hamiltonian system is the definition of Stokes-Dirac structure. Based on the theory of the generalized Hamiltonian system, we can define the boundary ports in the Stokes-Dirac structure.

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. The accuracy and capabilities of the methods developed are demonstrated by presenting several numerical experiments.