Academy of Mathematics and Systems Science, CAS
Colloquia & Seminars

Speaker:	Professor Yan Xu,University of Science and Technology of China
Inviter:	Professor Xiaoying Dai
Title:	Port-Hamiltonian systems of hyperbolic partial differential equations and their discretizations
Language:	Chinese
Time & Venue:	2024.03.29 09:00-10:00 N933
Abstract:	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.