School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, Province, P.R. China.
The Hamiltonian formulation describes the time-evolution of a system using a Hamiltonian functional representing the total energy of the system, and a Poisson bracket, which defines its dynamics. The Hamiltonian structure expresses important invariants and symmetries of the system and provides a deep insight into its behavior.
Many partial differential equations in mathematical physics have a deep mathematical structure that can be represented as a Hamiltonian system. For example, the Maxwell equations, which describe electromagnetic waves, the Euler equations of compressible gas dynamics, all have a Hamiltonian structure. But in general, a numerical discretization will not preserve the Hamiltonian structure of partial differential equations.
In this PhD project, we will focus on deriving a port-Hamiltonian discontinuous Galerkin discretization for the Euler equations of gas dynamics that preserves the underlying Hamiltonian structure of the partial differential equations. This requires a whole new paradigm to derive numerical discretizations of Hamiltonian systems. The development of this discretization requires the derivation of a weak formulation for the port-Hamiltonian equations and the Dirac structure. In this process, apart from the primary variables, we also need to discretize the differential forms for the dual variables. The key benefit of this approach is that we still obtain an element-based discretization without the need of using a staggered mesh, which would severely complicate a discontinuous Galerkin discretization. We will also perform a detailed error and stability analysis and verify if the Hamiltonian structure of the equations is also preserved by the discontinuous Galerkin discretization, in particular on locally refined meshes