We develop a Hamiltonian discontinuous finite element discretization of a generalized Hamiltonian system for linear hyperbolic systems, which include the rotating shallow water equations, the acoustic and Maxwell equations. These equations have a Hamiltonian structure with a bilinear Poisson bracket, and as a consequence the phase-space structure, “mass��? and energy are preserved. We discretize the bilinear Poisson bracket in each element with discontinuous elements and introduce numerical fluxes via integration by parts while preserving the skew-symmetry of the bracket. This automatically results in a mass and energy conservative discretization. When combined with a symplectic time integration method, energy is approximately conserved and shows no drift. For comparison, the discontinuous Galerkin method for this problem is also used. A variety numerical examples is shown to illustrate the accuracy and capability of the new method.
- Rotating shallow water equations · Acoustic equations · Maxwell equations ·Hamiltonian dynamics · Discontinuous Galerkin method · Numerical flux
Xu, Y., van der Vegt, J. J. W., & Bokhove, O. (2008). Discontinuous Hamiltonian finite element method for linear hyperbolic systems. Journal of scientific computing, 35(WP 08-02/2-3), 241-265. [10.1007/s10915-008-9191-y]. https://doi.org/10.1007/s10915-008-9191-y