Discontinuous Hamiltonian Finite Element Method for Linear Hyperbolic Systems

Authors:
Yan Xu;Jaap J. Der Vegt;Onno Bokhove
Affiliations:
Department of Mathematics, University of Science and Technology of China, Anhui, People's Republic of China 230026;Department of Applied Mathematics, University of Twente, Enschede, The Netherlands 7500 AE;Department of Applied Mathematics, University of Twente, Enschede, The Netherlands 7500 AE
Venue:
Journal of Scientific Computing
Year:
2008

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Abstract

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.