Discontinuous Hamiltonian Finite Element Method for Linear Hyperbolic Systems

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.

Hamiltonian discontinuous Galerkin FEM for linear, rotating incompressible Euler equations: Inertial waves

“…We first calculate the partial derivatives of the Hamiltonian (32) and use these in the RHS of (30a). Subsequently, we multiply the four equations in (30a) by ∂F/∂Ū i,j,k , ∂F/∂V i,j,k , ∂F/∂W i,j,k , and ∂F/∂R i,j,k , respectively, add them up, and sum over all cells.…”

“…Enforcing zero perturbation density strongly, leading to a constant total density, as constraint also in time leads to the incompressibility condition of zero divergence as secondary constraint. In [32], we derived a DGFEM for twodimensional linear Hamiltonian hyperbolic systems. A Hamiltonian DGFEM for the compressible Euler equations thus exists as the linear Euler equations are hyperbolic, and the extension to three dimensions is straightforward.…”

“…Consequently, this skew-symmetric bracket automatically satisfies the Jacobi identity. Guided by the requirement to preserve the skew-symmetry of the discrete bracket, the relevant fluxes at element interfaces were readily derived in [32]. Our first computational step is to test the algorithm and implementation of this three-dimensional Hamiltonian DGFEM.…”

Discrete and continuous hamiltonian systems for wave modelling

“…In this work we have chosen a DGFEM method for a symmetry and structure preserving discretisation of given a Hamiltonian formulation. The method described in this thesis adds a novel numerical model to the limited number of applications in fluid dynamics that use a variational or Hamiltonian formulation in combination with a (DG)FEM discretisation [5,109]. The motivation and advantages of this choice are discussed in corresponding chapters.…”

“…Unconventional is that the numerical flux is also acting on the test functions δF/δu h . We refer to [109] for a proof that the bracket (2.49) can be transformed to a classical, discontinuous Galerkin finite element weak formulation with alternating fluxes, provided θ = 1/2 at boundary faces and for constant material parameters. When material parameters are a function of space, then the Hamiltonian formulation with its division between bracket and Hamiltonian becomes crucial.…”