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In this paper we present energy-conserving, mixed discontinuous Galerkin (DG)and continuous Galerkin (CG) schemes for the solution of a broad class ofphysical systems described by Hamiltonian evolution equations. These systemsoften arise in fluid mechanics (incompressible Euler equations) and plasmaphysics (Vlasov--Poisson equations and gyrokinetic equations), for example. Thedynamics is described by a distribution function that evolves given aHamiltonian and a corresponding Poisson bracket operator, with the Hamiltonianitself computed from field equations. Hamiltonian systems have severalconserved quantities, including the quadratic invariants of total energy andthe $L_2$ norm of the distribution function. For accurate simulations one mustensure that these quadratic invariants are conserved by the discrete scheme. Weshow that using a discontinuous Galerkin scheme to evolve the distributionfunction and ensuring that the Hamiltonian lies in its continuous subspaceleads to an energy-conserving scheme in the continuous-time limit. Further, the$L_2$ norm is conserved if central fluxes are used to update the distributionfunction, but decays monotonically when using upwind fluxes. The conservationof density and $L_2$ norm is then used to show that the entropy is anon-decreasing function of time. The proofs shown here apply to any Hamiltoniansystem, including ones in which the Poisson bracket operator is non-canonical(for example, the gyrokinetic equations). We demonstrate the ability of thescheme to solve the Vlasov--Poisson and incompressible Euler equations in 2Dand provide references where we have applied these schemes to solve the muchmore complex 5D electrostatic and electromagnetic gyrokinetic equations.

Discontinuous Galerkin schemes for a class of Hamiltonian evolution equations with applications to plasma fluid and kinetic problems