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

Abstract

In this paper we present energy-conserving, mixed discontinuous Galerkin (DG) and continuous Galerkin (CG) schemes for the solution of a broad class of physical systems described by Hamiltonian evolution equations. These systems often arise in fluid mechanics (incompressible Euler equations) and plasma physics (Vlasov--Poisson equations and gyrokinetic equations), for example. The dynamics is described by a distribution function that evolves given a Hamiltonian and a corresponding Poisson bracket operator, with the Hamiltonian itself computed from field equations. Hamiltonian systems have several conserved quantities, including the quadratic invariants of total energy and the $L_2$ norm of the distribution function. For accurate simulations one must ensure that these quadratic invariants are conserved by the discrete scheme. We show that using a discontinuous Galerkin scheme to evolve the distribution function and ensuring that the Hamiltonian lies in its continuous subspace leads 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 distribution function, but decays monotonically when using upwind fluxes. The conservation of density and $L_2$ norm is then used to show that the entropy is a non-decreasing function of time. The proofs shown here apply to any Hamiltonian system, including ones in which the Poisson bracket operator is non-canonical (for example, the gyrokinetic equations). We demonstrate the ability of the scheme to solve the Vlasov--Poisson and incompressible Euler equations in 2D and provide references where we have applied these schemes to solve the much more complex 5D electrostatic and electromagnetic gyrokinetic equations.

References in corpus (14)

• A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov-Poisson equations
• The serendipity family of finite elements
• Discontinuous Galerkin algorithms for fully kinetic plasmas
• AstroGK: Astrophysical Gyrokinetics Code
• Energy-conserving discontinuous Galerkin methods for the Vlasov-Ampère system
• A discontinuous Galerkin solver for Boltzmann Poisson systems in nano devices
• A semi-Lagrangian Vlasov solver in tensor train format
• A general theory for gauge-free lifting
• Electromagnetic full-$f$ gyrokinetics in the tokamak edge with discontinuous Galerkin methods
• Laguerre-Hermite Pseudo-Spectral Velocity Formulation of Gyrokinetics
• Conservative Discontinuous Galerkin Schemes for Nonlinear Fokker-Planck Collision Operators
• High-order Discretization of a Gyrokinetic Vlasov Model in Edge Plasma Geometry
• A conservative discontinuous Galerkin scheme for the 2D incompressible Navier--Stokes equations
• Gyrokinetic Continuum Simulation of Turbulence in Open-Field-Line Plasmas

Cited by in corpus (5)

• Electromagnetic full-$f$ gyrokinetics in the tokamak edge with discontinuous Galerkin methods
• Fluid & Gyrokinetic turbulence in open field-line, helical plasmas
• Turbulent field fluctuations in gyrokinetic and fluid plasmas
• Magnetic Fluctuations in Gyrokinetic Simulations of Tokamak Scrape-Off Layer Turbulence
• Conservative DG Method for the Micro-Macro Decomposition of the Vlasov-Poisson-Lenard-Bernstein Model