Energy conserving discontinuous Galerkin spectral element method for the Vlasov–Poisson system

Abstract

Abstract. We propose a new, energy conserving, spectral element, discontinuous Galerkin method for the approximation of the Vlasov-Poisson system in arbitrary dimension, using Cartesian grids. The method is derived from the one proposed in [ACS12], with two modifications: energy conservation is obtained by a suitable projection operator acting on the solution of the Poisson problem, rather than by solving multiple Poisson problems, and all the integrals appearing in the finite element formulation are approximated with Gauss-Lobatto quadrature, thereby yielding a spectral element formulation. The resulting method has the following properties: exact energy conservation (up to errors introduced by the time discretization), stability (thanks to the use of upwind numerical fluxes), high order accuracy and high locality. For the time discretization, we consider both Runge-Kutta methods and exponential integrators, and show results for 1D and 2D cases (2D and 4D in phase space, respectively).

A dynamical adaptive tensor method for the Vlasov–Poisson system

“…For numerical purposes, equation (1) has to be solved on a truncated domain Ω = Ω x × Ω v . One can for instance impose periodic boundary conditions on ∂Ω x or ∂Ω v , as done in [33]. An alternative formulation would consist of imposing homogeneous boundary conditions on Ω v , the velocity domain.…”

“…The tensorized decomposition of c and l implies that each term appearing in (33) and (34) can be quickly evaluated, since they only involve forms defined on Hilbert spaces of functions depending on only one variable. Such tensorized decompositions are always naturally available in the Vlasov-Poisson context, and this crucial fact is at the heart of the efficiency of this approach.…”

“…where k = 0.5 is the wavenumber of the perturbation and the amplitude β = 0.01 set the problem in a linear Landau damping regime (see [33,29]). In such a configuration the analytical decay rate for the electric amplitude is γ ≈ 0.153.…”

“…The first test proposed is a standard linear Landau damping in a 1D-1D configuration, as proposed in [33]. The domain size is Ω…”

Metriplectic integrators for the Landau collision operator

“…If Coulomb collisions are neglected altogether, recent work on Vlasov-Poisson, Vlasov-Maxwell and related systems has provided algorithms that satisfy energy conservation and also preserve other invariants present in the system, such as the momentum and charge conservation, and the divergence-free nature of the magnetic field. For algorithms in the particle-in-cell framework see for example [4][5][6][7][8][9][10][11][12][13][14][15][16], for discontinuous Galerkin methods see [17][18][19][20][21][22][23][24][25] and for other grid-based methods see [26][27][28][29]. Realistic kinetic simulations of plasmas, expanding to macroscopic time scales, however, require also the effects of Coulomb collisions.…”