Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system

Abstract

We present a computational study for a family of discontinuous Galerkin methods for the one dimensional Vlasov-Poisson system, recently introduced in [4]. We introduce a slight modification of the methods to allow for feasible computations while preserving the properties of the original methods. We study numerically the verification of the theoretical and convergence analysis, discussing also the conservation properties of the schemes. The methods are validated through their application to some of the benchmarks in the simulation of plasma physics.Numerical simulation has become a major tool for understanding the complex behavior of a plasma or a particle beam in many situations. This is due not only to the large number of physical applications and technological implications of the behavior of plasmas, but also to the intrinsic difficulties of the models used to describe such behavior. In fact, it was recognized long time ago that there does not exist any fully satisfactory macroscopic model (fluid equations) which can be used to describe the particle interaction in laser-fusion problems. In contrast, microscopic models (kinetic equations) can provide a more accurate description of the plasmas.One of the simplest model problems that is currently used in the simulation of plasmas is the Vlasov-Poisson system. Such system describes the evolution of a plasma of charged particles (electrons and ions) under the effects of the transport and self-consistent electric field. The unknown, typically denoted by f (x, v, t) (with x standing for position, v for velocity and t for time), represents the distribution function of particles (ions, electrons, etc.) in the phase space. The coupling with a self-consistent electrostatic field (neglecting magnetic effects) is taken into account through the Poisson equation. The nonlinear structure of the system prevents from obtaining analytical solutions, except for a few academic cases (see the surveys [35,13,26] for a good description on the state of the art of the mathematical analysis of the problem). Therefore, numerical simulations have to be performed to study realistic physical phenomena.At the present time, there can be distinguished two main classes of numerical methods for simulating plasmas; Lagrangian (or probabilistic) and Eulerian (or deterministic) methods. The former class include all different types of particle methods [12,22,47,34,41,32,7] and has been a preferred choice since the beginnings of numerical simulations in plasma physics 1991 Mathematics Subject Classification. 82C80, 65M60, 65M12, 82A70.

An Adaptive Multiresolution Discontinuous Galerkin Method for Time-Dependent Transport Equations in Multidimensions

“…where u = 0.99, v t = 0.3. In the literature, RKDG schemes for the VP system [6,30,14] have been extensively studied. They are shown to have superior performance in conservation.…”

GEMPIC: geometric electromagnetic particle-in-cell methods

“…While the former guarantees exact local conservation of important quantities like mass, momentum, energy and the L 2 norm of the distribution function after a semidiscretization in space, the latter retains these properties even after the discretization in time. Recently, also various discretizations based on discontinuous Galerkin methods have been proposed for both, the Vlasov-Poisson [33,32,34,44,24,55] and the Vlasov-Maxwell system [25,26,27]. Even though these are usually not based on geometric principles, they tend to show good long-time conservation properties with respect to momentum and/or energy.…”

Arbitrary-order time-accurate semi-Lagrangian spectral approximations of the Vlasov–Poisson system

“…An alternative to PIC and Transform methods is offered by the class of Eulerian and Semi-Lagrangian methods, which discretize the Vlasov equation on a grid of the phase space. Common approaches for the implementation are: Finite Volume Methods [25,5], Discontinuous Galerkin [3,4,31], finite difference methods based on ENO and WENO polynomial reconstructions [20], or propagation of the solution along the characteristics in an operator splitting framework [1,16,19,27,26,44,22]. Semi-Lagrangian methods were first developed for meteorological applications in the early '90s [6,7,45].…”