Study of conservation and recurrence of Runge–Kutta discontinuous Galerkin schemes for Vlasov–Poisson systems

Abstract

In this paper we consider Runge-Kutta discontinuous Galerkin (RKDG) schemes for Vlasov-Poisson systems that model collisionless plasmas. One-dimensional systems are emphasized. The RKDG method, originally devised to solve conservation laws, is seen to have excellent conservation properties, be readily designed for arbitrary order of accuracy, and capable of being used with a positivity-preserving limiter that guarantees positivity of the distribution functions. The RKDG solver for the Vlasov equation is the main focus, while the electric field is obtained through the classical representation by Green's function for the Poisson equation. A rigorous study of recurrence of the DG methods is presented by Fourier analysis, and the impact of different polynomial spaces and the positivity-preserving limiters on the quality of the solutions is ascertained. Several benchmark test problems, such as Landau damping, the two-stream instability, and the KEEN (Kinetic Electro static Electron Nonlinear) wave, are given.

Nonlinear regime of electrostatic waves propagation in presence of electron-electron collisions

“…We described numerically the onset and nonlinear saturation of the bump-on-tail instability 22,23 (in its symmetric form) and the excitation and propagation of the so-called Kinetic Electrostatic Electron Nonlinear waves, 14,15,26 in situations of intermediate range of plasma collisionality. In this way, we get rid of the restrictive collision-free assumption, keeping, however, the system dynamics far from the strong collisional fluid regime, where the plasma always remains at thermodynamic equilibrium.…”

“…5(a)], we recover one of the typical features of the KEEN waves. 14,15,26 While the driver is turned on, the energy injected into the fundamental wavenumber component (black line) flows also to the higher spectral components (red, blue and yellow solid lines). After the driver has been turned off, the resulting electric signal is composed by many wavenumbers, in a stable ratio one with another, thus departing significantly from the purely sinusoidal spatial shape of the driver field.…”

A Kinetic Vlasov Model for Plasma Simulation Using Discontinuous Galerkin Method on Many-Core Architectures

“…The discontinuous Galerkin method is selected for its robustness to work well for kinetic equations in phase space, Maxwell's electromagnetic equations, and fluid equations in physical space with multiple contemporary examples for these application areas [12,13,14,15,16,17,18]. Additionally, the explicit method is straightforward to implement and well suited for emerging high performance computing architectures as described in Chapters 4 and 9.…”

An h-Adaptive RKDG Method for the Vlasov–Poisson System

“…However, it suffers statistical noise O (1/ √ N ) with N being the number of particles. There are highly accurate mesh-based semi-Lagrangian [5,11,17,[24][25][26]28,32,33] and Eulerian [1,8,15,21,23,34,35,37] methods, which have been shown to be advantageous due to their efficiency and effectiveness in resolving rich solution structures. The semi-Lagrangian method is designed by propagating information along characteristic curves.…”

“…Then the spatially discretized ODE systems are evolved with high order numerical time integrator such as the Runge-Kutta (RK) methods [19] via the method-of-line approach. These methods have been well-known for being highly accurate both in space and time, and being very robust as a black-box procedure in a truly multi-dimensional setting (without dimensional splitting) [8,21].…”