Implicitly extrapolated geometric multigrid on disk-like domains for the gyrokinetic Poisson equation from fusion plasma applications

In the context of Tokamak fusion plasma, the gyrokinetic Poisson equation has to be solved on disk-like domains which correspond to the poloidal cross-section of the Tokamak geometry. In its simpliest form, this cross section takes a circular form but deformed geometries were found to be advantageous and are more realistic. We propose a taylored solver for the gyrokinetic Poisson equation on disklike geometries described by curvilinear coordinates. Our solver also copes with an anisotropic, locally refined mesh to model the edge part of the Tokamak and a rapidly dropping density profile. Multigrid methods can achieve optimal complexity for many problems and are among the most efficient solvers for elliptic model problems. Multigrid methods for geometries described by curvilinear coordinates are however less common. We present a taylored geometric multigrid algorithm using optimized line smoothers to enable parallel scalability. We propose an implicit extrapolation scheme for our algorithm to increase the order of convergence by using nonstandard numerical integration rules for P1 finite elements. We also use problem-specific finite differences, which numerically show the same order of convergence as their finite elements' counterpart, enabling a matrix-free implementation with a low memory footprint

Theory of the Drift-Wave Instability at Arbitrary Collisionality

A numerically efficient framework that takes into account the effect of the Coulomb collision operator at arbitrary collisionalities is introduced. Such model is based on the expansion of the distribution function on a Hermite-Laguerre polynomial basis, to study the effects of collisions on magnetized plasma instabilities at arbitrary mean-free path. Focusing on the drift-wave instability, we show that our framework allows retrieving established collisional and collisionless limits. At the intermediate collisionalities relevant for present and future magnetic nuclear fusion devices, deviations with respect to collision operators used in state-of-the-art turbulence simulation codes show the need for retaining the full Coulomb operator in order to obtain both the correct instability growth rate and eigenmode spectrum, which, for example, may significantly impact quantitative predictions of transport. The exponential convergence of the spectral representation that we propose makes the representation of the velocity space dependence, including the full collision operator, more efficient than standard finite difference methods.Comment: 7 pages, 3 figures, accepted for publication on Physical Review Letter

Second order nonlinear gyrokinetic theory : From the particle to the gyrocenter

A gyrokinetic reduction is based on a specific ordering of the different small parameters characterizing the background magnetic field and the fluctuating electromagnetic fields. In this tutorial, we consider the following ordering of the small parameters: $\epsilon\_B=\epsilon\_\delta^2$ where $\epsilon\_B$ is the small parameter associated with spatial inhomogeneities of the background magnetic field and $\epsilon\_\delta$ characterizes the small amplitude of the fluctuating fields. In particular, we do not make any assumption on the amplitude of the background magnetic field. Given this choice of ordering, we describe a self-contained and systematic derivation which is particularly well suited for the gyrokinetic reduction, following a two-step procedure. We follow the approach developed in [Sugama, Physics of Plasmas 7, 466 (2000)]:In a first step, using a translation in velocity, we embed the transformation performed on the symplectic part of the gyrocentre reduction in the guiding-centre one. In a second step, using a canonical Lie transform, we eliminate the gyroangle dependence from the Hamiltonian. As a consequence, we explicitly derive the fully electromagnetic gyrokinetic equations at the second order in $\epsilon\_\delta$

Semi-Lagrangian Scheme with Arakawa Splitting for Gyro-kinetic Equations

The gyro-kinetic model is an approximation of the Vlasov-Maxwell system in a strongly magnetized magnetic field. We propose a new algorithm for solving it combining the Semi-Lagrangian (SL) method and the Arakawa (AKW) scheme with a time-integrator. Both methods are successfully used in practice for different kinds of applications, in our case, we combine them by first decomposing the problem into a fast (parallel) and a slow (perpendicular) dynamical system. The SL approach and the AKW scheme will be used to solve respectively the fast and the slow subsystems. Compared to the scheme in [1], where the entire model is solved using only the SL method, our goal is to replace the method used in the slow subsystem by the AKW scheme, in order to improve the conservation of the physical constants

Linear Theory of Electron-Plasma Waves at Arbitrary Collisionality

The dynamics of electron-plasma waves are described at arbitrary collisionality by considering the full Coulomb collision operator. The description is based on a Hermite-Laguerre decomposition of the velocity dependence of the electron distribution function. The damping rate, frequency, and eigenmode spectrum of electron-plasma waves are found as functions of the collision frequency and wavelength. A comparison is made between the collisionless Landau damping limit, the Lenard-Bernstein and Dougherty collision operators, and the electron-ion collision operator, finding large deviations in the damping rates and eigenmode spectra. A purely damped entropy mode, characteristic of a plasma where pitch-angle scattering effects are dominant with respect to collisionless effects, is shown to emerge numerically, and its dispersion relation is analytically derived. It is shown that such a mode is absent when simplified collision operators are used, and that like-particle collisions strongly influence the damping rate of the entropy mode.Comment: 23 pages, 10 figures, accepted for publication on Journal of Plasma Physic

Solving the Vlasov–Maxwell equations using Hamiltonian splitting

In this paper, the numerical discretizations based on Hamiltonian splitting for solving the Vlasov–Maxwell system are constructed. We reformulate the Vlasov–Maxwell system in Morrison–Marsden–Weinstein Poisson bracket form. Then the Hamiltonian of this system is split into five parts, with which five corresponding Hamiltonian subsystems are obtained. The splitting method in time is derived by composing the solutions to these five subsystems. Combining the splitting method in time with the Fourier spectral method and finite volume method in space gives the full numerical discretizations which possess good conservation for the conserved quantities including energy, momentum, charge, etc. In numerical experiments, we simulate the Landau damping, Weibel instability and Bernstein wave to verify the numerical algorithms

A "metric" semi-Lagrangian Vlasov-Poisson solver

We propose a new semi-Lagrangian Vlasov-Poisson solver. It employs elements of metric to follow locally the flow and its deformation, allowing one to find quickly and accurately the initial phase-space position $Q(P)$ of any test particle $P$, by expanding at second order the geometry of the motion in the vicinity of the closest element. It is thus possible to reconstruct accurately the phase-space distribution function at any time $t$ and position $P$ by proper interpolation of initial conditions, following Liouville theorem. When distorsion of the elements of metric becomes too large, it is necessary to create new initial conditions along with isotropic elements and repeat the procedure again until next resampling. To speed up the process, interpolation of the phase-space distribution is performed at second order during the transport phase, while third order splines are used at the moments of remapping. We also show how to compute accurately the region of influence of each element of metric with the proper percolation scheme. The algorithm is tested here in the framework of one-dimensional gravitational dynamics but is implemented in such a way that it can be extended easily to four or six-dimensional phase-space. It can also be trivially generalised to plasmas.Comment: 32 pages, 14 figures, accepted for publication in Journal of Plasma Physics, Special issue: The Vlasov equation, from space to laboratory plasma