Variational formulation of particle algorithms for kinetic plasma simulations

Abstract

Common time-explicit numerical methods for kinetic simulations of plasmas in the lowcollisions limit fall into two classes of algorithms: momentum conserving [also known as Particle-In-Cell (PIC)] and energy conserving. Each has certain drawbacks. The PIC algorithm does not conserve total energy, which may lead to spurious numerical heating (grid heating). Its overall accuracy is at most second due to the nature of the force interpolation between grid and particle position. Energy-conserving algorithms do not exhibit grid heating, but because their formulation uses potentials, computationally undesirable matrix inversions may be necessary. In addition, compared to PIC algorithms for the same accuracy, these algorithms have higher numerical noise due to the restricted choice of particle shapes. Here we formulate time-explicit, finite-size particle algorithms using particular reductions of the particle distribution function. These reductions are used in two variational principles, a Lagrangian-based and a Hamiltonian-based in conjunction with a non-canonical Poisson bracket. The Lagrangian formulations here generalize previous such formulations. The Hamiltonian formulation is presented here for the first time. Many drawbacks of the two classes of particle methods are mitigated. For example, restrictions on particle shapes are relaxed in energy conserving algorithms, which allows to decrease the numerical noise in these methods. The Hamiltonian formulation of particle algorithms is done in terms of fields instead of potentials, thus avoiding solving Poisson's equation. An algorithm that conserves both energy and momentum is presented. Other features of the algorithms include a natural way to perform coordinate transformations, the use of various time integrating methods, and the ability to increase the overall accuracy beyond second order, including all generalizations. For simplicity, we restrict our discussion to one-dimensional, non-relativistic, unmagnetized, electrostatic plasmas.

On the velocity space discretization for the Vlasov–Poisson system: Comparison between implicit Hermite spectral and Particle-in-Cell methods

“…The PIC code employs linear interpolation, usually referred to as 'Cloud-in-Cell' (CIC) [1]. Note that higher order interpolation schemes have been proposed, for instance, in [77,78]. In order to make the comparison as fair as possible, we keep the same number of grid points for FH and PIC.…”

Variational formulation of macro-particle plasma simulation algorithms

“…1 For simplicity, we consider a single mobile plasma species and restrict our discussion to the electrostatic case. We will consider an electromagnetic extension in Sec.…”

“…Following Ref. 1, we now consider a general discrete spatial representation using a truncated basis. Let U m (x), m ¼ 1, 2, …, M be the first M elements of a complete basis.…”

“…It appears that the only way to satisfy (50) is for U m to be a truncated Fourier basis. 1 Thus, using a Fourier representation of the fields, we expect the macro-particle model will have exact gauge invariance as well as charge and momentum conservation. Computationally, the Fourier representation appears to be sufficiently expensive compared to the finite-difference representation such as to be unsuitable for production work.…”

“…Recently, a variational formulation of macro-particle plasma models, based on a reduction of the distribution function, has been developed. 1 As in the usual Particle-In-Cell (PIC) formulation, 2-4 these macro-particles have a definite momentum and are spatially extended. The primary advantage of this approach is the preservation of the link between symmetries and conservation laws.…”