Spectral Solver for Multi-scale Plasma Physics Simulations with Dynamically Adaptive Number of Moments

Abstract

A spectral method for kinetic plasma simulations based on the expansion of the velocity distribution function in a variable number of Hermite polynomials is presented. The method is based on a set of non-linear equations that is solved to determine the coefficients of the Hermite expansion satisfying the Vlasov and Poisson equations. In this paper, we first show that this technique combines the fluid and kinetic approaches into one framework. Second, we present an adaptive strategy to increase and decrease the number of Hermite functions dynamically during the simulation. The technique is applied to the Landau damping and two-stream instability test problems. Performance results show 21% and 47% saving of total simulation time in the Landau and two-stream instability test cases, respectively.

A Semi-Lagrangian Spectral Method for the Vlasov–Poisson System Based on Fourier, Legendre and Hermite Polynomials

“…Within this approach, exact conservation laws in the discrete setting, i.e., discrete invariants in time for number of particles (mass, charge), momentum and energy, can be constructed from Hermite expansion's coefficients. Moreover, as pointed out in [57,56,24,45,46], expansions in Hermite basis functions are intrinsically multiscale, providing a natural connection between low-order moments of the plasma distribution function and typical fluid moments.…”

“…Semi-Lagrangian methods have been proposed in different frameworks such as the Finite Volume method [26,4], Discontinuous Galerkin method [2,3,39], finite difference methods based on ENO and WENO polynomial reconstructions [21], as well as in the propagation of solutions along the characteristics in an operator splitting context [1,17,20,28,27,52,23]. These methods offer an alternative to Particle-in-Cell (PIC) methods [22,58,59,12,18,19,42,43,47,53] and to the so called Transform methods based on spectral approximations [45,48,41,57,56]. PIC methods are very popular in the plasma physics community and are the most widely used methods because of their robustness and relative simplicity [6].…”

“…in(31), when the Fourier-Fourier method (top) and the Fourier-Hermite method (bottom) with N = M = 2 4 , ∆t = 0.01 and α = 1 are used. If we compare withFigure 4, the two plots behave rather differently, being the second much better that the first one.In all the numerical experiments the number of particles Q(k) N,M and the momentum P (k) N,M defined in(53)and(57) are preserved. The situation is slightly different concerning the energy, where a slight growth is observed, with a rate depending on the size of the time step.…”

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

“…Based on the seminal paper [30], an alternative approach, called the Transform method, was developed at the end of the '60s, which uses a spectral decomposition of the distribution function and leads to a truncated set of moment equations for the expansion coefficients [2]. To this end, Hermite basis functions are used for unbounded domains, Legendre basis functions for bounded domains, and Fourier basis functions for periodic domains, see, e.g., [36,40,33,48,47]. These techniques can outperform PIC [13,14] in Vlasov-Poisson simulations.…”

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

“…that describe the plasma macroscopically and are usually the physical quantities of interest. As a consequence, the use of the Hermite basis allows a smooth transition from a fluid to a kinetic description by simply increasing the number of coefficients retained [53]. This is an important and crucial feature of this method in approaching multi-scale problems and in assessing the importance of kinetic effects, which is not available in PIC or Vlasov methods that are forced to treat the full distribution function.…”