Fluid preconditioning for Newton–Krylov-based, fully implicit, electrostatic particle-in-cell simulations

Abstract

A recent proof-of-principle study proposes an energy-and charge-conserving, nonlinearly implicit electrostatic particle-in-cell (PIC) algorithm in one dimension [Chen et al, J. Comput. Phys., 230 (2011) 7018].The algorithm in the reference employs an unpreconditioned Jacobian-free Newton-Krylov method, which ensures nonlinear convergence at every timestep (resolving the dynamical timescale of interest). Kinetic enslavement, which is one key component of the algorithm, not only enables fully implicit PIC a practical approach, but also allows preconditioning the kinetic solver with a fluid approximation. This study proposes such a preconditioner, in which the linearized moment equations are closed with moments computed from particles. Effective acceleration of the linear GMRES solve is demonstrated, on both uniform and non-uniform meshes. The algorithm performance is largely insensitive to the electron-ion mass ratio.Numerical experiments are performed on a 1D multi-scale ion acoustic wave test problem.

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

“…Of course, the grid spacing introduces a numerical error in the current accumulation and interpolation routines. Finally, although a preconditioned version of the fully-implicit PIC has been presented in [60], we use the unpreconditioned version here, so that both method are unpreconditioned. Since the focus of this work is on the discretization in velocity space, i.e.…”

“…Using these techniques, the numerical stability is greatly improved (with respect to explicit PIC), but energy is still not conserved exactly and, at each time step, there is a small inconsistency between the charge and current densities calculated from the particles and the one that is used for advancing the fields. Some authors have suggested that such limitations are responsible for the accumulation of numerical errors that preclude semi-implicit PIC simulations to run for long time intervals [54,60]. Recently, however, [61] and [54] have formulated and successfully implemented a fully-implicit, one-dimensional PIC code (see also [62] for an electromagnetic extension).…”

A curvilinear, fully implicit, conservative electromagnetic PIC algorithm in multiple dimensions

“…The first one is a piston-driven 1D slow magnetosonic wave shock problem, with a grid packed near the piston. Unlike earlier studies in implicit PIC algorithms [41,43,32,1], this example features an open system with non-periodic boundary conditions (inflow and perfect conductor). The second one is a 2D Weibel instability on a fully curvilinear grid based on a sinusoidal map deformation.…”

Implicit Temporal Discretization and Exact Energy Conservation for Particle Methods Applied to the Poisson–Boltzmann Equation

“…The main advantage of this method is the reduction of the memory requirements of the Krylov method because the particles have been brought out of the Krylov loop and only the filed quantities matter when computing the memory requirement. This approach is especially suitable to hybrid architectures (Chen et al, 2012) and can be made most competitive when fluid-based preconditioning is used Chen et al (2014). However, in the present case of a low dimensionality problem run on standard CPU computers, particle hiding is neither needed nor competitive.…”