On Variational Fourier Particle Methods

In this article we describe a unifying framework for variational electromagnetic particle schemes of spectral type, and we propose a novel spectral Particle-In-Cell (PIC) scheme that preserves a discrete Hamiltonian structure. Our work is based on a new abstract variational derivation of particle schemes which builds on a de Rham complex where Low’s Lagrangian is discretized using a particle approximation of the distribution function. In this framework, which extends the recent Finite Element based Geometric Electromagnetic PIC (GEMPIC) method to a wide variety of field solvers, the discretization of the electromagnetic potentials and fields is represented by a de Rham sequence of compatible spaces, and the particle-field coupling procedure is described by approximation operators that commute with the differential operators involved in the sequence. For spectral Maxwell solvers these compatible spaces are spanned by a finite number of Fourier modes, and using L² projections as commuting operators leads to the gridless Particle-In-Fourier method which involves exact Fourier coefficients and continuous convolutions in the particle-field coupling. Our new variational PIC method, which we call Fourier-GEMPIC, is obtained by using a new sequence of commuting projections which combine discrete Fourier transforms (DFT), differentiation of particle shape functions and antiderivative filtering operators in Fourier space. As the resulting particle-field coupling essentially involves pointwise evaluations of shape functions and FFT (or DFT) algorithms on a sampling grid this method resembles usual spectral PIC methods, moreover a fully discrete scheme is derived using a directional Hamiltonian splitting procedure. The corresponding time steps are then given in closed form: they preserve the Gauss laws and the discrete Poisson bracket associated with the Hamiltonian structure. These explicit steps are found to share many similarities with a standard spectral PIC method that appears as a Gauss and momentum-preserving variant of the variational method. As arbitrary filters are allowed in our framework, we also discuss aliasing errors and study a natural back-filtering procedure to mitigate the damping caused by anti-aliasing smoothing particle shapes.