Efficient Fourier Basis Particle Simulation

The standard particle-in-cell algorithm suffers from finite grid errors which break energy conservation, cause numerical dispersion, and create numerical instabilities. There exists a gridless alternative which bypasses the deposition step and calculates each Fourier mode of the charge density directly from the particle positions. We show that a gridless method can be computed efficiently through the use of an Unequally Spaced Fast Fourier Transform (USFFT) algorithm. After a spectral field solve, the forces on the particles are calculated via the inverse USFFT (a rapid solution of an approximate linear system). We provide a one dimensional implementation of this algorithm with an asymptotic runtime of $O(N_p + N_m \log N_m)$ for each iteration, identical to the standard PIC algorithm (where $N_p$ is the number of particles and $N_m$ is the number of Fourier modes). Higher dimensional formulations would scale as $O(N_p + N_m^D \log N_m^D)$, where $D$ is the spatial dimensionality of the problem. We demonstrate superior energy conservation and reduced noise, as well as convergence of the energy conservation at small time steps.