Particle Simulation in Fourier Space

The standard particle-in-cell algorithm suffers from finite grid effects which break energy conservation, cause numerical dispersion, and create numerical instabilities. We present a gridless alternative, bypassing the deposition step and calculating each Fourier mode of the charge density directly from the particle positions. This can be done efficiently through the use of an Unequally Spaced Fast Fourier Transform (USFFT) algorithm [1,2]. After a spectral field solve, the forces on the particles are calculated via the inverse USFFT (a rapid solution of an approximate linear system). The asymptotic runtime of this approach is 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 modes). We provide implementations of this algorithm and apply them to several test cases, demonstrating comparable performance and identical scaling, as well as superior energy conservation and elimination of the finite grid instability. We prove energy conservation in the continuous-time limit, as well as momentum conservation.

[1] G. Beylkin, Applied and Computational Harmonic Analysis 2, 363 (1995).

[2] A. Dutt and V. Rokhlin, SIAM J. Sci. Comput. 14, 1368 (1993).