PPPS-2013: Topic 1.2: Numerical stability of the pseudospectral EM PIC algorithm

Abstract

Summary form only given. Pseudo-spectral methods, which advance fields in Fourier space, offer a number of advantages over more common Finite Difference Time Domain (FDTD) PIC algorithms. In particular, Haber's Pseudo-Spectral Analytical Time-Domain (PSATD) algorithm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> exhibits dispersion-free propagation and no Courant limit in vacuum. Moreover, Vay's recent research <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> suggests that it can be parallelized about as well as FDTD algorithms. As the research presented here shows, it also has superior numerical stability properties, at least for relativistic beam transport. Typically the most serious numerical instability in PIC simulations of particle beams is the numerical Cherenkov instability <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> , arising from coupling between electromagnetic and nonphysical beam modes. Due to its improved dispersion properties, the PSATD algorithm is less prone to this instability. Nonetheless, the instability still occurs, either directly for time steps larger than the FDTD Courant limit or through nonphysical beam mode aliases at any time step. In this talk we derive the PSATD numerical dispersion relation, present illustrative numerical solutions of it, and compare them with simulation results from the WARP-FFT PIC code. Additionally, we compare these findings with our recent analysis of numerical instabilities in FDTD PIC beam codes <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> and, as time permits, with stability properties of other pseudo-spectral algorithms.