Techniques for Robust Nonlinear Delta-F Simulations of Beams

We describe means by which the range of applicability of nonlinear-characteristic δf methods to beams may be enhanced, so as to faithfully describe a sharp-edged beam or a smooth beam whose edge moves by more than its scale length. As others have done, we follow a population of Lagrangian characteristic “marker” particles in the total (equilibrium plus perturbation) field. However, in contrast with usual practice, our marker distribution is not proportional to the physical particle distribution. We introduce “ghost” particles: a population of markers loaded into regions of phase space where the equilibrium f0 is zero or very small. In addition, we do not numerically evolve either δf or a “weight” w, but rather we use knowledge of the marker positions in phase space and of the functional form of f0 to evaluate δf anew at each timestep for each marker. We describe the application of our formalism to the model problem of an oscillating displaced beam. We show that marker loading with phasespace density proportional to f0 leads to inconsistencies, while our modified marker loading is consistent and (for modest displacements) can be statistically “quieter” than conventional particle-in-cell simulation.