Weight growth due to resonant simulation particles and a modified δf algorithm with smooth switching between δf and total- f methods

When applying the standard δf particle-in-cell simulation method to simulate linear and nonlinear collective instabilities with coherent structures, wave-particle interaction may result in large weight growth for resonant or nearly resonant simulation particles. In this paper, we demonstrate that the large noise associated with the large weight of nearly resonant simulation particles can produce significant error fields at the nonlinear stage of the instability. To overcome this deleterious effect, we have developed a modified δf method that contains a smooth switching algorithm between the δf and total- f methods. Before the switch, the simulation effectively makes use of the desirable low-noise feature of the δf method for small weight to accurately follow unstable mode structures. When the weight function becomes large during the nonlinear phase, the low-noise advantage of the δf method ceases to be significant and the simulation is switched to the total- f method to avoid the large noise induced by nearly resonant simulation particles. This algorithm has been successfully applied to simulation studies of the electrostatic Harris instability driven by large temperature anisotropy in high-intensity charged particle beams typical of applications in high current accelerators, including high-energy density physics and heavy ion fusion.