Equilibrium of an Arbitrary Bunch Train with Cavity Resonators and Short Range Wake: Enhanced Iterative Solution with Anderson Acceleration

Abstract

This paper continues the work of two previous treatments of bunch lengthening by a passive harmonic cavity in an electron storage ring. Such cavities, intended to reduce the effect of Touschek scattering, are a feature of fourth generation synchrotron light sources. The charge densities in the equilibrium state are given by solutions of coupled Ha\"issinski equations, which are nonlinear integral equations. If the only wake fields are from cavity resonators, the unknowns can be the Fourier transforms of bunch densities at the resonator frequencies. The solution scheme based on this choice of unknowns proved to be deficient at the design current when multiple resonators were included. Here we return to the conventional formulation of Ha\"issinski equations in coordinate space, the unknowns being charge densities at mesh points on a fine grid. This system would be awkward to solve by the Newton method used previously, because the Jacobian matrix is very large. Here a new solution is described, which is both Jacobian-free and much simpler. It is based on an elementary fixed point iteration, accelerated by Anderson's method. The scheme is notably fast and robust, accommodating even the case of extreme over-stretching at current far beyond the design value. The Anderson method is promising for many problems in accelerator theory and beyond, since it is quite simple and can be used to attack all kinds of nonlinear and linear integral and differential equations. Results are presented for ALS-U, with updated design parameters. The model includes harmonic and main r.f. cavities, compensation of beam loading of the main cavity by adjustment of the generator voltage, and a realistic short range wake field (rather than the broad-band resonator wake invoked previously).