SLAC, Menlo Park, California
UNM, Albuquerque, New Mexico
The goal is to construct a symplectic evolution map for a large section of an accelerator, say a full turn of a large ring or a long wiggler. We start with an accurate tracking algorithm for single particles, which is allowed to be slightly non-symplectic. By tracking many particles for a distance S one acquires sufficient data to construct the mixed-variable generator of a symplectic map for evolution over S. Two ways to find the generator are considered: (i) Find its gradient from tracking data, then the generator itself as a line integral *. (ii) Compute Hamilton's principal function on many orbits. The generator is given finally as an interpolatory C2 function, say through B-splines or Shepard's meshless interpolation. A test of method (i) is given in a hard example: a full turn map for an electron ring with strong sextupoles. The method succeeds where Taylor maps fail, but there are technical difficulties near the dynamic aperture due to oddly shaped interpolation domains. Method (ii) looks more promising in strongly nonlinear cases. We also explore explicit maps from direct fits of tracking data, with symplecticity imposed on local interpolating functions.
The University of Liverpool, Liverpool
UNM, Albuquerque, New Mexico
We consider a 2D bunch represented by \mathcal N simulation particles moving on arbitrary planar orbits. The mean field of the bunch is computed from Maxwell's equations in the lab frame with a smoothed charge/current density, using retarded potentials. The particles are tracked in beam frame, thus requiring a transformation of densities from lab to beam frame. We seek improvements in speed and practicality in two directions: (a) choice of integration variables and quadrature rules for the field calculation; and (b) finding smooth densities from scattered data. For item (a) we compare a singularity-free formula with the retarded time as integration variable, which we used previously, with a formula based on Frenet-Serret coordinates. The latter suggests good approximations in different regions of the retardation distance, for instance a multipole expansion which could save both time and storage. For item (b) we compare various ideas from mathematical statistics and numerical analysis, e.g., quasi-random vs. pseudo-random sampling, Fourier vs. kernel smoothing, etc. Implementations in a parallel code with \mathcal N up to a billion will be given, for a chicane bunch compressor.