On FFT-based convolutions and correlations, with application to solving   Poisson's equation in an open rectangular pipe
  A new method is presented for solving Poisson's equation inside an open-ended rectangular pipe. The method uses Fast Fourier Transforms (FFTs) to perform mixed convolutions and correlations of the charge density with the Green function. Descriptions are provided for algorithms based on the ordinary Green function and for an integrated Green function (IGF). Due to its similarity to the widely used Hockney algorithm for solving Poisson's equation in free space, this capability can be easily implemented in many existing particle-in-cell beam dynamics codes. 
Robert D. Ryne
physics.acc-ph physics.comp-ph