An adaptive algorithm for n-body field expansions
  An expansion of a density field or particle distribution in basis functions which solve the Poisson equation both provides an easily parallelized n-body force algorithm and simplifies perturbation theories. The expansion converges quickly and provides the highest computational advantage if the lowest-order potential-density pair in the basis looks like the unperturbed galaxy or stellar system. Unfortunately, there are only a handful of such basis in the literature which limits this advantage. This paper presents an algorithm for deriving these bases to match a wide variety of galaxy models. The method is based on efficient numerical solution of the Sturm-Liouville equation and can be used for any geometry with a separable Laplacian. Two cases are described in detail. First for the spherical case, the lowest order basis function pair may be chosen to be exactly that of the underlying model. The profile may be cuspy or have a core and truncated or of infinite extent. Secondly, the method yields a three-dimensional cylindrical basis appropriate for studying galaxian disks. In this case, the vertical and radial bases are coupled; the lowest order radial part of the basis function can be chosen to match the underlying profile only in the disk plane. Practically, this basis is still a very good match to the overall disk profile and converges in a small number of terms. 
Martin D. Weinberg (Dept. of Physics & Astronomy, University of   Massachuestts, Amherst, MA, USA)