  A fourth-order time-splitting Laguerre-Hermite pseudospectral method is
introduced for Bose-Einstein condensates (BEC) in 3-D with cylindrical
symmetry.
  The method is explicit, unconditionally stable, time reversible and time
transverse invariant. It conserves the position density, and is spectral
accurate in space and fourth-order accurate in time. Moreover, the new method
has two other important advantages: (i) it reduces a 3-D problem with
cylindrical symmetry to an effective 2-D problem; (ii) it solves the problem in
the whole space instead of in a truncated artificial computational domain. The
method is applied to vector Gross-Pitaevskii equations (VGPEs) for
multi-component BECs. Extensive numerical tests are presented for 1-D GPE, 2-D
GPE with radial symmetry, 3-D GPE with cylindrical symmetry as well as 3-D
VGPEs for two-component BECs to show the efficiency and accuracy of the new
numerical method.
