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A fourth-order time-splitting Laguerre--Hermite pseudospectral method is introduced for Bose--Einstein condensates (BECs) in three dimensions with cylindrical symmetry. The method is explicit, 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 three-dimensional (3-D) problem with cylindrical symmetry to an effective two-dimensional (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 multicomponent BECs. Extensive numerical tests are presented for the one-dimensional (1-D) GPE, the 2-D GPE with radial symmetry, the 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.

A Fourth-Order Time-Splitting Laguerre--Hermite Pseudospectral Method for Bose--Einstein Condensates