Split-step methods for the solution of the nonlinear Schro¨dinger equation
SIAM Journal on Numerical Analysis
Pseudo-spectral solution of nonlinear Schro¨dinger equations
Journal of Computational Physics
Regular Article: Computing Fourier Transforms and Convolutions on the 2-Sphere
Advances in Applied Mathematics
Stable and Efficient Spectral Methods in Unbounded Domains Using Laguerre Functions
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Higher-order split-step Fourier schemes for the generalized nonlinear Schrödinger equation
Mathematics and Computers in Simulation
A Fourth-Order Time-Splitting Laguerre--Hermite Pseudospectral Method for Bose--Einstein Condensates
SIAM Journal on Scientific Computing
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Journal of Computational Physics
Journal of Computational and Applied Mathematics
SIAM Journal on Scientific Computing
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We present a method for numerically solving a Gross-Pitaevskii system of equations with a harmonic and a toroidal external potential that governs the dynamics of one- and two-component Bose-Einstein condensates. The method we develop maintains spectral accuracy by employing Fourier or spherical harmonics in the angular coordinates combined with generalised-Laguerre basis functions in the radial direction. Using an error analysis, we show that the method presented leads to more accurate results than one based on a sine transform in the radial direction when combined with a time-splitting method for integrating the equations forward in time. In contrast to a number of previous studies, no assumptions of radial or cylindrical symmetry is assumed allowing the method to be applied to 2D and 3D time-dependent simulations. This is accomplished by developing an efficient algorithm that accurately performs the generalised-Laguerre transforms of rotating Bose-Einstein condensates for different orders of the Laguerre polynomials. Using this spatial discretisation together with a second order Strang time-splitting method, we illustrate the scheme on a number of 2D and 3D computations of the ground state of a non-rotating and rotating condensate. Comparisons between previously derived theoretical results for these ground state solutions and our numerical computations show excellent agreement for these benchmark problems. The method is further applied to simulate a number of time-dependent problems including the Kelvin-Helmholtz instability in a two-component rotating condensate and the motion of quantised vortices in a 3D condensate.