Efficient and spectrally accurate numerical methods for computing ground and first excited states in Bose–Einstein condensates

In this paper, we present two efficient and spectrally accurate numerical methods for computing the ground and first excited states in Bose–Einstein condensates (BECs). We begin with a review on the gradient flow with discrete normalization (GFDN) for computing stationary states of a nonconvex minimization problem and show how to choose initial data effectively for the GFDN. For discretizing the gradient flow, we use sine-pseudospectral method for spatial derivatives and either backward Euler scheme (BESP) or backward/forward Euler schemes for linear/nonlinear terms (BFSP) for temporal derivatives. Both BESP and BFSP are spectral order accurate for computing the ground and first excited states in BEC. Of course, they have their own advantages: (i) for linear case, BESP is energy diminishing for any time step size where BFSP is energy diminishing under a constraint on the time step size; (ii) at every time step, the linear system in BFSP can be solved directly via fast sine transform (FST) and thus it is extremely efficient, and in BESP it needs to be solved iteratively via FST by introducing a stabilization term and thus it could be efficient too. Comparisons between BESP and BFSP as well as other existing numerical methods are reported in terms of accuracy and total computational time. Our numerical results show that both BESP and BFSP are much more accurate and efficient than those existing numerical methods in the literature. Finally our new numerical methods are applied to compute the ground and first excited states in BEC in one dimension (1D), 2D and 3D with a combined harmonic and optical lattice potential for demonstrating their efficiency and high resolution.