An Efficient Compact Finite Difference Method for the Solution of the Gross-Pitaevskii Equation

Abstract EN

We present an efficient, unconditionally stable, and accurate numerical method for the solution of the Gross-Pitaevskii equation.

We begin with an introduction on the gradient flow with discrete normalization (GFDN) for computing stationary states of a nonconvex minimization problem.

Then we present a new numerical method, CFDM-AIF method, which combines compact finite difference method (CFDM) in space and array-representation integration factor (AIF) method in time.

The key features of our methods are as follows: (i) the fourth-order accuracy in space and rth ( r ≥ 2 ) accuracy in time which can be achieved and (ii) the significant reduction of storage and CPU cost because of array-representation technique for efficient handling of exponentialmatrices.

The CFDM-AIF method is implemented to investigate the ground and first excited state solutions of the Gross-Pitaevskii equation in two-dimensional (2D) and three-dimensional (3D) Bose-Einstein condensates (BECs).

Numerical results are presented to demonstrate the validity, accuracy, and efficiency of the CFDM-AIF method.