A splitting compact finite difference method for computing the dynamics of dipolar Bose–Einstein condensate

ABSTRACT

We numerically study the nonlocal Gross–Pitaevskii equation (NGPE) which describes the dynamics of Bose–Einstein condensates (BEC) with dipole–dipole interaction at extremely low temperature. In preparation for the numerics, first we reformulate the dimensionless NGPE into a Schrödinger–Poisson system. Then, we discretize the three-dimensional Schrödinger–Poisson system in space by a sixth-order compact finite difference method and in time by a splitting technique. By means of three-dimensional discrete fast Sine transform, we develop a fast solver for the resulting discretized system. Finally, we present numerical examples in three dimensions to demonstrate the power of the numerical methods and to discuss some physics of dipolar BEC. The merits of the proposed method for the NGPE are that it is fast and unconditionally stable. Moreover, the method is of spectral-like accuracy in space, and conserves the particle number and the energy of the system in the discretized level.

KEYWORDS:

• Nonlocal Gross–Pitaevskii equation
• dipolar Bose–Einstein condensate
• splitting
• compact finite difference method
• fast Sine transform

CLASSIFICATION:

• 65M06
• 65M22

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

The research of H. Wang is supported in part by the National Natural Science Foundation of China [grant number 11261065 and 91430103].