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
Bose-Einstein condensates (BECs) were first experimentally observed in 1995 [
One of the fundamental problems in numerical simulation of BECs is to find its ground state so as to compare the numerical results with experimental observations and to prepare initial data for studying the dynamics of BEC. To find a stationary solution of (
From mathematical point of view, the ground state
In recent years, a great deal of approaches have been proposed for computing the ground state. In fact, these approaches were divided into two types: one method is either based on solving the nonlinear eigenvalue problem [
The imaginary time method has been used extensively by the physics community and has proved to be powerful. In this work we choose this approach for computing the ground and excited states of BEC. There are many methods discretizing the normalized gradient flow in the imaginary time. These methods include spectral (pseudospectral) methods [
Recently, there has been growing interest in high-order compact methods for solving Gross-Pitaevskii equation [
The backward Euler and Crank-Nicolson method were usually applied in the time discretization. However large global nonlinear systems need to be solved at each time step. Therefore, number of operation for the nonlinear scheme may be large. Besides that, these time integration methods are limited to second-order accuracy. In this paper, we use integration factor method for time discretization in which an array representation for the linear differential operators is introduced [
Finally we combine the compact finite difference method in space and array-representation integration factor method in time to get the CFDM-AIF method for solving the GFDN. This method yields fourth-order accuracy in space and
In this section, we first review the compact finite difference method. Then we will propose the array-representation integration factor method for fully discretizing the GFDN. We will construct the scheme in 3D and the procedure in 2D will follow the similar procedure.
To build the compact finite difference method for the gradient flow equation (
Setting
By using a Taylor expansion, we get
In this subsection, we will give the time discretization method for the ODEs (
In a typical representation of the linear differential operator, the matrix
We define the matrices
Because the linear mapping
For the 2D case the second order CFDM-AIF scheme is similar as (
In this section, we apply the CFDM-AIF scheme to compute the ground and excited states of BEC with harmonic-plus-optical lattice potential in two and three dimensions. In our computation, the stationary state is reached when
The CFDM-AIF method (
We consider the following equation on
Error, order of accuracy, and CPU time with CFDM-AIF2 and RK2 scheme for Example
|
CFDM-AIF2, |
RK2, |
||||
---|---|---|---|---|---|---|
|
Order | CPU (s) |
|
Order | CPU (s) | |
|
5.50 × 10−5 | — | 0.05 | 6.95 × 10−3 | — | 0.09 |
32 × 32 × 32 | 3.59 × 10−6 | 3.94 | 0.43 | 1.74 × 10−3 | 2.00 | 2.06 |
64 × 64 × 64 | 2.65 × 10−7 | 3.76 | 5.93 | 4.36 × 10−4 | 2.00 | 61.9 |
128 × 128 × 128 | 2.69 × 10−8 | 3.30 | 127 | 1.09 × 10−4 | 2.00 | 2448 |
We compute the ground and first excited states with different parameters in the harmonic-plus-optical lattice potential by our method. The potential function in (
We first compute the ground state with different parameters
For fixed
The ground state (top row) and time evolution of energy (bottom row) with fixed
The ground state with fixed
Then we compute the first excited state by taking
The first excited states in
Finally, we apply CFDM-AIF method to compute the ground and first excited states in 3D under a combined harmonic and optical lattice potential; that is,
Figure
The isosurface plots of ground state in Example
The isosurface plots of first excited state in Example
In this paper we have presented the CFDM-AIF method for computing the ground and first excited states in BECs. The method is based on the compact finite difference method in space and array-representation integration factor method in time. This method can reduce the computational cost significantly in both storage and CPUs and is an efficient and attractive method for 2D and 3D GPE.
The authors declare that there is no conflict of interests regarding the publication of this paper.
Rongpei Zhang and Guozhong Zhao’s work was supported by the National Nature Science Foundation of China (11261035). The authors are grateful to the referees for their invaluable suggestions and comments regarding the improvement of the paper.