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

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Rongpei Zhang ◽  
Jia Liu ◽  
Guozhong Zhao
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Numerical Method ◽  
Difference Method ◽  
Three Dimensional ◽  
Gradient Flow ◽  
Pitaevskii Equation ◽  
Compact Finite Difference ◽  
Compact Finite Difference Method ◽  
Representation Technique

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 andrth (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 exponential matrices. 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.

A Stable Compact Finite Difference Method With Skew-Symmetric Form for LES of Variable Density Flows

2011 ◽  
Author(s):  
Kota Takeichi ◽  
Shinji Tamano ◽  
Yohei Morinishi
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Numerical Method ◽  
Difference Method ◽  
Variable Density ◽  
Channel Flows ◽  
Symmetric Form ◽  
Compact Finite Difference ◽  
Compact Finite Difference Method ◽  
Turbulent Channel Flows

The objective of this study is to develop a reliable high-order numerical method for the large eddy simulation (LES) of variable density flows. To improve both the numerical accuracy and stability, we use the compact finite difference method (compact FDM) for the transport equation of compressible flows, in which the skew-symmetric form of the convection term is adopted. In the LES of turbulent flows, the reliability of computational results depends strongly on both the reliability of the subgrid scale (SGS) model and the accuracy of the numerical method. First, the reliability is investigated by performing numerical simulations with unresolved grid resolution for compressible turbulent channel flows without SGS models. Then, we perform the LES of compressible turbulent channel flows with several dynamic and non-dynamic SGS models, and compare turbulence statistics with the corresponding DNS data.

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