求解薛定谔–泊松方程组的时间分裂紧致差分格式
Time-Splitting Compact Difference Scheme for Solving Schrodinger-Poisson Equations
摘要: 本文通过应用高阶紧致差分格式，时间分裂法与Crank-Nicolson法等方法求解非线性薛定谔–泊松方程组。基于快速Sine变换，我们设计了一种求解离散系统的快速算法。我们用该算法分别求解一维、二维、三维的非线性薛定谔–泊松方程组。我们列举具体的数值例子，使用MATLAB软件编写出算法程序，并且通过程序计算近似误差与画出近似解的图像。数值计算结果证实了该算法在空间方向具有谱精度，也证实了该算法的高效性与稳定性。
Abstract: In this paper, we have introduced fourth-order compact finite difference, the time splitting method and the Crank-Nicolson method to solve the nonlinear Schrödinger-Poisson equations. Based on fast Sine transform, we construct a fast solver for the fully discretized system. The presented numerical algorithm has been used to solve one-dimensional, two-dimensional and three-dimensional nonlinear Schrödinger-Poisson equations. We provide specific numerical examples. Through the MATLAB software, we write matlab programs based on the presented numerical algorithm, calcu-late approximated error and draw the approximated numerical solution. The numerical results prove that the presented algorithm has spectral accuracy in space direction. They also confirm its efficiency and stability.
文章引用：姜珊, 刘荣华, 马秀, 王汉权. 求解薛定谔–泊松方程组的时间分裂紧致差分格式[J]. 应用数学进展, 2019, 8(1): 7-25. https://doi.org/10.12677/AAM.2019.81002

参考文献

[1] Anderson, D., et al. (2002) Statistical Effects in the Multistream Model for Quantum Plasmas. Physical Review E, 65, 046417. [Google Scholar] [CrossRef] 
[2] Ben Abdallah, N. (2000) On a Multidimensional Schrödinger-Poisson Scattering Model for Semiconductors. Journal of Mathematical Physics, 41, 4241-4261. [Google Scholar] [CrossRef] 
[3] Shukla, P.K. and Stenflo, L. (2006) Stimulated Scattering Instabilities of Electromagnetic Waves in an Ultracold Quantum Plasma. Physics of Plasmas, 13, Article ID: 044505. [Google Scholar] [CrossRef] 
[4] Shukla, P.K. and Eliasson, B. (2007) Nonlinear Interactions between Electromagnetic Waves and Electron Plasma Oscillations in Quantum Plasmas. Physical Review Letters, 99, Article ID: 096401.
[5] Ashcroft, N.W. and Mermin, N.D. (1976) Solid State Physics. Saunders College Publishing, Orlan-do.
[6] Lange, H., Toomire, B. and Zweifel, P.F. (1995) An Overview of Schrödinger-Poisson Problems. Reports on Mathematical Physics, 36, 331-345. [Google Scholar] [CrossRef] 
[7] Markowich, P.A., Ringhofer, C.A. and Schmeiser, C. (1990) Semiconductor Equations. Springer, Berlin. [Google Scholar] [CrossRef] 
[8] Davydov, A.S. (1979) Solitons in Molecular Systems. Physica Scripta, 20, 387-394. [Google Scholar] [CrossRef] 
[9] Cheng, C., Liu, Q., Lee, J. and Massoud, H.Z. (2004) Spectral Element Method for the Schrödinger-Poisson System. Journal of Computational Electronics, 3, 417-421. [Google Scholar] [CrossRef] 
[10] Dong, X. (2011) A Short Note on Simplified Pseudospectral Methods for Computing Ground State and Dynamics of Spherically Symmetric Schrödinger-Poisson-Slater System. Journal of Computational Physics, 230, 7917-7922. [Google Scholar] [CrossRef] 
[11] Shaikh, D. and Shukla, P.K. (2008) 3D Electron Fluid Turbulence at Nanoscales in Dense Plasmas. New Journal of Physics, 10, Article ID: 083007. [Google Scholar] [CrossRef] 
[12] Mauser, N.J. and Zhang, Y. (2014) Exact Artificial Boundary Condition for the Poisson Equation in the Simulation of the 2D Schrödinger-Poisson System. Computer Physics Communications, 16, 764-780. [Google Scholar] [CrossRef] 
[13] Moroz, I., Penrose, R. and Tod, P. (1998) Spherical-ly-Symmetric Solutions of the Schrödinger-Newton Equations. Classical and Quantum Gravity, 15, C2733-C2742. [Google Scholar] [CrossRef] 
[14] Brezzi, F. and Markowich, P.A. (1991) The Three-Dimensional Wigner-Poisson Problem: Existence, Uniqueness and Approximation. Mathematical Methods in the Applied Sciences, 14, 35-61. [Google Scholar] [CrossRef] 
[15] Castella, F. (1997) L2 Solutions to the Schrödinger-Poisson System: Existence, Uniqueness, Time Behavior, and Smoothing Effects. Mathematical Models and Methods in Applied Sciences, 7, 1051-1083. [Google Scholar] [CrossRef] 
[16] Bao, W. and Cai, Y. (2013) Mathematical Theory and Numerical Methods for Bose-Einstein Condensation. Kinetic and Related Models, 6, 1-135. [Google Scholar] [CrossRef] 
[17] Bao, W., Mauser, N.J. and Stimming, H.P. (2003) Effective One Particle Quantum Dynamics of Electrons: A Numerical Study of the Schrödinger-Poisson- Model. Communications in Mathematical Sciences, 1, 809-831. [Google Scholar] [CrossRef] 
[18] Haas, F., Manfredi, G. and Feix, M. (2000) Multistream Model for Quantum Plasmas. Physical Review B, 62, 2763. [Google Scholar] [CrossRef] 
[19] Manfredi, G., Haas, F. (2001) Self-Consistent Fluid Model for a Quantum Electron Gas. Physical Review B, 64, 075316. [Google Scholar] [CrossRef] 
[20] 王芳芳. 紧致差分格式的理论及其分析[D]: [硕士学位论文]. 沈阳: 东北大学, 2010: 5-17.
[21] Zhang, Y. (2013) Optimal Error Estimates of Compact Finite Difference Discretizations for the Schrödinger-Poisson System. Communications in Computational Physics, 13, 1357-1388. [Google Scholar] [CrossRef] 
[22] 张勇. 薛定谔–泊松方程组的数值计算和分析及其应用[D]: [博士学位论文]. 北京: 清华大学, 2012: 2-23.
[23] Lele, S. (1992) Compact Finite Difference Scheme with Spectral-Like Resolution. Journal of Computational Physics, 103, 16-42.
[24] Wang, Y. and Lu, X. (2014) Modulational Instability of Electrostatic Acoustic Waves in an Electron-Hole Semiconductor Quantum Plasma. Physics of Plasmas, 21, Article ID: 022107. [Google Scholar] [CrossRef] 
[25] Wang, H., Zhang, Y., Ma, X., Qiu, J. and Liang, Y. (2016) An Efficient Implementation of Fourth-Order Compact Finite Difference Scheme for Poisson Equation with Dirichlet Boundary Conditions. Computers & Mathematics with Applications, 71, C1843-C1860. [Google Scholar] [CrossRef] 

为你推荐