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提出了一种新的算法高阶辛时域有限差分法(SFDTD(3, 4): symplectic finite-difference time-domain)求解含时薛定谔方程.在时间上采用三阶辛积分格式离散, 空间上采用四阶精度的同位差分格式离散, 建立了求解含时薛定谔方程的高阶离散辛框架;探讨了高阶辛算法的稳定性及数值色散性.通过理论上的分析及数值算例表明:当空间采用高阶同位差分格式时, 辛积分可提高算法的稳定度;SFDTD(3, 4)法和FDTD(2, 4)法较传统的FDTD(2, 2)法数值色散性明显改善.对二维量子阱和谐振子的仿真结果表明: SFDTD(3, 4)法较传统的FDTD(2, 2)法及高阶FDTD(2, 4)法有着更好的计算精度和收敛性, 且SFDTD(3, 4)法能够保持量子系统的能量守恒, 适用于长时间仿真.Using three-order symplectic integrators and fourth-order collocated spatial differences, a high-order symplectic finite-difference time-domain (SFDTD(3, 4)) scheme is proposed to solve the time-dependent Schrdinger equation. First, high-order symplectic framework for discretizing the Schrdinger equation is described. The numerical stability and dispersion analyses are provided for the FDTD(2, 2), FDTD(2, 4) and SFDTD(3, 4) schemes. The results are demonstrated in terms of theoretical analyses and numerical simulations. The spatial high-order collocated difference reduces the stability that can be improved by the high-order symplectic integrators. The SFDTD(3, 4) scheme and FDTD(2, 4) approach show better numerical dispersion than the traditional FDTD(2, 2) method. The simulation results of a two-dimensional quantum well and harmonic oscillator strongly confirm the advantages of the SFDTD(3, 4) scheme over the traditional FDTD(2, 2) method and other high-order approaches. The explicit SFDTD(3, 4) scheme, which is high-order-accurate and energy-conserving, is well suited for long-term simulation.
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Keywords:
- symplectic integrator /
- high-order collocated difference /
- Schrdinger equation /
- numerical stability and dispersion
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[16] Wang T C, Nie T, Zhang L M 2009 J. Comput. Appl. Math. 231 745
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[18] Taflove A, Hagness S C 2005 Computational Electrodynamics: the Finite-Difference Time-Domain Method (3rd Ed.) (Boston: Artech House)
[19] Yoshida H 1990 Phys. Lett. A 150 262
[20] Sha W, Huang Z X, Chen M S, Wu X L 2008 IEEE Trans. Antennas Propag. 56 493
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[1] Datta S 2005 Quantum Transport: Atom to Transistor (New York: Cambridge University Press)
[2] Griffiths D J 2004 Introduction to Quantum Mechanics (Second Edition Addison-Wesley) (Boston)
[3] Joe Y S, Satanin A M, Kim C S 2006 Phys. Scr. 74 259
[4] Soriano A, Navarro E A, Porti J A, Such V 2004 J. Appl. Phys. 95 8011
[5] Sullivan D M, Citrin D S 2005 J. Appl. Phys. p97
[6] Sanz-Serna J M, Calvo M P 1994 Numerical Hamiltonian Problems (London: Chapman & Hall)
[7] Wen G Y 1999 Journal of Microwave 15(1) 68 (in Chinese) [文舸一 1999 微波学报 15(1) 68]
[8] Feng K 2003 Symplectic Geometric Algorithms for Hamiltonian Systems (Hangzhou: Zhejiang Science and Technology Publishing House) p358-359 (in Chinese) [冯康 2003 哈密尔顿系统的辛几何算法(杭州:浙江科学技术出版社) 第358—359页]
[9] Gray S K, Manolopoulos D E 1996 J. Chem. Phys. 104 7099
[10] Liu X Y, Ding P Z, Hong J L, Wang L J 2005 Comput. Math. Appl. 50 637
[11] Blanes S, Casas F, Murua A 2006 J. Chem. Phys. p124
[12] Chin S A, Chen C R 2002 J. Chem. Phys. 117 1409
[13] Liu X S, Liu X Y, Zhou Z Y, Ding P Z, Pan S F 2000 Int. J. Quantum. Chem. 79 343
[14] Monovasilis T, Kalogiratou Z, Simos T E 2008 Phys. Lett. A 372 569
[15] Islas A L, Karpeev D A, Schober C M 2001 J. Comput. Phys. 173 116
[16] Wang T C, Nie T, Zhang L M 2009 J. Comput. Appl. Math. 231 745
[17] Sullivan D M 2000 Electromagnetic Simulation Using the FDTD Method (New York: IEEE Press)
[18] Taflove A, Hagness S C 2005 Computational Electrodynamics: the Finite-Difference Time-Domain Method (3rd Ed.) (Boston: Artech House)
[19] Yoshida H 1990 Phys. Lett. A 150 262
[20] Sha W, Huang Z X, Chen M S, Wu X L 2008 IEEE Trans. Antennas Propag. 56 493
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