本論文主要的研究目的為利用高解析度之時空守恆法(CESE method)模擬薛丁格方程式以探討自由粒子，單一孤立波和一對孤立波碰撞後的機率分布。薛丁格方程式能解釋粒子在量子狀態下的機率分布，所謂的自由粒子是指粒子在運動過程中不受外力的干擾；孤立波是一種在傳播過程中波形，振幅和速度都能保持不變的波，且兩個孤立波互相碰撞後仍能保持各自的形狀，振幅及速度不變。
時空守恆法是以物理觀念為出發點，利用統御方程式滿足物理通量的守恆定律，在時間和空間上均具有二階準確度的新數值方法。數值解與自由粒子之解析解比較顯示，時空守恆法模擬變化劇烈的機率分布具有相當精確的結果；並且在模擬非線性的兩個孤立波相撞後仍提供合理準確的數值解。此外，本論文還以von Neumann穩定性分析探討時空守恆法在處理線性薛丁格方程式穩定性的參數限制。

In this paper, the purpose of investigation is to apply high accuracy numerical scheme – the “space-time conservation element and solution element (CESE) method” to simulate the wave function of free particle, single soliton and collision of two solitons obeying the Schrödinger equation. The Schrödinger equation explains the wave function of the probability distribution of the quantum phenomena. Free particles move according to the Schrödinger equation without external forces. The characters of a soliton posses invariabilities of wave shape, amplitude and velocity. Even after colliding of two solitons their wave shapes, amplitudes and velocities remain unchanged. The CESE method satisfies physical concept and casts the governing equation in integral form obeyed conservation law, which has a second – order accuracy in both space and time. Comparing the numerical results with the exact solutions for the free particle case, we showed that the CESE method had excellent results in simulating the variation of the wave function. Moreover, the CESE method had been applied to simulate the collision of two solitons. The result provided a reasonable numerical solution. Besides, this study presented the restriction of numerical parameters of the CESE method for solving the Schrödinger equation by using the von Neumann stability analysis.

[1] N. Watanabe, M. Tsukada, “Finite Element Approach for Simulating Quantum Electron Dynamics in a Magnetic Field”, J. Phys. Soc. Japan, 69, 9, 2962, 2000.

[2] J. M. Sanz-Serna, J. G. Verwer, “Conservative and Nonconservative Scheme for the Solution of Nonlinear Schrödinger Equation”, IMA J. Numer. Anal., 6, 25-42, 1986.

[3] J. W. Miles, “An Envelope Soliton Problem”, SIAM. J. Appl. Math., 41, 2, 1981.

[4] Q. Chang, E. Jia, W. sun, “Difference Scheme for Solving the Generalized Nonlinear Schrödinger Equation”, J. Comput. Phys., 148, 397-415, 1999.

[5] Q. Sheng, A. Q. M. Khaliq, E. A. AI-said, ”Solving the Generalized Nonlinear Schrödinger Equation Via Quartic Spline Approximation”, J. Comput. Phys., 166, 400-417, 2001.

[6] S. Yu, S. Zhao, G. W. Wei, “Local Spectral Time Splitting Method for First and Second Order Partial Differential Equation”, J. Comput. Phys., 206, 727-780, 2005.

[7] Y. Xu, C. W. Shu, “Local Discontinuous Galerkin Methods for Nonlinear Schrödinger Equation”, J. Comput. Phys., 205, 72-97, 2005.

[8] İ. Daĝ, “A Quadratic B-Spline Finite Element Method for Solving Nonlinear Schrödinger Equation”, J. Comput. Phys., 174, 247-258, 1999.

[9] A. S. Sehra, “Finite Element Analysis of the Schrödinger Equation”, University of Wales Swansea, 2006.

[10] S. C. Chang, “The Method of Space-Time Conservation Element and Solution Element-a New Approach for Solving the Navire-Stokes and Euler Equation”, J. Comput. Phys., 199, 295-342, 1995.

[11] S. C. Chang, X. Y. Wang, W. M. To, “Application of the Space - Time Conservation Element and Solution Element Method to One- Dimensional Convection-Diffusion Problems”, J. Comput. Phys., 165, 189-215, 2000.

[12] S. T. Yu, S. C. Chang, P. C. E. Jorgenson, S. J. Park, M. C. Lai, “Treating Stiff Source Terms in Conservation Laws by the Space- Time Conservation Element and Solution Element Method”, AIAA, 97-0435.

[13] P. A. Tipler, R. A. Llewellyn, “Modern Physics”, W. H. Freeman and Company, 2003.

[14] 李修良， ”我懂了! 量子力學”， 商周出版社， 2004

[15] 林琦焜， ”孤立波淺談”， 數學傳播， 18卷三期， 2000年3月