【作者】 邵和助；

【导师】 汪仲诚；

【作者基本信息】 上海大学 ， 无线电物理， 2010， 博士

【摘要】 本文致力于数值求解Schrodinger方程的差分方法的研究,主要包含针对一维Schrodinger方程的Obrechkoff方法和针对含时Schrodinger方程的时间空间离散方法两部分内容。第一部分是应用Obrechkoff方法数值求解一维Schrodinger方程,在第二章中研究了用Obrechkoff单步法来离散一维Schrodinger方程,利用Mathematica的多精度计算软件包来求得任意精度的数值解,着重探讨了指数拟合方法对于求解Schrodinger方程的束缚态和共振态本征值的精度的影响,数值实验表明对于高能级的共振态,指数拟合Obrechkoff单步法比不拟合的情形在精度和效率上都有很大的提高。第三章中本文提出一种基于Obrechkoff单步法组合的P稳定两步法,其特点是在差分格式中,加入连续高阶微商,而这种结构仍能保持P稳定性,同M. Van Daele和G. Vanden Berghe的方法相比,当两种方法在使用相同的最高阶的微商时,本文提出的P稳定两步法在精度上很大地超越了他们的方法。在第四章中,本文继续第三章关于单步法组合成为两步法的讨论,得到了更高精度的两步方法,进而对其稳定性进行了研究,提出了一种普遍的适用于两步法的相位延迟(phase-lag)公式,对这种新的两步格式进行三角函数拟合得到了相位延迟阶数为无穷的三角函数拟合P稳定两步法,本文应用这种两步法求解Schrodinger方程的数值解,显示出它的精度和效率的优越性。在第二部分,本文关注含时Schrodinger方程的数值解法。文中改进了其他小组特别是W. van Dijk和F. M. Toyama对于含时Schrodinger方程的空间离散方法,在他们的结构上充分加入离散各点波函数的两阶微商,从而将方法的精度从O(h2l)提高到O(h4l),同时采用Pade近似来计算时间演化算符,从而在时间演化算符计算方面可以达到相当高的精度。本文用LU分解来求解对时空离散后得到的含(2l+1)对角矩阵的矩阵方程,从而有效的得到高精度的数值解。更多还原

【Abstract】 This dissertation is devoted to numerical methods for the Schrodinger equation. The following two parts are included:The first is about how to overcome the barrier for a finite difference method of the one-dimensional Schroinger equation defined on the infinite integration interval to obtain the nu-merical solutions accurate than the standard precision in most computing systems of 10-16. In chapter 2, the Obrechkoff one-step method implemented in the multi precision mode is employed to obtain the numerical solutions of the Woods-Saxon potential with errors less than 10-50 and 10-30 for the bound and resonant state, respectively, within a reasonable ef-ficiency. In this dissertation, I also investigate that how to use exponentially-fitting to im-prove the Obrechkoff one-step method in finding the numerical solutions of both the bound and resonant states of the Schroinger equation. The numerical experiments show that the exponentially-fitted method, when the number of fitted coefficients is not so much, can im-prove the precision of the eigenvalues in the bound and resonant state in the lower-order case, nevertheless the advantage in the precision is not so remarkable in the higher-order case. If the coefficients of the Obrechkoff one-step method are full fitted by the exponential function as shown in our work, these methods will surpass the non-fitted Obrechkoff one-step method in accuracy and efficiency considerably for finding out the numerical solutions of the high level resonant states. In chapter 3, a new P-stable Obrechkoff two-step method, which is based on the Obrechkoff one-step method, is presented. Compared to the scheme of M. Van Daele and G. Vanden Berghe, the present one, which contains the high-order derivatives of both even and odd order, can greatly reduce the error. Five numerical examples, which includes the Woods-Saxon potential, the Morse potential, the modified Poschl-Teller potential, the Stiefel-Betis problem, and the Duffing equation, are given to illustrate the performance of this method. Chapter 4 concerns the trigonometrically-fitted two-step method with multi-derivative for the numerical solution to the one-dimensional Shrodinger equation. In this chapter, a general formula of the Phase-lag for the Obrechkoff two-step method is presented.The second is for accurate numerical solutions of the time-dependent Schrodinger equa-tion (TDSE). We present an improved space discretization scheme for the numerical solutions of the TDSE. Compared to the scheme of van Dijk and Toyama,the present one, which con- tains more terms of second-order partial derivatives, can greatly reduce the error resulting from the integration over the space. For a (2l+1)-point formula with (2l+1) terms of second-order partial derivatives, the local truncation error can decrease from the order of (△x)2l to (△x)4l, while the previous one contains only one term of second-order partial derivative. In addition, we employ the high-order Pade approximant for the time evolution operator. Two well-known numerical examples and the corresponding error analysis demonstrate that the present scheme has the advantage in the precision and efficiency over the previous one.更多还原