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Improving the accuracy of the Generalized FDTD-Q scheme for solving the linear time-dependent Schrodinger equation.

机译：提高广义FDTD-Q格式求解线性时变薛定inger方程的准确性。

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摘要

This dissertation improves the accuracy of the Generalized Finite Difference Time Domain (FDTD) scheme by determining a differential operator that is capable of achieving reasonable accuracy when used to obtain even-order derivatives up to order fourteen. The Generalized FDTD scheme is an explicit, scheme used to solve the time-dependent Schrodinger equation, and being an explicit scheme, it must utilize a carefully devised ratio of the temporal step to the spatial step to maintain numerical stability. This ratio is called the mesh ratio, and the Generalized FDTD scheme allows this ratio to be significantly relaxed. As the mesh ratio increases, the generalized scheme requires the evaluation of increasingly high-order spatial derivatives.;In Chapter 3, two classes of differential operators are considered, the first being the repeated application of a central difference approximation of the Laplace operator using various orders of accuracy, and the second class being the differentiated Lagrange interpolating polynomials. This approach intentionally avoids attempting to approximate such derivatives using increasingly high-order finite differences, as the number of uncomputable points becomes very large as the order of the derivative increases.;Based on the conclusions from Chapter 3, a sixth-order accurate central difference operator is chosen to approximate the Laplace operator, and in Chapter 4 the order of accuracy is determined. The numerical stability is analyzed using the Von Neumann analysis and a stability condition is shown.;The validity of the analysis performed in Chapter 4 is verified by solving a Schrodinger equation with exact solution, and observing the numerical error and stability. The order of accuracy of the scheme is also verified through experimentation, it is shown both theoretically and empirically that the chosen differential operator is both stable and accurate when used to solve the time-dependent Schrodinger equation using the Generalized FDTD method.
机译：本文通过确定一种微分算子，提高了FDTD算法的精度。该算子在获得偶数阶导数到14阶时能够达到合理的精度。通用FDTD方案是用于解决与时间相关的Schrodinger方程的显式方案，并且作为显式方案，它必须利用精心设计的时间步长与空间步长之比来保持数值稳定性。该比率称为网格比率，并且通用FDTD方案使该比率显着放松。随着网格比率的增加，广义方案需要评估越来越高阶的空间导数。在第三章中，考虑了两类微分算子，第一类是使用各种拉普拉斯算子的中心差分近似的重复应用。精度等级，第二类是微分的拉格朗日插值多项式。该方法有意避免尝试使用越来越高的有限差分来逼近此类导数，因为随着导数阶数的增加，不可计算的点数变得非常大。;基于第三章的结论，六阶精确中心差选择算子来近似拉普拉斯算子，并在第4章中确定精度的顺序。使用冯·诺依曼分析法对数值稳定性进行了分析，并给出了稳定性条件。第四章中的分析是通过用精确解求解薛定inger方程，并观察数值误差和稳定性来验证的。通过实验也验证了该方案的精度顺序，从理论和经验上都表明，当使用广义FDTD方法求解与时间有关的薛定inger方程时，所选择的微分算子既稳定又准确。

著录项

  • 作者

    Elliott, James John, III.;

  • 作者单位

    Louisiana Tech University.;

  • 授予单位 Louisiana Tech University.;
  • 学科 Applied Mathematics.;Computer Science.
  • 学位 Ph.D.
  • 年度 2011
  • 页码 144 p.
  • 总页数 144
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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