麦克斯韦方程几种时域有限差分算法能量范数守恒研究

发布时间：2018-06-16 16:34

本文选题：麦克斯韦方程 + 时域有限差分法(FDTD)　；参考：《大连理工大学》2015年博士论文

【摘要】：在电磁场数值算法中,时域有限差分法(Finite-Difference Time-Domain(FDTD))在电磁场的各个相关领域中具有重要的实际应用背景和理论探讨价值.用能量范数分析时域有限差分法的各种性质一直是国内外学者重视的课题.本文以周期边界条件下的交替方向隐式时域有限差分法(Alternating Direction Implicit Finite-Difference Time-Domain (ADI-FDTD) method)和分裂算子时域有限差分法(Splitting Finite-Difference Time-Domain (S-FDTD))为研究对象,用能量范数分析算法性质,主要取得以下成果：1.第二章分别对二维和三维麦克斯韦方程组在矩形域周期边界条件下建立了H1和H2范数的意义下的能量范数恒等式.2.第三章对应用在矩形域周期边界条件下的二维ADI-FDTD算法,推导出了离散H1范数意义下的能量范数恒等式.在能量范数恒等式基础上证明了二维ADI-FDTD算法的无条件稳定性和超收敛性.3.第四章对应用在矩形域周期边界条件下的三维ADI-FDTD算法,推导出了离散H1和H2范数意义下的能量范数恒等式.在能量范数恒等式基础上证明了三维ADI-FDTD算法的无条件稳定性和超收敛性.数值算例表明三维ADI-FDT D算法在离散L2,H1和H2范数意义下的能量范数保持稳定并且收敛阶约为2.4.第五章对应用在矩形域周期边界条件下的二维S-FDTD算法,推导出了离散H1和H2范数意义下的能量范数恒等式.在能量范数恒等式基础上证明了二维S-FDTD算法的无条件稳定性和超收敛性.数值算例表明二维ADI-FDTD算法和二维S-FDTD算法在离散L2,H1和H2范数意义下的能量范数保持稳定并且收敛阶约为2.5.第六章在FDTD算法和高阶Taylor法的基础上推出了一种自适应时间步长时域有限差分法(Adaptive Time Step Finite-Difference Time-Domain (ATS-FDTD)).这种算法的时间步长、空间步长及泰勒展开的阶数由一个基于稳定性分析的准则产生.数值算例显示,与ADI-FDTD算法相比,ATS-FDTD算法在离散L2范数意义下的能量范数稳定,绝对误差小,收敛阶高,数值散度小.
[Abstract]:Finite-Difference Time-Domain method (Finite-Difference FDTD) has important practical application background and theoretical value in various fields of electromagnetic field. Energy norm analysis of various properties of finite-difference time-domain method has been paid attention to by domestic and foreign scholars. In this paper, the alternating direction implicit finite difference time-domain finite difference method (Alternating Direction implicit Finite-Difference Time-Domain / ADI-FDTD) method and split-operator finite-difference time-domain method (Splitting Finite-Difference Time-Domain S-FDTDG) are used to analyze the properties of the algorithm with energy norm, and the following results are obtained: 1. In chapter 2, the energy norm identities of H _ 1 and H _ 2 norm in the sense of H _ 1 and H _ 2 norm are established for two-dimensional and three-dimensional Maxwell equations respectively under the periodic boundary condition in rectangular domain. In chapter 3, the energy norm identity in the sense of discrete H _ 1 norm is derived for the two-dimensional ADI-FDTD algorithm, which is applied to the periodic boundary condition in a rectangular region. Based on the energy norm identity, the unconditional stability and superconvergence of the two-dimensional ADI-FDTD algorithm are proved. In chapter 4, the energy norm identities in the sense of discrete H _ 1 and H _ 2 norm are derived for the 3D ADI-FDTD algorithm which is applied to the periodic boundary condition in rectangular domain. Based on the energy norm identity, the unconditional stability and superconvergence of 3D ADI-FDTD algorithm are proved. Numerical examples show that the energy norm of 3D ADI-FDT algorithm is stable and convergent order is about 2.4 in the sense of discrete L _ 2H _ 1 and H _ 2 norm. In chapter 5, the energy norm identities in the sense of H _ 1 and H _ 2 norm are derived for the 2-D S-FDTD algorithm which is applied to the periodic boundary condition in rectangular domain. Based on the energy norm identity, the unconditional stability and superconvergence of the 2-D S-FDTD algorithm are proved. Numerical examples show that the energy norm of the two-dimensional ADI-FDTD algorithm and the two-dimensional S-FDTD algorithm are stable and convergent order is about 2.5 in the sense of discrete L _ 2H _ 1 and H _ 2 norm. In chapter 6, based on the FDTD algorithm and the higher order Taylor method, an adaptive time step finite-difference Time-Domain (ATS-FDTD) method is derived. The time step size, space step size and Taylor expansion order of the algorithm are generated by a criterion based on stability analysis. Numerical examples show that the energy norm of ATS-FDTD algorithm is stable, the absolute error is small, the convergence order is high, and the numerical divergence is small compared with the ADI-FDTD algorithm in the sense of discrete L2 norm.
【学位授予单位】：大连理工大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O241.82

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