本文選題：麥克斯韋方程 + 時域有限差分法(FDTD)�。� 參考：《大連理工大學(xué)》2015年博士論文

【摘要】：在電磁場數(shù)值算法中,時域有限差分法(Finite-Difference Time-Domain(FDTD))在電磁場的各個相關(guān)領(lǐng)域中具有重要的實際應(yīng)用背景和理論探討價值.用能量范數(shù)分析時域有限差分法的各種性質(zhì)一直是國內(nèi)外學(xué)者重視的課題.本文以周期邊界條件下的交替方向隱式時域有限差分法(Alternating Direction Implicit Finite-Difference Time-Domain (ADI-FDTD) method)和分裂算子時域有限差分法(Splitting Finite-Difference Time-Domain (S-FDTD))為研究對象,用能量范數(shù)分析算法性質(zhì),主要取得以下成果：1.第二章分別對二維和三維麥克斯韋方程組在矩形域周期邊界條件下建立了H1和H2范數(shù)的意義下的能量范數(shù)恒等式.2.第三章對應(yīng)用在矩形域周期邊界條件下的二維ADI-FDTD算法,推導(dǎo)出了離散H1范數(shù)意義下的能量范數(shù)恒等式.在能量范數(shù)恒等式基礎(chǔ)上證明了二維ADI-FDTD算法的無條件穩(wěn)定性和超收斂性.3.第四章對應(yīng)用在矩形域周期邊界條件下的三維ADI-FDTD算法,推導(dǎo)出了離散H1和H2范數(shù)意義下的能量范數(shù)恒等式.在能量范數(shù)恒等式基礎(chǔ)上證明了三維ADI-FDTD算法的無條件穩(wěn)定性和超收斂性.數(shù)值算例表明三維ADI-FDT D算法在離散L2,H1和H2范數(shù)意義下的能量范數(shù)保持穩(wěn)定并且收斂階約為2.4.第五章對應(yīng)用在矩形域周期邊界條件下的二維S-FDTD算法,推導(dǎo)出了離散H1和H2范數(shù)意義下的能量范數(shù)恒等式.在能量范數(shù)恒等式基礎(chǔ)上證明了二維S-FDTD算法的無條件穩(wěn)定性和超收斂性.數(shù)值算例表明二維ADI-FDTD算法和二維S-FDTD算法在離散L2,H1和H2范數(shù)意義下的能量范數(shù)保持穩(wěn)定并且收斂階約為2.5.第六章在FDTD算法和高階Taylor法的基礎(chǔ)上推出了一種自適應(yīng)時間步長時域有限差分法(Adaptive Time Step Finite-Difference Time-Domain (ATS-FDTD)).這種算法的時間步長、空間步長及泰勒展開的階數(shù)由一個基于穩(wěn)定性分析的準則產(chǎn)生.數(shù)值算例顯示,與ADI-FDTD算法相比,ATS-FDTD算法在離散L2范數(shù)意義下的能量范數(shù)穩(wěn)定,絕對誤差小,收斂階高,數(shù)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.
【學(xué)位授予單位】：大連理工大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2015
【分類號】：O241.82

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