無界區(qū)域上熱傳導(dǎo)方程的混合譜方法
發(fā)布時間：2019-03-07 12:46
【摘要】：工程領(lǐng)域中的許多問題都可以轉(zhuǎn)化成無界區(qū)域上熱傳導(dǎo)方程的求解問題。由于求解域為無界區(qū)域,因此,給數(shù)值模擬帶來了較大難度。解決此類問題,最簡單的方法是設(shè)定一個人工的邊界條件,然后再在有限區(qū)間上用有限差分法、有限元法或者譜方法求解。但是這樣的方法會產(chǎn)生相應(yīng)的誤差。為了避免這種不必要的誤差,人們提出了一些直接的方法,比如Hermite譜方法。已有文獻用到的權(quán)函數(shù)χ(x)=eαx2,α?=0會造成許多理論分析和數(shù)值計算上的麻煩。因此,本文將用權(quán)函數(shù)為χ(x)≡1的帶伸縮因子的Hermite函數(shù)來作為基函數(shù),可以更好的匹配解的漸進行為,提高數(shù)值解的精度。本文主要研究無界區(qū)域上的熱傳導(dǎo)方程的譜方法。首先,在第二章中介紹一些一維Hermite正交逼近結(jié)果和一維Legendre正交逼近的基本結(jié)果。這些結(jié)果是本文建立無界區(qū)域中的正交多項式或正交函數(shù)系為基底的正交逼近理論的數(shù)學(xué)基礎(chǔ)。在第三章中,首先以帶伸縮因子的廣義Hermite函數(shù)為基函數(shù)展開全直線上線性熱傳導(dǎo)方程的數(shù)值解,逼近全直線上的線性熱傳導(dǎo)方程的正確解。給出算法格式和收斂性分析,數(shù)值結(jié)果表明所提算法格式的有效性和高精度。然后,在此基礎(chǔ)上結(jié)合Legendre正交逼近研究無窮帶狀區(qū)域上的線性熱傳導(dǎo)問題的混合譜方法,建立一些混合的廣義Hermite-Legendre正交逼近結(jié)果。構(gòu)造無界區(qū)域上各向異性的熱傳導(dǎo)方程混合的廣義Hermite-Legendre譜格式,并證明其收斂性,數(shù)值結(jié)果驗證所提算法格式的有效性和高精度。本文所用方法與已有文獻所用的Hermite-Legendre譜方法相比較,能得到更加精確的數(shù)值解。在第四章中,對于具有不同漸進行為的非線性方程提出了全離散格式的廣義Hermite譜方法。通過使用在時間上的二階有限差分格式得到了全離散的譜格式,分析了所提算法格式的收斂性和穩(wěn)定性,數(shù)值試驗驗證了理論分析的正確性,并且顯示了所提方法的有效性。第五章對全文進行了總結(jié),并且提出了有待進一步深入研究的一些問題。
[Abstract]:Many problems in the engineering field can be transformed into the problem of solving the heat conduction equation in the unbounded region. Because the solution domain is unbounded, it brings great difficulty to the numerical simulation. The simplest way to solve this kind of problem is to set an artificial boundary condition, and then use finite difference method, finite element method or spectral method to solve the problem on the finite interval. But such an approach would produce a corresponding error. In order to avoid this unnecessary error, some direct methods, such as Hermite spectral method, have been proposed. The weight function 蠂 (x) = e 偽 x 2, 偽? = 0 used in the literature will cause a lot of troubles in theoretical analysis and numerical calculation. Therefore, in this paper, we use the Hermite function with stretching factor as the basis function, which is the weight function 蠂 (x) = 1, which can better match the asymptotic behavior of the solution and improve the accuracy of the numerical solution. In this paper, the spectral method of heat conduction equation on unbounded domain is studied. Firstly, some results of one-dimensional Hermite orthogonal approximation and one-dimensional Legendre orthogonal approximation are introduced in chapter 2. These results are the mathematical basis of the orthogonal approximation theory based on orthogonal polynomials or orthogonal function systems in unbounded domains. In chapter 3, the numerical solution of the linear heat conduction equation on the whole line is expanded by using the generalized Hermite function with the expansion factor as the basic function, and the correct solution of the linear heat conduction equation on the whole line is approximated to that of the linear heat conduction equation on the whole line. The algorithm scheme and convergence analysis are given. Numerical results show that the proposed scheme is effective and accurate. Then, combined with the mixed spectral method of Legendre orthogonal approximation for linear heat conduction problems on infinite banded domains, some mixed generalized Hermite-Legendre orthogonal approximation results are established. The generalized Hermite-Legendre spectral scheme for anisotropic heat conduction equations on unbounded domains is constructed and its convergence is proved. The numerical results show that the proposed scheme is effective and accurate. Compared with the Hermite-Legendre spectrum method used in the literature, the numerical solution can be obtained more accurately. In chapter 4, the generalized Hermite spectral method of fully discrete scheme is proposed for nonlinear equations with different asymptotic behavior. The fully discrete spectral scheme is obtained by using the second-order finite difference scheme in time. The convergence and stability of the proposed scheme are analyzed. Numerical experiments verify the correctness of the theoretical analysis and show the effectiveness of the proposed method. The fifth chapter summarizes the full text, and puts forward some problems that need to be further in-depth study.
【學(xué)位授予單位】：河南科技大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2015
【分類號】：O551.3;O241.8
本文編號：2436136
[Abstract]:Many problems in the engineering field can be transformed into the problem of solving the heat conduction equation in the unbounded region. Because the solution domain is unbounded, it brings great difficulty to the numerical simulation. The simplest way to solve this kind of problem is to set an artificial boundary condition, and then use finite difference method, finite element method or spectral method to solve the problem on the finite interval. But such an approach would produce a corresponding error. In order to avoid this unnecessary error, some direct methods, such as Hermite spectral method, have been proposed. The weight function 蠂 (x) = e 偽 x 2, 偽? = 0 used in the literature will cause a lot of troubles in theoretical analysis and numerical calculation. Therefore, in this paper, we use the Hermite function with stretching factor as the basis function, which is the weight function 蠂 (x) = 1, which can better match the asymptotic behavior of the solution and improve the accuracy of the numerical solution. In this paper, the spectral method of heat conduction equation on unbounded domain is studied. Firstly, some results of one-dimensional Hermite orthogonal approximation and one-dimensional Legendre orthogonal approximation are introduced in chapter 2. These results are the mathematical basis of the orthogonal approximation theory based on orthogonal polynomials or orthogonal function systems in unbounded domains. In chapter 3, the numerical solution of the linear heat conduction equation on the whole line is expanded by using the generalized Hermite function with the expansion factor as the basic function, and the correct solution of the linear heat conduction equation on the whole line is approximated to that of the linear heat conduction equation on the whole line. The algorithm scheme and convergence analysis are given. Numerical results show that the proposed scheme is effective and accurate. Then, combined with the mixed spectral method of Legendre orthogonal approximation for linear heat conduction problems on infinite banded domains, some mixed generalized Hermite-Legendre orthogonal approximation results are established. The generalized Hermite-Legendre spectral scheme for anisotropic heat conduction equations on unbounded domains is constructed and its convergence is proved. The numerical results show that the proposed scheme is effective and accurate. Compared with the Hermite-Legendre spectrum method used in the literature, the numerical solution can be obtained more accurately. In chapter 4, the generalized Hermite spectral method of fully discrete scheme is proposed for nonlinear equations with different asymptotic behavior. The fully discrete spectral scheme is obtained by using the second-order finite difference scheme in time. The convergence and stability of the proposed scheme are analyzed. Numerical experiments verify the correctness of the theoretical analysis and show the effectiveness of the proposed method. The fifth chapter summarizes the full text, and puts forward some problems that need to be further in-depth study.
【學(xué)位授予單位】：河南科技大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2015
【分類號】：O551.3;O241.8
【參考文獻】
相關(guān)期刊論文 前1條
1 楊繼明;;熱傳導(dǎo)方程初邊值問題的譜方法[J];湖南工程學(xué)院學(xué)報(自然科學(xué)版);2007年02期
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