无界区域上热传导方程的混合谱方法
发布时间：2019-03-07 12:46
【摘要】：工程领域中的许多问题都可以转化成无界区域上热传导方程的求解问题。由于求解域为无界区域,因此,给数值模拟带来了较大难度。解决此类问题,最简单的方法是设定一个人工的边界条件,然后再在有限区间上用有限差分法、有限元法或者谱方法求解。但是这样的方法会产生相应的误差。为了避免这种不必要的误差,人们提出了一些直接的方法,比如Hermite谱方法。已有文献用到的权函数χ(x)=eαx2,α?=0会造成许多理论分析和数值计算上的麻烦。因此,本文将用权函数为χ(x)≡1的带伸缩因子的Hermite函数来作为基函数,可以更好的匹配解的渐进行为,提高数值解的精度。本文主要研究无界区域上的热传导方程的谱方法。首先,在第二章中介绍一些一维Hermite正交逼近结果和一维Legendre正交逼近的基本结果。这些结果是本文建立无界区域中的正交多项式或正交函数系为基底的正交逼近理论的数学基础。在第三章中,首先以带伸缩因子的广义Hermite函数为基函数展开全直线上线性热传导方程的数值解,逼近全直线上的线性热传导方程的正确解。给出算法格式和收敛性分析,数值结果表明所提算法格式的有效性和高精度。然后,在此基础上结合Legendre正交逼近研究无穷带状区域上的线性热传导问题的混合谱方法,建立一些混合的广义Hermite-Legendre正交逼近结果。构造无界区域上各向异性的热传导方程混合的广义Hermite-Legendre谱格式,并证明其收敛性,数值结果验证所提算法格式的有效性和高精度。本文所用方法与已有文献所用的Hermite-Legendre谱方法相比较,能得到更加精确的数值解。在第四章中,对于具有不同渐进行为的非线性方程提出了全离散格式的广义Hermite谱方法。通过使用在时间上的二阶有限差分格式得到了全离散的谱格式,分析了所提算法格式的收敛性和稳定性,数值试验验证了理论分析的正确性,并且显示了所提方法的有效性。第五章对全文进行了总结,并且提出了有待进一步深入研究的一些问题。
[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 伪 x 2, 伪? = 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.
【学位授予单位】：河南科技大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O551.3;O241.8
本文编号：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 伪 x 2, 伪? = 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.
【学位授予单位】：河南科技大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O551.3;O241.8
【参考文献】
相关期刊论文 前1条
1 杨继明;;热传导方程初边值问题的谱方法[J];湖南工程学院学报(自然科学版);2007年02期
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