无界区域上非齐次抛物型方程的人工边界条件法

发布时间：2018-05-06 10:34

本文选题：非齐次热传导方程 + 非齐次Burgers方程　；参考：《北方工业大学》2017年硕士论文

【摘要】：抛物型偏微分方程(PDE)是对热、声、磁场、气体等具有传播扩散特性的基本模型的模拟。科学与工程计算领域中大量的实际问题,举例来说,假设管道是无限长的,流体在管道中的流动问题,在空间中电磁波、声波的传播等是用无界区域上的抛物型PDE来描述的。因为实际问题的复杂性以及物理区域的无界性,特别是对于非齐次问题,其在理论上的精确解不易得到,或者其真解计算量巨大,所以寻找计算量相对较少、误差阶数相对较高、相对稳定的数值算法,有重要的研究意义和现实应用价值。热传导方程和Burgers方程是两类经典的抛物型PDE。通过众多科研学者的不懈努力,这两种方程在有界区域上的数值解的研究取得了很多有价值的成果,但是,目前对于无界区域上的方程的数值解的研究相对较少。本论文结合当前的研究现状,运用人工边界条件法(ABM)和有限差分法(FDM)解决问题。以下是本文的研究内容和创新点:第一部分求解了热传导方程,其物理区域是无界区域,维数是一维,具有非齐次和非线性的特性。与半无界研究相似,我们将原问题进行转化,这需要人工边界条件来实现,且边界条件应该是精确的。在使用降阶的基础上,将热传导方程和边界条件进行离散,构造了差分格式。该理论的稳定性和误差阶Q(τ3/2+h2)被证明。差分格式的精确性通过非齐次的数值算例被验证。第二部分求解了 Burgers方程,其物理区域是无界区域,维数是一维,具有非齐次和非线性的特性。利用非线性人工边界条件来转化原问题。不同于热传导方程的是Burgers方程和其人工边界条件具有非线性特性,如此需要我们引入适当的函数变换,将非线性特性变换为线性特性。在使用降阶法的基础上,我们对方程和边界条件进行离散化,构造了差分格式,该差分格式比较新颖、具有更加简洁的特性。该方法的唯一可解性、无条件稳定性以及在空间方向上的2阶精度和时间方向上的3/2阶精度被严格证明。理论算法的有效性和精确性通过三个非齐次的数值算例被一一验证。与前人的研究成果相比,此方法不仅避免了解决非线性问题的困难,而且消除了中间变量,从而大大地节约了计算时间,降低了计算成本。
[Abstract]:The parabolic partial differential equation (PDE) is a simulation of the basic models of heat, sound, magnetic field, gas and so on, which have the characteristics of propagation and diffusion. There are many practical problems in the field of scientific and engineering computation. For example, assuming that the pipe is infinite, the flow of fluid in the pipeline, the electromagnetic wave in space, the propagation of sound wave, etc., are described by parabolic PDE in the unbounded region. Because of the complexity of the practical problems and the unboundedness of the physical region, especially for the inhomogeneous problems, the exact solutions in theory are not easy to be obtained, or the true solutions are computationally large, so the amount of searching calculation is relatively small. The numerical algorithm with relatively high error order and relative stability has important research significance and practical application value. Heat conduction equation and Burgers equation are two kinds of classical parabolic PDE. Through the unremitting efforts of many researchers, many valuable results have been obtained in the study of the numerical solutions of these two equations in the bounded region. However, at present, the research on the numerical solutions of the equations in the unbounded region is relatively rare. In this paper, the artificial boundary condition method (ABM) and the finite difference method (FDM) are used to solve the problem. The following are the contents and innovations of this paper: in the first part, the heat conduction equation is solved. The physical region is unbounded, and the dimension is one-dimensional, which is nonhomogeneous and nonlinear. Similar to the semi-boundless study, we transform the original problem, which requires artificial boundary conditions, and the boundary conditions should be accurate. On the basis of order reduction, the heat conduction equation and boundary conditions are discretized and the difference scheme is constructed. The stability and error order Q (蟿 3 / 2 h 2) of the theory are proved. The accuracy of the difference scheme is verified by non-homogeneous numerical examples. In the second part, the Burgers equation is solved. The physical domain is unbounded and the dimension is one-dimensional, which has the properties of nonhomogeneous and nonlinear. The nonlinear artificial boundary condition is used to transform the original problem. Unlike the heat conduction equation, the Burgers equation and its artificial boundary conditions have nonlinear properties, so we need to introduce proper functional transformation to transform the nonlinear characteristics into linear properties. On the basis of using the reduced order method, we discretize the equation and boundary conditions, and construct the difference scheme. The difference scheme is novel and more concise. The unique solvability, unconditional stability and the second order accuracy in the space direction and the 3 / 2 order accuracy in the time direction are strictly proved. The validity and accuracy of the theoretical algorithm are verified by three nonhomogeneous numerical examples. Compared with the previous research results, this method not only avoids the difficulty of solving nonlinear problems, but also eliminates the intermediate variables, thus greatly saves the calculation time and reduces the calculation cost.
【学位授予单位】：北方工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8

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