基于B样条小波的反演方法及地下水污染源识别应用研究

发布时间：2018-01-09 17:54

  本文关键词：基于B样条小波的反演方法及地下水污染源识别应用研究　出处：《哈尔滨工业大学》2016年硕士论文　论文类型：学位论文

  更多相关文章： 污染源反演 B样条小波方法 Laplace方程反问题 扩散方程反问题 对流-扩散反应方程反问题

【摘要】：近年来,我国很多城市和地区的地下水受到污染,并且污染程度不断增加。地下水占人类饮用水的三分之一,地下水污染造成供水紧张,已经严重影响了人们的生活。解决有关地下水治理的问题已迫在眉睫。治理地下水的前提是查清污染源。污染物在地下水中运动的控制方程是扩散方程或对流-扩散反应方程。因此,一些学者在数学模型研究的基础上,用应用数学的方法反演污染源。目前关于地下水污染源参数识别的方法有很多,大致可以分为迭代法和直接法两类。直接法是建立在无网格基础上的,具有较高的计算效率和精度。直接法中研究比较成熟的是径向基配点法,但对于地下水污染源反演问题,其所得数值解较解析解误差较大。考虑到小波理论具有的局部紧支撑等良好特性,本文基于三次B样条小波尺度函数,借鉴径向基配点法处理反问题的思路,提出了处理偏微分方程初、边值反演的B样条小波方法,并将之应用于地下水污染数学模型的源项反演研究,得到了较好的研究结果。本文首先介绍B样条小波的性质及特点,给出小波多分辨率分析的构造方法,构造B样条小波尺度函数,将方程离散化。其次,用B样条小波方法反演空间域上的Laplace方程的未知边界条件,将此反问题转变为一个大规模的代数方程组求解的不适定问题,并采用最小二乘方法求解。然后,引入时间域,在假定时间域和空间域相互独立的条件下,采用B样条小波方法求解时空域上一维扩散方程反问题。最后,考虑了增加对流项的一维对流-扩散反应方程反问题的求解。通过对算法的数值结果的比较和分析,可以得出:当待反演函数光滑时,B样条小波方法可以有效地处理Laplace方程反问题、一维扩散方程反问题和一维对流-扩散反应方程反问题。相比径向基配点法和改进的径向基配点法,B样条小波方法得到的数值解精度更高。
[Abstract]:In recent years, groundwater in many cities and regions of China has been polluted, and the degree of pollution is increasing. Groundwater accounts for 1/3 of human drinking water, groundwater pollution causes water supply shortage. It is urgent to solve the problem of groundwater treatment. The premise of groundwater treatment is to find out the source of pollution. The governing equation of pollutant movement in groundwater is diffusion equation or convection. The diffusion reaction equation. Based on the research of mathematical model, some scholars use the method of applying mathematics to retrieve the pollution sources. There are many methods for identifying the parameters of groundwater pollution sources. The direct method is based on the meshless method, which has high computational efficiency and accuracy. The radial basis collocation method is the more mature one in the direct method. However, for the problem of groundwater pollution source inversion, the error of numerical solution is larger than that of analytical solution. Considering the good characteristics of wavelet theory, such as local tight support, this paper based on cubic B-spline wavelet scaling function. A B-spline wavelet method for initial and boundary inversion of partial differential equations is proposed by using the idea of radial basis collocation method to deal with the inverse problem, and it is applied to the source term inversion of the mathematical model of groundwater pollution. First of all, this paper introduces the properties and characteristics of B-spline wavelet, gives the construction method of wavelet multi-resolution analysis, constructs B-spline wavelet scale function, and discretizes the equation. The B-spline wavelet method is used to inverse the unknown boundary conditions of the Laplace equation in spatial domain, and the inverse problem is transformed into an ill-posed problem for solving a large scale algebraic equations. Then the time domain is introduced and the B-spline wavelet method is used to solve the inverse problem of one-dimensional diffusion equation in space-time domain under the assumption that the time domain and the space domain are independent. The inverse problem of one-dimensional convection-diffusion reaction equation with increasing the convection term is considered. Through the comparison and analysis of the numerical results of the algorithm, it can be concluded that when the inversion function is smooth. B-spline wavelet method can deal with the inverse problem of Laplace equation effectively. The inverse problem of one-dimensional diffusion equation and the inverse problem of one-dimensional convection-diffusion reaction equation are more accurate than the radial basis collocation method and the modified B-spline wavelet method.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：X523;O241.82
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本文编号：1402034