本文關(guān)鍵詞：基于B樣條小波的反演方法及地下水污染源識(shí)別應(yīng)用研究　出處：《哈爾濱工業(yè)大學(xué)》2016年碩士論文　論文類型：學(xué)位論文

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【摘要】：近年來,我國很多城市和地區(qū)的地下水受到污染,并且污染程度不斷增加。地下水占人類飲用水的三分之一,地下水污染造成供水緊張,已經(jīng)嚴(yán)重影響了人們的生活。解決有關(guān)地下水治理的問題已迫在眉睫。治理地下水的前提是查清污染源。污染物在地下水中運(yùn)動(dòng)的控制方程是擴(kuò)散方程或?qū)α?擴(kuò)散反應(yīng)方程。因此,一些學(xué)者在數(shù)學(xué)模型研究的基礎(chǔ)上,用應(yīng)用數(shù)學(xué)的方法反演污染源。目前關(guān)于地下水污染源參數(shù)識(shí)別的方法有很多,大致可以分為迭代法和直接法兩類。直接法是建立在無網(wǎng)格基礎(chǔ)上的,具有較高的計(jì)算效率和精度。直接法中研究比較成熟的是徑向基配點(diǎn)法,但對于地下水污染源反演問題,其所得數(shù)值解較解析解誤差較大。考慮到小波理論具有的局部緊支撐等良好特性,本文基于三次B樣條小波尺度函數(shù),借鑒徑向基配點(diǎn)法處理反問題的思路,提出了處理偏微分方程初、邊值反演的B樣條小波方法,并將之應(yīng)用于地下水污染數(shù)學(xué)模型的源項(xiàng)反演研究,得到了較好的研究結(jié)果。本文首先介紹B樣條小波的性質(zhì)及特點(diǎn),給出小波多分辨率分析的構(gòu)造方法,構(gòu)造B樣條小波尺度函數(shù),將方程離散化。其次,用B樣條小波方法反演空間域上的Laplace方程的未知邊界條件,將此反問題轉(zhuǎn)變?yōu)橐粋�(gè)大規(guī)模的代數(shù)方程組求解的不適定問題,并采用最小二乘方法求解。然后,引入時(shí)間域,在假定時(shí)間域和空間域相互獨(dú)立的條件下,采用B樣條小波方法求解時(shí)空域上一維擴(kuò)散方程反問題。最后,考慮了增加對流項(xiàng)的一維對流-擴(kuò)散反應(yīng)方程反問題的求解。通過對算法的數(shù)值結(jié)果的比較和分析,可以得出:當(dāng)待反演函數(shù)光滑時(shí),B樣條小波方法可以有效地處理Laplace方程反問題、一維擴(kuò)散方程反問題和一維對流-擴(kuò)散反應(yīng)方程反問題。相比徑向基配點(diǎn)法和改進(jìn)的徑向基配點(diǎn)法,B樣條小波方法得到的數(shù)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.
【學(xué)位授予單位】：哈爾濱工業(yè)大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2016
【分類號(hào)】：X523;O241.82
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本文編號(hào)：1402034