本文選題：非齊次熱傳導(dǎo)方程 + 非齊次Burgers方程　； 參考：《北方工業(yè)大學(xué)》2017年碩士論文

【摘要】：拋物型偏微分方程(PDE)是對(duì)熱、聲、磁場(chǎng)、氣體等具有傳播擴(kuò)散特性的基本模型的模擬。科學(xué)與工程計(jì)算領(lǐng)域中大量的實(shí)際問(wèn)題,舉例來(lái)說(shuō),假設(shè)管道是無(wú)限長(zhǎng)的,流體在管道中的流動(dòng)問(wèn)題,在空間中電磁波、聲波的傳播等是用無(wú)界區(qū)域上的拋物型PDE來(lái)描述的。因?yàn)閷?shí)際問(wèn)題的復(fù)雜性以及物理區(qū)域的無(wú)界性,特別是對(duì)于非齊次問(wèn)題,其在理論上的精確解不易得到,或者其真解計(jì)算量巨大,所以尋找計(jì)算量相對(duì)較少、誤差階數(shù)相對(duì)較高、相對(duì)穩(wěn)定的數(shù)值算法,有重要的研究意義和現(xiàn)實(shí)應(yīng)用價(jià)值。熱傳導(dǎo)方程和Burgers方程是兩類(lèi)經(jīng)典的拋物型PDE。通過(guò)眾多科研學(xué)者的不懈努力,這兩種方程在有界區(qū)域上的數(shù)值解的研究取得了很多有價(jià)值的成果,但是,目前對(duì)于無(wú)界區(qū)域上的方程的數(shù)值解的研究相對(duì)較少。本論文結(jié)合當(dāng)前的研究現(xiàn)狀,運(yùn)用人工邊界條件法(ABM)和有限差分法(FDM)解決問(wèn)題。以下是本文的研究?jī)?nèi)容和創(chuàng)新點(diǎn):第一部分求解了熱傳導(dǎo)方程,其物理區(qū)域是無(wú)界區(qū)域,維數(shù)是一維,具有非齊次和非線(xiàn)性的特性。與半無(wú)界研究相似,我們將原問(wèn)題進(jìn)行轉(zhuǎn)化,這需要人工邊界條件來(lái)實(shí)現(xiàn),且邊界條件應(yīng)該是精確的。在使用降階的基礎(chǔ)上,將熱傳導(dǎo)方程和邊界條件進(jìn)行離散,構(gòu)造了差分格式。該理論的穩(wěn)定性和誤差階Q(τ3/2+h2)被證明。差分格式的精確性通過(guò)非齊次的數(shù)值算例被驗(yàn)證。第二部分求解了 Burgers方程,其物理區(qū)域是無(wú)界區(qū)域,維數(shù)是一維,具有非齊次和非線(xiàn)性的特性。利用非線(xiàn)性人工邊界條件來(lái)轉(zhuǎn)化原問(wèn)題。不同于熱傳導(dǎo)方程的是Burgers方程和其人工邊界條件具有非線(xiàn)性特性,如此需要我們引入適當(dāng)?shù)暮瘮?shù)變換,將非線(xiàn)性特性變換為線(xiàn)性特性。在使用降階法的基礎(chǔ)上,我們對(duì)方程和邊界條件進(jìn)行離散化,構(gòu)造了差分格式,該差分格式比較新穎、具有更加簡(jiǎn)潔的特性。該方法的唯一可解性、無(wú)條件穩(wěn)定性以及在空間方向上的2階精度和時(shí)間方向上的3/2階精度被嚴(yán)格證明。理論算法的有效性和精確性通過(guò)三個(gè)非齊次的數(shù)值算例被一一驗(yàn)證。與前人的研究成果相比,此方法不僅避免了解決非線(xiàn)性問(wèn)題的困難,而且消除了中間變量,從而大大地節(jié)約了計(jì)算時(shí)間,降低了計(jì)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.
【學(xué)位授予單位】：北方工業(yè)大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O241.8

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