本文選題：Burgers方程 + 高階人工邊界條件��； 參考：《北方工業(yè)大學》2017年碩士論文

【摘要】：本文主要研究了無界域的Burgers方程的高階人工邊界方法。第一章首先介紹了 Burgers方程的研究價值,解決無界域問題的常用方法,以及人工邊界條件法的應(yīng)用。第二章應(yīng)用積分型人工邊界條件來求無界域的一維Burgers方程的數(shù)值解。使用Hopf-Cole變換將原問題轉(zhuǎn)換為無界域中的熱傳導方程,再引入兩個積分型的人工邊界條件,將得到的熱傳導方程簡化為有界的計算域中的等價方程,之后用降階法為這個等價方程構(gòu)建有限差分格式,求解線性方程組即可得到所得熱傳導方程的數(shù)值解,進而得到原Burgers方程的數(shù)值解。該方法被證明是唯一可解、無條件穩(wěn)定的,且具有空間上的2階收斂和時間的3/2階收斂,并用算例加以驗證。第三章應(yīng)用高階人工邊界條件來求無界域的Burgers方程的求解。首先,同樣通過Hopf-Cole變換將原來的Burgers方程(非線性)轉(zhuǎn)化為無界域中的熱傳導方程(線性),即克服了Burgers方程本身的非線性問題。然后,通過使用Pade逼近、Laplace變換及其逆變換給出高階人工邊界條件將所得的熱傳導方程限制在有限的計算域上。之后,我們證明了所得的熱傳導方程與Burgers方程解的穩(wěn)定性。之后,將應(yīng)用高階人工邊界條件而得到有限計算域的熱傳導方程,只在空間方向上通過Taylor展開進行離散。對于所得空間方向上離散的熱傳導方程,在理論上證明了求得的半離散解的穩(wěn)定性及收斂性。最后,我們在這個有界的計算域上建立了 Burgers方程的有限差分格式,并用兩個數(shù)值例子說明了該方法的穩(wěn)定性、有效性,且在空間方向上二階收斂,在時間方向上約為二階收斂。
[Abstract]:In this paper, the higher order artificial boundary method for unbounded Burgers equation is studied. In the first chapter, the research value of Burgers equation, the common methods to solve the unbounded domain problem and the application of artificial boundary condition method are introduced. In chapter 2, the integral artificial boundary condition is used to obtain the numerical solution of one-dimensional Burgers equation in unbounded domain. The original problem is transformed into the heat conduction equation in the unbounded domain by Hopf-Cole transform. Two artificial boundary conditions of integral type are introduced, and the obtained heat conduction equation is simplified to the equivalent equation in the bounded computational domain. Then the finite difference scheme is constructed for the equivalent equation by using the reduced order method, and the numerical solution of the heat conduction equation is obtained by solving the linear equation system, and the numerical solution of the original Burgers equation is obtained. It is proved that the method is solvable, unconditionally stable, and has the convergence of order 2 in space and the convergence of order 3 / 2 in time, which is verified by an example. In chapter 3, the higher order artificial boundary condition is used to solve the Burgers equation in unbounded domain. Firstly, the original Burgers equation (nonlinear) is also transformed into the heat conduction equation in the unbounded domain by Hopf-Cole transformation (linear equation), which overcomes the nonlinear problem of Burgers equation itself. Then, by using the Pade approximation Laplace transform and its inverse transformation, the higher order artificial boundary conditions are given to limit the heat conduction equation to the finite computational domain. Then we prove the stability of the solutions of the heat conduction equation and the Burgers equation. After that, the heat conduction equations in finite computational domain are obtained by using higher order artificial boundary conditions, and are discretized only in the spatial direction by Taylor expansion. The stability and convergence of the obtained semi-discrete solutions are theoretically proved for the discrete heat conduction equations in the direction of the obtained space. Finally, we establish the finite difference scheme of Burgers equation in this bounded computational domain. Two numerical examples are given to illustrate the stability and validity of the method, and the second order convergence in the space direction. In the time direction, the convergence is about second order.
【學位授予單位】：北方工業(yè)大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O241.8

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本文編號：2044034