本文選題：發(fā)展方程　切入點(diǎn)：指數(shù)時(shí)間差分法　出處：《南昌航空大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：本文主要研究幾類發(fā)展方程的緊致差分法,并對(duì)設(shè)計(jì)的相應(yīng)數(shù)值格式進(jìn)行理論分析,通過一些數(shù)值算例來驗(yàn)證數(shù)值算法的準(zhǔn)確性和有效性。本文共五章,具體的研究工作如下:第二章主要致力于一維Burgers方程的兩種高階數(shù)值求解方法的發(fā)展和應(yīng)用,這兩種方法在時(shí)空方向均有四階精度。其中,方法一在時(shí)間方向使用Crank-Nicolson格式和Richardson外推法,在空間方向用四階緊致差分法逼近;方法二采取基于padé逼近的時(shí)間步方法和空間四階的緊致差分法。另外,我們運(yùn)用矩陣分析法分別研究了這兩種方法的穩(wěn)定性。數(shù)值實(shí)驗(yàn)證實(shí)了新算法的合理性和高效性。第三章研究了一維非線性常延遲反應(yīng)擴(kuò)散方程的緊致差分法,并運(yùn)用能量法證明差分解在最大范數(shù)意義下具有O(t~2+h~4)的收斂階。接著,在時(shí)間方向運(yùn)用Richardson外推法,獲得了O(t~4+h~4)外推解。然后,將該數(shù)值方法推廣到其它復(fù)雜的延遲問題。最后,數(shù)值算例驗(yàn)證了算法的計(jì)算精度和有效性。第四章對(duì)一維粘性波動(dòng)方程,構(gòu)造一個(gè)三層緊致差分格式,并運(yùn)用能量法進(jìn)行誤差分析,證明差分格式在最大范數(shù)意義下有O(t~2+h~4)的收斂階。利用Richardson外推法,得到O(t~4+h~4)的外推解。最后,給出一個(gè)數(shù)值算例,證實(shí)該差分格式的收斂階和有效性。第五章對(duì)全文進(jìn)行了總結(jié)、展望。
[Abstract]:In this paper, the compact difference method for several kinds of evolution equations is studied, and the corresponding numerical schemes are theoretically analyzed. Some numerical examples are used to verify the accuracy and validity of the numerical algorithm. The specific research work is as follows: in chapter 2, we focus on the development and application of two high-order numerical methods for solving one-dimensional Burgers equations, both of which have fourth-order accuracy in the space-time direction. Methods one is to use the Crank-Nicolson scheme and Richardson extrapolation in the time direction, the other is to approximate the space direction by the fourth-order compact difference method, the second method is the time step method based on pad 茅 approximation and the space fourth-order compact difference method. We use matrix analysis method to study the stability of these two methods. Numerical experiments show that the new algorithm is reasonable and efficient. In chapter 3, we study the compact difference method for nonlinear constant delay reaction diffusion equation. The energy method is used to prove the order of convergence of the difference decomposition with 2 h ~ 4) in the sense of maximum norm. Then, using the Richardson extrapolation method in the time direction, the extrapolation solution is obtained. Then, the numerical method is extended to other complex delay problems. Numerical examples verify the accuracy and validity of the algorithm. In Chapter 4th, a three-layer compact difference scheme is constructed for one-dimensional viscous wave equation, and the error is analyzed by energy method. It is proved that the difference scheme has the order of convergence in the sense of the maximum norm. By using the Richardson extrapolation method, the extrapolation solution of OT4) is obtained. Finally, a numerical example is given. The convergence order and validity of the difference scheme are confirmed. Chapter 5th summarizes and prospects the full text.
【學(xué)位授予單位】：南昌航空大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O241.8

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