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博碩士論文 etd-0025118-225133 詳細資訊
Title page for etd-0025118-225133
論文名稱
Title
求解伯格斯方程之高階徑向基底數值法
On the high order methods for Burgers’ equation using RBFs.
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
21
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2018-01-24
繳交日期
Date of Submission
2018-01-25
關鍵字
Keywords
MQ擬插值函數、有限差分法、伯格斯方程
multiquadric quasi-interpolation, Burgers equation, finite-difference methods
統計
Statistics
本論文已被瀏覽 5707 次,被下載 188
The thesis/dissertation has been browsed 5707 times, has been downloaded 188 times.
中文摘要
本文介紹有限差分法與MQ擬插值函數在求解伯格斯方程之應用,並且使用高階的有限差分法結合MQ擬插值函數得到兩種延伸的高階徑向基底數值法。在一開始我們先介紹有關伯格 斯方程的相關知識,接著我們提供四種不同的演算法為將來求解伯格斯方程做準備,並且將 其應用在後面的章節,透過數值模擬可以看到這四種方法在求解伯格斯方程上的不同表現, 最後給出高階徑向基底數值法將有效改善MQ擬插值函數在伯格斯方程上的表現。
Abstract
In this article, we introduce solving Burgers’s equations using finite-difference methods and multiquadric quasi-interpolation. By combining high-order finite-difference methods with multiquadric quasi-interpolation, we obtain high-order radial basis function numerical method. First we introduce the knowledge about Burgers equation, then we provide four different algorithms for solving Burgers’ s equations. And the numerical experiments justify the performance of these four methods. Finally, we conclude high-order radial basis function numerical method effectively improves the performance of multiquadric quasi-interpolation on solving Burgers’ s equations.
目次 Table of Contents
論文審定書i
摘要ii
Abstract iii
1 引言1
2 伯格斯方程2
3 數值方法3
3.1 有限差分法. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.2 MQ擬插值函數. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.3 二階徑向基底數值法. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.4 四階徑向基底數值法. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4 數值實驗7
5 結論12
Reference 13
參考文獻 References
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[2] C. S. Chen, M. Ganesh, M. A. Golberg, and A. H. D. Cheng, Multilevel compact radial func- tion based computational schemes for some ellipitic problems, Comput Math Appl 43 (2002), 359–378.
[3] G. E. Fasshauer, Solving differential equations with radial basis functions: Multilevel methods and smoothing, Adv Comput Math 11 (1999), 139–159.
[4] G.E.Fasshauer,Newton iteration with multiquadrics for the solution of nonlinear PDEs, Comput- Math Appl 43 (2002), 423–438.
[5] R. L. Hardy, Theory, and applications of the multiquadric-biharmonic method, 20 years of dis- covery 1968–1988, Comput Math Appl 19 (1990), 163–208.
[6] R. K. Beatson and M. J. D. Powell, Univariate multiquadric approximation: quasi-interpolation to scattered data, Constr Approx 8 (1992), 275–288.
[7] R. K. Beatson and N. Dyn, Multiquadric B-splines, J Approx Theory 87 (1996), 1–24.
[8] Z.M.Wu,CompactlysupportedradialfunctionsandtheStrang-Fixcondition,ApplMathComput 84 (1997), 115–124.
[9] Z. M. Wu and R. Schaback, Local error estimates for radial basis function interpolation of scat- tered data, IMA J Numer Anal 13 (1993), 13–27.
[10] Z. M. Wu and R. Schaback, Shape preserving properties and convergence of univariate multi- quadric quasi-interpolation, ACTA Math Appl Sinica 10 (1994), 441–446.
[11] CHEN Ronghua, WU Zongmin. Solving hyperbolic conservation laws using multiquadric quasi- interpolation [J]. Numer. Methods Partial Differential Equations, 2006, 22(4): 776–796.
[12]Courant,R.;Friedrichs,K.;Lewy,H.(1928),”U ̈berdiepartiellenDifferenzengleichungender mathematischen Physik”, Mathematische Annalen (in German), 100 (1): 32–74,
[13] Fornberg, Bengt, Generation of Finite Difference Formulas on Arbitrarily Spaced Grids, Mathe- matics of Computation, 1988, 51 (184): 699–706
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