# 臺灣博碩士論文加值系統

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 摘要 本論文著重在研究數值方法在解常微分方程的初值問題時的穩定區域圖形上,用預測修正法在重複修正過程中(Predictor-Corrector Methods with Self-Correcting Process)穩定區域圖形產生一些戲劇性的變化,並提出一種有效率的作圖方法 ( BDS),並藉此方法來探討區域預測自我修正的穩定性問題,此外,以此方法,也發現過去文獻中某些最佳穩定截距的錯誤 關鍵詞: 區域式預測修正法, 絕對穩定, 自我修正過程
 Abstract In this thesis, we are interested in studying the configuration of stability regions of numerical schemes when solving an initial value problem in ODEs. By applying the predictor-corrector scheme with repeated correcting processes, the variation of stability region could be changed drastically. We propose a delicate algorithm on which the boundary of the stability region could be located efficiently. By using the algorithm, the stability of some block predictor-corrector schemes with repeated self-correction process have been studied. Moreover, some incorrect results of optimal stability intercept in some literatures have also been found. Key words: Block predictor-corrector Methods, Absolute Stability, Self-Correction Process
 Table of Contents 1 Introduction………………………………………………………………… 1 2 Convergence………………………………………………………..………. 3 2.1 General form of BPC Scheme………………………………………….. 3 2.2 Convergence Theory of 2-step schemes……………………………….. 5 3 Stability…………………………………………………………………….. 6 3.1 General Stability………………………………………………………. 6 3.2 Row-wise root condition……………………………………………… 6 3.3 Region of Absolute Stability…………………………………………… 7 3.4 Bubble detection scheme for root-locus plots…………………………. 9 4 The Stability Region of EP(EC)j for Chu and Hamilton Scheme………..… 12 4.1 Evolution matrix Pj(H)………………………………………………… 12 4.2 Region of Stability……………………………………………………... 15 5 Concluding Remark………………………………………………..………. 18 6 Appendix : 2-block schemes of Chu & Hamilton…………………..……. 19 7 References………………………………………………………………….. 21
 1. J.B. Rosser, A Runge-Kutta for all seasons, SIAM Rev. Vol.9, July 1967.2. H. J. Fang, On Adaptive Block Predictor-Corrector Methods for ODEs, Doctoral Dissertation, NCCU,2001.3. W.L. Miranker and W.M. Liniger, Parallel Methods for The Numerical Integration of Ordinary Differential \ Equations, Math. Comp., Vol. 21 , 19674. I.N.Karz, M.A.Franklin and A.Sen, Optimally Stable Parallel Predictors for Adams-Moulton Correctors, Comp. Math. Apply., Vol. 3, 1977, pp.217-233.5. L.F. Shampine and H.A. Watts, Block implicit one-step methods, Math. Comp., Vol. 23, 1969.6. P.B. Worland, Parallel methods for the numerical solution of ordinary differential equations, IEEE Trans. \ \ Comp., Vol. 25, 1976.7. K.E. Atkinson, An Introduction to Numerical Analysis, New York, 1978.8. R.L. Burden and J.D. Faires, Numerical Analysis, Prindle, Weber \& Schmidt, 1981.9. L.G. Birta and O. Abou-Rabia, Parallel block predictor-corrector methods for ODEs,, IEEE Trans. Comp., \ \ \ \ \ \ Vol. 36, No.3, Mar. 1987.10. M.T. Chu and H. Hamilton, Parallel solution of ODEs by multi-block methods, SIAM J. Sci. Stat. Comp.,\ Vol.8, No.3, 1987, pp.342-348.11. H.J. Fang, On Numerical Algorithms for Solving IVP in ODEs by Applied on A Four-Processor Machine, \ \ \ Master Thesis, NCTU, 1988.12. S.Y. Yang, On Stability of Parallel Block Methods for ODEs, Master Thesis, NCTU, 1989.13. PC-Mathlab for MS-Dos Personal Computers, Copyright 1985-89, by The Mathworks, Inc., 198914. C. William Gear, Numerical Initial Value Problems In Ordinary Differential Equations, Prentice-Hall, Inc. 1971.
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