Selecting Explicit Runge-Kutta Methods with Improved Stability Properties

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• 关键词：Ordinary differential equations ; Explicit Runge ; Kutta methods ; Richardson extrapolation ; Absolute stability regions ; Order conditions
• 刊名：Lecture Notes in Computer Science
• 出版年：2015
• 出版时间：2015
• 年：2015
• 卷：9374
• 期：1
• 页码：409-416
• 全文大小：303 KB
• 参考文献：1.Butcher, J.C.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley, New York (2003)MATH CrossRef
  2.Fehlberg, E.: New high-order Runge-Kutta formulas with an arbitrary small truncation error. Z. Angew. Math. Mech. 46, 1–15 (1966)MATH MathSciNet CrossRef
  3.Lambert, J.D.: Numerical Methods for Ordinary Differential Equations. Wiley, New York (1991)
  4.Richardson, L.F.: The deferred approach to the limit. I-Single Lattice Philos. Trans. R. Soc. Lond., Ser. A 226, 299–349 (1927)MATH CrossRef
  5.Zlatev, Z., Farago, I., Havasi, A.: Stability of the Richardson extrapolation applied together with the \(\theta \) - method. J. Comput. Appl. Math. 235(2), 507–520 (2010)MATH MathSciNet CrossRef
  6.Zlatev, Z., Georgiev, K., Dimov, I.: Influence of climatic changes on pollution levels in the Balkan Peninsula. Comput. Math. Appl. 65(3), 544–562 (2013)MathSciNet CrossRef
  7.Zlatev, Z., Georgiev, K., Dimov, I.: Improving the absolute stability properties of some explicit Runge-Kutta methods and their combinations with Richardson extrapolation, Talk presented at the NM&A14 Conference, Borovets, Bulgaria, August (2014)
  8.Zlatev, Z., Georgiev, K., Dimov, I.: Studying absolute stability properties of the Richardson extrapolation combined with explicit Runge-Kutta methods. Comput. Math. Appl. 67, 2294–2307 (2014)MathSciNet CrossRef
  
• 作者单位：Zahari Zlatev (16)
  Krassimir Georgiev (17)
  Ivan Dimov (17)
  
  16. Department of Environmental Science, Aarhus University, Frederiksborgvej 399, P.O. 358, 4000, Roskilde, Denmark
  17. Institute of Information and Communication Technologies, BAS, Acad. G. Bonchev Str., Bl. 25-A, 1113, Sofia, Bulgaria
  
• 丛书名：Large-Scale Scientific Computing
• ISBN：978-3-319-26520-9
• 刊物类别：Computer Science
• 刊物主题：Artificial Intelligence and Robotics
  Computer Communication Networks
  Software Engineering
  Data Encryption
  Database Management
  Computation by Abstract Devices
  Algorithm Analysis and Problem Complexity
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1611-3349

文摘

Explicit Runge-Kutta methods can efficiently be used in the numerical integration of initial value problems for non-stiff systems of ordinary differential equations (ODEs). Let m and p be the number of stages and the order of a given explicit Runge-Kutta method. We have proved in a previous paper [8] that the combination of any explicit Runge-Kutta method with \(m=p\) and the Richardson Extrapolation leads always to a considerable improvement of the absolute stability properties. We have shown in [7] (talk presented at the NM&A14 conference in Borovets, Bulgaria, August 2014) that the absolute stability regions can be further increased when \(p<m\) is assumed. For two particular cases, \(p=3 \wedge m=4\) and \(p=4 \wedge m=6\) it is demonstrated that(a) the absolute stability regions of the new methods are larger than those of the corresponding explicit Runge-Kutta methods with \(p=m\), and