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The stability of modified Runge-Kutta methods for the pantograph equation


Authors: M. Z. Liu, Z. W. Yang and Y. Xu
Journal: Math. Comp. 75 (2006), 1201-1215
MSC (2000): Primary 65L02, 65L05; Secondary 65L20
DOI: https://doi.org/10.1090/S0025-5718-06-01844-8
Published electronically: May 3, 2006
MathSciNet review: 2219025
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Abstract: In the present paper, the modified Runge-Kutta method is constructed, and it is proved that the modified Runge-Kutta method preserves the order of accuracy of the original one. The necessary and sufficient conditions under which the modified Runge-Kutta methods with the variable mesh are asymptotically stable are given. As a result, the $ \theta$-methods with $ \tfrac12\leq\theta\leq 1$, the odd stage Gauss-Legendre methods and the even stage Lobatto IIIA and IIIB methods are asymptotically stable. Some experiments are given.


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Additional Information

M. Z. Liu
Affiliation: Department of Mathematics, Harbin Institute of Technology, Harbin 150001, People’s Republic of China
Email: mzliu@hope.hit.edu.cn

Z. W. Yang
Affiliation: Department of Mathematics, Harbin Institute of Technology, Harbin 150001, People’s Republic of China

Y. Xu
Affiliation: Department of Mathematics, Harbin Institute of Technology, Harbin 150001, People’s Republic of China

DOI: https://doi.org/10.1090/S0025-5718-06-01844-8
Keywords: Pantograph equation, asymptotical stability, Runge-Kutta methods.
Received by editor(s): September 13, 2004
Published electronically: May 3, 2006
Additional Notes: This paper was supported by the National Natural Science Foundation of China (10271036).
Article copyright: © Copyright 2006 American Mathematical Society