Advertisement

Applied Mathematics and Mechanics

, Volume 25, Issue 12, pp 1405–1411 | Cite as

GP-stability of rosenbrock methods for system of delay differential equation

  • Cong Yu-hao
  • Cai Jia-ning
  • Xiang Jia-xiang
Article

Abstract

The stability analysis of the Rosenbrock method for the numerical solutions of system of delay differential equations was studied. The stability behavior of Rosenbrock method was analyzed for the solutions of linear test equation. The result that the Rosenbrock method is GP-stable if and only if it is A-stable is obtained.

Key words

delay differential equation Rosenbrock method GP-stability 

Chinese Library Classification

O241.8 

2000 Mathematics Subject Classification

65L20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Bellen A, Jackiewicz Z, Zennaro M. Stability analysis of one-step methods for neutral delay-differential equations[J].Numerische Mathematic, 1988,52(3):605–619.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    LIU Ming-zhu, Spijker M N. The stability of θ-methods in the numerical solution[J].IMA Journal of Numerical Analysis, 1990,10(1):31–48.MathSciNetGoogle Scholar
  3. [3]
    in't Hout K J. The stability of θ-methods for systems of delay differential equations[J].Annals of Numerical Mathematics, 1994,1(3):323–334.MATHMathSciNetGoogle Scholar
  4. [4]
    Koto T. A stability property of A-stable natural Runge-Kutta methods for systems of delay differential equations[J].BIT, 1994,34(2):262–267.MATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    HU Guang-da, Mitsui T. Stability of numerical methods for systems of nautral delay differential equations[J].BIT, 1995,35(4):504–515.MathSciNetCrossRefGoogle Scholar
  6. [6]
    Hairer E, Nørsett S P, Wanner G.Solving Ordinary Differential Equations[M]. New York: Springer-Verlag, 2000, 103–117.Google Scholar
  7. [7]
    CAO Xue-nian, LIU De-gui, LI Shou-fo. Asymptotic stability of Rosenbrock methods for delay differential equations[J].Journal of System Simulation, 2002,14(3):290–292. (in Chinese)Google Scholar
  8. [8]
    Lambert J D.Computational Methods in Ordinary Differentail Equations[M]. New York: John-Willy, 1990.Google Scholar
  9. [9]
    KUANG Jiao-xun, TIAN Hong-jiong. The asymptotic behaviour of theoretical and numerical solutions for nonlinear differential systems with several delay terms[J].Journal of Shanghai Teachers University (Natural Sciences), 1995,24(1):1–7.Google Scholar
  10. [10]
    in't Hout K J. A new interpolation procedure for adapting Runge-Kutta methods to delay differential equations[J].BIT, 1992,32(4):634–649.MATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    KUANG Jiao-xun. The PL-stability of block θ-methods[J].Math Numer Sinica, 1997,15(2): 135–140. (in Chinese)Google Scholar
  12. [12]
    YANG Biao, QIU Lin, KUANG Jiao-xun. The GPL-stability of Runge-Kutta methods for delay differential systems[J].J Comput Math, 2000,18(1):75–82.MathSciNetGoogle Scholar
  13. [13]
    Huang C M, Li S F, Fu H Y,et al. Stability and error analysis of one-leg methods for nonlinear delay differential equations[J].Journal of Computational and Applied Mathematics, 1999,103(2):263–279.MATHMathSciNetCrossRefGoogle Scholar
  14. [14]
    CHEN Li-rong, LIU De-gui. Combined RK-Rosenbrock methods and their stability[J].Mathematica Numerica Sinica, 2000,22(3):319–332.MathSciNetGoogle Scholar
  15. [15]
    LI Shou-fu. Nonlinear stability of general linear methods[J].Journal of Computational Mathematics, 1991,9(2):97–104.MathSciNetGoogle Scholar
  16. [16]
    Robert Piché. An L-stable Rosenbrock method for step-by-step time integration in structural dynamics[J].Computer Methods in Applied Mechanics and Engineering, 1995,126(3/4): 343–354.MathSciNetGoogle Scholar
  17. [17]
    SUN Geng. A class of single step methods with a large interval of absolute stability[J].J Comput Math, 1991,9(2):185–193.MathSciNetGoogle Scholar
  18. [18]
    Barwell V K. Special stability problems for functional differential equations[J].BIT, 1975,15 (2):130–135.MATHMathSciNetCrossRefGoogle Scholar
  19. [19]
    ZHANG Cheng-jian, ZHOU Shu-zi. Nonlinear stability and D-convergence of Runge-Kutta methods for delay differential equations[J].Journal of Computational and Applied Mathematics, 1997,85(2):225–237.MathSciNetCrossRefGoogle Scholar

Copyright information

© Editorial Committee of Applied Mathematics and Mechanics 2004

Authors and Affiliations

  • Cong Yu-hao
    • 1
  • Cai Jia-ning
    • 1
  • Xiang Jia-xiang
    • 1
  1. 1.Department of MathematicsShanghai Normal UniversityShanghaiPR China

Personalised recommendations