Advertisement

Initial-value problems: multi-step methods

  • L. Fox
  • D. F. Mayers

Abstract

The one-step finite-difference methods of the previous chapter, with the possible exception of the trapezoidal rule method, are basically rather too uneconomic for general use. Their local truncation error is of rather low order, with the result that to achieve good accuracy we need to use either: (a) a rather small step-by-step interval, involving many steps to cover a specified range; or (b) one of our correcting devices which involve some additional computation and more computer programming. The special one-step methods of the previous chapter, the Taylor series method and the explicit Runge-Kutta methods, do not share these disadvantages, but they involve extra computer storage, the Taylor series method involves a possibly large amount of non-automatic differentiation, and with the Runge—Kutta methods we have to compute possibly rather complicated expressions several times in each step. Moreover, both these methods have rather poor partial stability properties.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [11]
    J.D. Lambert (1973), Computational Methods in Ordinary Differential Equations, WileyGoogle Scholar
  2. [12]
    C.W. Gear (1971), Numerical Initial-value Problems in Ordinary Differential Equations, Prentice-Hall.Google Scholar
  3. [13]
    L.F. Shampine and M.K. Gordon (1975), Computer Solution of Ordinary Differential Equations: The Initial-value Problem, Freeman.Google Scholar
  4. [14]
    G. Hall and J.M. Watt (Eds.) (1976), Modern Numerical Methods for Ordinary Differential Equations (proceedings of Liverpool-Manchester Summer School), Clarendon Press, OxfordGoogle Scholar
  5. [15]
    I. Gladwell and D. Sayers (Eds.) (1980), Computational Techniques for Ordinary Differential Equations (proceedings of an I.M.A. Symposium), Academic Press.Google Scholar
  6. [16]
    A.R. Curtis (1980), The FACSIMILE numerical integrator for stiff initial- value problems.Google Scholar
  7. [17]
    D. Barton, J.M. Willers and R.V.M. Zahar (1972), Taylor series method for ordinary differential equations (in Mathematical Software, Ed. J.R. Rice), Academic Press.Google Scholar
  8. [18]
    D. Barton (1980), On Taylor series and stiff systems. TOMS, 6, 280–294.CrossRefGoogle Scholar
  9. [19]
    G. Corliss and Y.F. Chang (1982) Solving ordinary differential equations using Taylor series. TOMS, 8, 114–144.Google Scholar
  10. [20]
    J.R. Cash (1983), Block Runge-Kutta methods for the numerical integration of initial-value problems in ordinary differential equations. I The non-stiff case. Math. Comp ., 40, 175–191.Google Scholar
  11. [21]
    J.R. Cash (1983), Block Runge-Kutta methods for the numerical integration of initial-value problems in order differential equations. II The stiff case. Math. Comp ., 40, 193–206.Google Scholar
  12. [22]
    J.C. Butcher, K. Burrage and F.H. Chipman (1979), STRIDE: Stable Runge-Kutta integration for differential equations. Report Series No. 150 (Computational Mathematics No. 20), Dept. of Mathematics, University of Auckland.Google Scholar
  13. [23]
    L. Fox and E.T. Goodwin (1949), Some new methods for the numerical integration of ordinary differential equations. Proc. Camb. Phil Soc ., 45, 373–388.CrossRefGoogle Scholar
  14. [24]
    R.K. Jain, N.S. Kambo and R. Goel (1984), A sixth-order P-stable symmetric multi-step method for periodic initial-value problems of second- order differential equations. I.M.A.J. Num. Anal., 4, 117–125.Google Scholar
  15. [25]
    L. Fox, D.F. Mayers, J.R. Ockendon and A.B. Taylor (1971), On a functional differential equation. J. Inst. Maths. Applies., 8, 271–307.CrossRefGoogle Scholar
  16. [26]
    C.W. Cryer (1972), Numerical methods for functional differential equations, pp. 97–102 of Delay and Functional Differential Equations and Their Applications. (Ed. K. Schmitt), Academic Press.Google Scholar
  17. [27]
    E. Hansen (Ed.) (1969), Topics in Interval Analysis, Clarendon Press, Oxford.Google Scholar
  18. [28]
    Kruckeberg (1969), Ordinary differential equations (from reference [27]).Google Scholar
  19. [29]
    K.L.E. Nickel (1980),Interval Mathematics. Academic Press.Google Scholar

Copyright information

© Fox and Mayers 1987

Authors and Affiliations

  • L. Fox
    • 1
  • D. F. Mayers
    • 1
  1. 1.Oxford UniversityUK

Personalised recommendations