Advertisement

BIT Numerical Mathematics

, Volume 21, Issue 2, pp 175–189 | Cite as

A generalization of singly-implicit methods

  • J. C. Butcher
Part II Numerical Mathematics

Abstract

Singly-implicit Runge-Kutta methods are characterized by a one-point spectrum property of the coefficient matrix. If a method of this type is also a collocation method, then its abscissae are proportional to the zeros of a Laguerre polynomial. The generalization introduced here is a multistep method in the style of Nordsieck and also a multistage method under the one-point spectrum constraint. It is found that much of the theory of singly-implicit methods carries over but with Laguerre polynomials replaced by their usual generalizations. Amongst the formal properties of the new method which are studied is a derivation of the similarity transformations which allow their efficient implementation. A preliminary investigation is made of the stability of the new methods.

CR classification

5.17 

AMS classification

65L05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    K. Burrage,A special family of Runge-Kutta methods for solving stiff differential equations, BIT 18 (1978), 22–41.Google Scholar
  2. 2.
    K. Burrage, J. C. Butcher and F. H. Chipman,An implementation of singly-implicit Runge-Kutta methods, BIT 20 (1980), 326–340.Google Scholar
  3. 3.
    J. C. Butcher,On the implementation of implicit Runge-Kutta methods, BIT 16 (1976), 237–240.Google Scholar
  4. 4.
    J. C. Butcher,A transformed implicit Runge-Kutta method, J. Assoc. Comput. Mach. 26 (1979), 731–738.Google Scholar
  5. 5.
    J. C. Butcher,A generalization of singly implicit methods, Auckland Computer Science Report No. 22, University of Auckland, 1980.Google Scholar
  6. 6.
    J. C. Butcher, K. Burrage and F. H. Chipman,STRIDE: stable Runge-Kutta integrator for differential equations, Computational Mathematics Report No. 20, University of Auckland, 1979.Google Scholar
  7. 7.
    U. W. Hochstrasser,Orthogonal Polynomials, inHandbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, Dover, New York (1965), pp 771–802.Google Scholar
  8. 8.
    G. Wanner, E. Hairer and S. P. Nørsett,Order stars and stability theorems, BIT 18 (1978), 475–489.Google Scholar

Copyright information

© BIT Foundations 1981

Authors and Affiliations

  • J. C. Butcher
    • 1
  1. 1.Department of Computer ScienceUniversity of AucklandAucklandNew Zealand

Personalised recommendations