BIT Numerical Mathematics

, Volume 29, Issue 1, pp 126–139 | Cite as

A note on error expressions for reflected and averaged implicit Runge-Kutta methods

  • P. H. Muir
  • P. W. Beame
Part II Numerical Mathematics


In addition to their usefulness in the numerical solution of initial value ODE's, the implicit Runge-Kutta (IRK) methods are also important for the solution of two-point boundary value problems. Recently, several classes of modified IRK methods which improve significantly on the efficiency of the standard IRK methods in this application have been presented. One such class is the Averaged IRK methods; a member of the class is obtained by applying an averaging operation to a non-symmetric IRK method and its reflection. In this paper we investigate the forms of the error expressions for reflected and averaged IRK methods. Our first result relates the expression for the local error of the reflected method to that of the original method. The main result of this paper relates the error expression of an averaged method to that of the method upon which it is based. We apply these results to show that for each member of the class of the averaged methods, there exists an embedded lower order method which can be used for error estimation, in a formula-pair fashion.

AMS (MOS) subject classification

65L05 65L10 


implicit Runge-Kutta methods error coefficients reflected methods averaged methods 


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Copyright information

© BIT Foundations 1989

Authors and Affiliations

  • P. H. Muir
    • 1
    • 2
  • P. W. Beame
    • 1
    • 2
  1. 1.Department of Mathematics and Computing ScienceSaint Mary's UniversityHalifaxCanada
  2. 2.Department of Computer ScienceUniversity of WashingtonSeattleU.S.A.

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