A note on error expressions for reflected and averaged implicit Runge-Kutta methods
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 classification65L05 65L10
Keywordsimplicit Runge-Kutta methods error coefficients reflected methods averaged methods
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- 1.U. Ascher and R. Weiss,Collocation for singular perturbation problems I: first order systems with constant coefficients, SIAM J. Numer. Anal. 20 (1983), 537–557.Google Scholar
- 2.U. Ascher and R. Weiss,Collocation for singular perturbation problems III: nonlinear problems without turning points, SIAM J. Sci. Stat. Comp. 5 (1984), 811–829.Google Scholar
- 3.J. C. Butcher,The Numerical Analysis of Ordinary Differential Equations, John Wiley and Sons, Toronto, 1987.Google Scholar
- 4.J. R. Cash and A. Singhal,High order methods for the numerical solution of two-point boundary value problems, BIT 22 (1982), 184–199.Google Scholar
- 5.W. H. Enright and P. H. Muir,Efficient classes of Runge-Kutta methods for two-point boundary value problems, Computing 37 (1986), 315–334.Google Scholar
- 6.S. Gupta,An adaptive boundary value Runge-Kutta solver for first order boundary value problems, SIAM J. Numer. Anal. 22 (1985), 114–126.Google Scholar
- 7.E. Hairer, S. P. Nørsett, and G. Wanner,Solving Ordinary Differential Equations I: Non-Stiff Problems, Springer-Verlag, New York, 1987.Google Scholar
- 8.P. H. Muir and W. H. Enright,Relationships among some classes of implicit Runge-Kutta methods and their stability functions, BIT 27 (1987), 403–423.Google Scholar
- 9.R. Scherer and H. Türke,Reflected and transposed methods, BIT 23 (1983), 262–266.Google Scholar