W. H. Enright
Abstract
A general procedure for the construction of interpolants for Runge-Kutta (RK) formulas is presented. As illustrations, this approach is used to develop interpolants for three explicit RK formulas, including those employed in the well-known subroutines RKF45 and DVERK. A typical result is that no extra function evaluations are required to obtain an interpolant with O(h5) local truncation error for the fifth-order RK formula used in RKF45; two extra function evaluations per step are required to obtain an interpolant with O(h6) local truncation error for this RK formula.
- 1 BELLEN, A., AND ZENNARO, M. Stability properties of interpolants for Runge-Kutta methods. Res. Rep. 109, Instituto di Mathematica, Universit~ degli Studi di Trieste, Trieste, Italy, 1985. Submitted for publication.Google Scholar
- 2 BUTCHER, J.C. Coefficients for the study of Runge-Kutta integration processes. J. Australian Math. Soc. 3 (1963), 185-201.Google ScholarCross Ref
- 3 DORMAND, J. R., AND PRINCE, P. J. Runge-Kutta triples. Manuscript, Mathematics Dept., Teesside Polytechnic, Middlesbrough, U.K., 1986.Google Scholar
- 4 ENRIGHT, W. H., AND PRYCE, J.D. Two Fortran packages for assessing initial value methods. Rep. 167/83, Dept. of Computer Science, Univ. of Toronto, Toronto, Canada, 1983.Google Scholar
- 5 ENRIGHT, W. H., AND JACKSON, K. R., NORSETr, S. P., AND THOMSEN, P.G. Effective solution of discontinuous IVPs using a Runge-Kutta formula pair with interpolants. Rep. 113, Mathematics Dept., Univ. of Manchester, U.K., 1986. Submitted for publication.Google Scholar
- 6 ENRIGHT, W. H., JACKSON, K. R., N~RSETT, S. P., AND THOMSEN, P.G. Interpolants for Runge-Kutta formulas. Rep. 180/85, Dept. of Computer Science, Univ. of Toronto, Toronto, Canada, 1985.Google Scholar
- 7 FEHLBERG, E. Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kotrolle und ihre Anwendung auf W/irmeleitungsprobleme. Computing 6, 1-2 (1970), 61-71.Google ScholarCross Ref
- 8 GEAR, C. W. Runge-Kutta starters for multistep methods. ACM Trans. Math. Softw. 6, 3 (Sept. 1980), 263-279. Google ScholarDigital Library
- 9 GLADWELL, I., SHAMPINE, L. F., BACA, L. S., AND BRANKIN, R. W. Practical aspects of interpolation in Runge-Kutta codes. Rep. 102, Mathematics Dept., Univ. of Manchester, Manchester, U.K., 1985.Google Scholar
- 10 HORN, M.K. Fourth- and fifth-order, scaled Runge-Kutta algorithms for treating dense output. SIAM J. Numer. Anal. 20, 3 (June 1983), 558-568.Google ScholarCross Ref
- 11 HORN, M. K. Scaled Runge-Kutta algorithms for handling dense output. Rep. DFVLR-FB 81-13, Deutsche Forschungs- und Versuchsanstalt f(ir Luft- und Raumfahrt, Oberpfaffenhofen, West Germany, 1981.Google Scholar
- 12 HULL, T. E., ENRIGHT, W. H., AND JACKSON, K.R. User's guide for DVERK--A subroutine for solving nonstiff ODE's. Rep. 100, Dept. of Computer Science, Univ. of Toronto, Toronto, Canada, 1976.Google Scholar
- 13 HULL, T. E., ENRIGHT, W. H., FELLEN, B. M., AND SEDGWICK, A.E. Comparing numerical methods for ordinary differential equations. SIAM J. Numer. Anal. 9, 4 (Dec. 1972), 603-637.Google ScholarCross Ref
- 14 SHAMPINE, L. F., ANn WATTS, H. A. DEPAC--Design of a user-oriented package of ODE solvers. Rep. SAND79-2374, Sandia National Laboratories, Albuquerque, N.M., 1980.Google Scholar
- 15 SHAMP{NE, L. F., AND WATTS, H.A. Practical solution of ordinary differential equations by Runge-Kutta methods. Rep. SAND76-0585, Sandia National Laboratories, Albuquerque, N.M., 1976.Google Scholar
- 16 SHAMPINE, L. F. Interpolation for Runge-Kutta methods. SIAM J. Numer. Anal. 22, 5 (Oct. 1985), 1014-1027.Google ScholarCross Ref
- 17 SHAMPINE, L.F. Some practical Runge-Kutta formulas. Math. Comput. 46, 173 (Jan. 1986), 135-150. Google ScholarDigital Library
- 18 STETTER, I-I. J. Interpolation and error estimation in Adams PC-codes. SIAM J. Numer. Anal. 16, 2 (Apr. 1979), 311-323.Google Scholar
- 19 ZENNARO, M. Natural continuous extensions of Runge-Kutta methods. Math. Comput. 46, 173 (Jan. 1986), 119-133. Google ScholarDigital Library
Index Terms
- Interpolants for Runge-Kutta formulas
Recommendations
Highly continuous Runge-Kutta interpolants
To augment the discrete Runge-Kutta solutlon to the mitlal value problem, piecewlse Hermite interpolants have been used to provide a continuous approximation with a continuous first derivative We show that it M possible to construct mterpolants with ...
Analysis of a Chebyshev-based backward differentiation formulae and relation with Runge-Kutta collocation methods
The construction of a class of backward differentiation formulae based on intra-step Chebyshev-Gauss-Lobatto nodal points, suitable for the approximate numerical integration of initial-value problems of first-order ordinary differential equations, is ...
The NUMOL solution of time-dependent PDEs using DESI Runge-Kutta formulae
In a recent publication Butcher and Cash introduced a new class of L-stable implicit Runge-Kutta formulae in an attempt to eliminate the disadvantages of SIRK and DIRK formulae and to combine their advantages. These formulae, namely Diagonally Extended ...
Reviews
John Charles Butcher
This paper considers the enhancement of a given Runge-Kutta method by the addition of an interpolation formula. The interpolation formula proposed makes use of the same derivative values as are used in the main method. It also possibly uses the derivative found at the end of the step and the derivative at an additional, appropriately chosen off-step point. The aim is to maintain an accuracy for interpolated results that is as close as possible to that of the principal method on which it is based. Such extended Runge-Kutt- a methods as these can be used to provide dense output and to enable discontinuities to be located efficiently. For three representative methods, each of which is already provided with a built-in error estimator, formulas are found for providing this interpolation and each extended method is tested with a collection of standard problems. At best, the results achieved are all that could be hoped for, with off-step interpolated results of comparable accuracy to that of the main method. However, there is a wide spread of observed behaviors; at worst, the errors can be much larger than those at the step values.
Access critical reviews of Computing literature here
Become a reviewer for Computing Reviews.
Comments