Summary
In this paper we describe a method for the estimation of global errors. An heuristic condition of validity of the method is given and several applications are described in detail for problems of ordinary differential equations with either initial or two point boundary conditions solved by finite difference formulas. The main idea of the method can be extended to other type of problems and applications to a problem solved by spline functions and to some partial differential equations solved by finite differences methods are outlined.
Similar content being viewed by others
References
Alfeld, P.: A Survey of Zadunaisky's Device Applied to Ordinary Differential Equations, M. Sc. Dissertation, Univ. of Dundee, Scotland, 1975
Fyfe, D. J.: The Use of Cubic Splines in the Solution of Two Point Boundary Value Problems. Comput. J.12, 188–192 (1969)
Fehlberg, E.: Classical Fifth-, Sixth-, Seventh- and Eighth Order Runge-Kutta Formulas with Stepsize Control. NASA TR R-287; also Computing4, 93 (1969)
Frank, R.: Schätzungen des Globalen Diskretisierungsfehlers bei Runge-Kutta Methoden. ISNM 27. Bassel, Stuttgart: Birkhäuser 1975, pp. 45–70
Frank, R.: The Method of Iterated Defect-Correction and its Application for Two-Point Boundary Value Problem. Institut für Numerische Mathematik, Technische Hochschule Wien, Report No. 8, 1975
Henrici, P.: Discrete Variable Methods in Ordinary Differntial Equations. New York: John Wiley & Sons 1962
Henrici, P.: Error Propagation for Difference Methods. New York: John Wiley & Sons 1963
Hockney, R. W.: The Potential Calculation and Some Applications. Methods in Computational Physics. vol. 9, New York, London: Acad. Press 1969
Lanczos, C.: Linear Differential Operators. New York: van Nostrand 1961
Lawson, J. D., Ehle, B. L.: Asymptotic Error Estimation for One-Step Methods Based on Quadrature, Aequationes Mathematicae, vol. 5, 1970
Pereyra, V.: Variable Order Variable Step Finite Difference Methods for Non Linear Boundary Value Problems. Conference on the Numerical Solution of Differential Equations, Dundee 1973. Lecture Notes in Mathematics 363. Berlin, Heidelberg, New York: Springer 1974
Rademacher, Hans A.: On the Accumulation of Errors in Processes of Integration on High Speed Calculating Machines. In: Proceedings of a Symposium on Large-Scale Digital Calculating Machinery, Cambridge, Mass.: Harvard Univ. Press 1948
Runge, C.: Über empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten. Z. Math. Phys.XLVI, 229 (1901)
Sterne, T. E.: The accuracy of Numerical Solutions of Ordinary Differential Equations. In Math. Tables and Other Aids to ComputationVII, 43, 159–164 (1953)
Stetter, H. J.: Economical Global Error Estimation. In: Proceedings of Symposium on Stiff Differential Systems, Wildbad, Oct. 1973, New York: Plenum Publ. Co., 1974 (IBM-Research Symposia Series)
Zadunaisky, P. E.: The Motion of Halley's Comet During the Return of 1910. Astr. J.71, 20–27 (1966)
Zadunaisky, P. E.: A Method for the Estimation of Errors Propagated in the Numerical Solution of a System of Ordinary Differential Equations. In: Proc. Intern. Astron. Union, Symposium No. 25, Thessaloniki, 1964, New York: Academic Press 1966
Zadunaisky, P. E.: On the Accuracy in the Numerical Computation of Orbits. In: ”Periodic Orbits, Stability and Resonances”, Symposium at the Univ. of Sao Paulo, Brasil 1969. Dordrecht: D. Reidel Public. Co. 1970
Zadunaisky, P. E.: On the Determination of Non-Gravitational Forces Acting on Comets. In: Proc. Intern. Astron. Union Symposium No. 45, Leningrad, U.S.S.R., 1970. Dordrecht: D. Reidel Publ. Co. 1972
Zani, R. C.: A Computer Study of the Estimated Propagation of Errors in the Numerical Integration of Ordinary Differential Equations; Thesis, Air Force Inst. of Technology, Wright-Patterson Air Force Base, Ohio, 1967
Author information
Authors and Affiliations
Additional information
Some results of the present work have been reported at the Conference on the Numerical Solution of Differential Equations, held at the University of Dundee, Scotland in 1973
Rights and permissions
About this article
Cite this article
Zadunaisky, P.E. On the estimation of errors propagated in the numerical integration of ordinary differential equations. Numer. Math. 27, 21–39 (1976). https://doi.org/10.1007/BF01399082
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01399082