Numerische Mathematik

, Volume 27, Issue 1, pp 21–39 | Cite as

On the estimation of errors propagated in the numerical integration of ordinary differential equations

  • Pedro E. Zadunaisky


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.


Boundary Condition Differential Equation Partial Differential Equation Ordinary Differential Equation Finite Difference 
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  1. 1.
    Alfeld, P.: A Survey of Zadunaisky's Device Applied to Ordinary Differential Equations, M. Sc. Dissertation, Univ. of Dundee, Scotland, 1975Google Scholar
  2. 2.
    Fyfe, D. J.: The Use of Cubic Splines in the Solution of Two Point Boundary Value Problems. Comput. J.12, 188–192 (1969)Google Scholar
  3. 3.
    Fehlberg, E.: Classical Fifth-, Sixth-, Seventh- and Eighth Order Runge-Kutta Formulas with Stepsize Control. NASA TR R-287; also Computing4, 93 (1969)Google Scholar
  4. 4.
    Frank, R.: Schätzungen des Globalen Diskretisierungsfehlers bei Runge-Kutta Methoden. ISNM 27. Bassel, Stuttgart: Birkhäuser 1975, pp. 45–70Google Scholar
  5. 5.
    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, 1975Google Scholar
  6. 6.
    Henrici, P.: Discrete Variable Methods in Ordinary Differntial Equations. New York: John Wiley & Sons 1962Google Scholar
  7. 7.
    Henrici, P.: Error Propagation for Difference Methods. New York: John Wiley & Sons 1963Google Scholar
  8. 8.
    Hockney, R. W.: The Potential Calculation and Some Applications. Methods in Computational Physics. vol. 9, New York, London: Acad. Press 1969Google Scholar
  9. 9.
    Lanczos, C.: Linear Differential Operators. New York: van Nostrand 1961Google Scholar
  10. 10.
    Lawson, J. D., Ehle, B. L.: Asymptotic Error Estimation for One-Step Methods Based on Quadrature, Aequationes Mathematicae, vol. 5, 1970Google Scholar
  11. 11.
    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 1974Google Scholar
  12. 12.
    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 1948Google Scholar
  13. 13.
    Runge, C.: Über empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten. Z. Math. Phys.XLVI, 229 (1901)Google Scholar
  14. 14.
    Sterne, T. E.: The accuracy of Numerical Solutions of Ordinary Differential Equations. In Math. Tables and Other Aids to ComputationVII, 43, 159–164 (1953)Google Scholar
  15. 15.
    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)Google Scholar
  16. 16.
    Zadunaisky, P. E.: The Motion of Halley's Comet During the Return of 1910. Astr. J.71, 20–27 (1966)Google Scholar
  17. 17.
    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 1966Google Scholar
  18. 18.
    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. 1970Google Scholar
  19. 19.
    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. 1972Google Scholar
  20. 20.
    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, 1967Google Scholar

Copyright information

© Springer-Verlag 1976

Authors and Affiliations

  • Pedro E. Zadunaisky
    • 1
  1. 1.Departamento de Matemática AplicadaObservatorio Nacional de Física CósmicaSan Miguel (FCNGSM)Argentina

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