Reducing the Uncertainty When Approximating the Solution of ODEs

  • Wayne H. Enright
Part of the IFIP Advances in Information and Communication Technology book series (IFIPAICT, volume 377)


One can reduce the uncertainty in the quality of an approximate solution of an ordinary differential equation (ODE) by implementing methods which have a more rigorous error control strategy and which deliver an approximate solution that is much more likely to satisfy the expectations of the user. We have developed such a class of ODE methods as well as a collection of software tools that will deliver a piecewise polynomial as the approximate solution and facilitate the investigation of various aspects of the problem that are often of as much interest as the approximate solution itself. We will introduce measures that can be used to quantify the reliability of an approximate solution and discuss how one can implement methods that, at some extra cost, can produce very reliable approximate solutions and therefore significantly reduce the uncertainty in the computed results.


Numerical methods initial value problems ODEs reliable methods defect control 


  1. 1.
    Enright, W.: A new error-control for initial value solvers. App. Math. Comp. 31, 288–301 (1989)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Enright, W.: The relative efficiency of alternative defect control schemes for high-order continuous Runge-Kutta formulas. SIAM Journal on Numerical Analysis 30(5), 1419–1445 (1993)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Enright, W., Jackson, W., Nφrsett, S., Thomsen, P.: Interpolants for Runge-Kutta formulas. ACM Transactions on Mathematical Software 12(3), 193–218 (1986)MATHCrossRefGoogle Scholar
  4. 4.
    Enright, W., Yan, L.: The Reliability/Cost trade-off for a class of ODE solvers. Numerical Algorithms 53(2), 239–260 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Enright, W., Muir, P.: New Interpolants for Asymptotically Correct Defect Control of BVODEs. Numerical Algorithms 53(2), 219–238 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Enright, W., Pryce, J.: Two FORTRAN packages for assessing initial value methods. ACM Transactions on Mathematical Software 13(1), 1–27 (1987)MATHCrossRefGoogle Scholar
  7. 7.
    Gladwell, I., Shampine, L., Baca, L., Brankin, R.: Practical aspects of interpolation in Runge-Kutta codes. SIAM Journal of Scientific and Statistical Computing (8), 322–341 (1987)Google Scholar
  8. 8.
    Hairer, E., Nφrsett, S., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, Berlin (1987)MATHGoogle Scholar
  9. 9.
    Shakourifar, M., Enright, W.: Reliable Approximate Solution of Systems of Volterra Integro-Differential Equations with Time Dependent Delays. SIAM Journal of Sc. Comp. 33, 1134–1158 (2011)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Shampine, L.: Interpolation for Runge-Kutta methods. SIAM Journal of Numererical Analysis (22), 1014–1027 (1985)Google Scholar
  11. 11.
    Shampine, L.: Solving ODEs and DDEs with residual control. Applied Numererical Mathematics (52), 113–127 (2005)Google Scholar
  12. 12.
    Zivaripiran, H., Enright, W.: An Efficient Unified Approach for the Numerical Solution of Delay Differential Equations. Numerical Algorithms 53(2), 397–417 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2012

Authors and Affiliations

  • Wayne H. Enright
    • 1
  1. 1.Department of Computer ScienceUniversity of TorontoCanada

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