Towards Improved Error Estimates for Higher Order Time Integration of ODEs with Non-Smooth Right Hand Side

Conference paper


The classical convergence analysis of higher order ODE time integration methods is based on rather strong smoothness assumptions on the right hand side that are typically not satisfied in technical applications since spline approximations of input functions and look-up tables result in frequent discontinuities in derivatives of the right hand side. Practical experience shows, however, that nevertheless the resulting non-smooth model equations may often be solved efficiently by higher order ODE time integration methods. For one typical problem class, the present paper gives a theoretical explanation of this behaviour. The results of the theoretical analysis are illustrated by a series of numerical tests for the simplified model of an agricultural device that moves along a track being defined by the spline approximation of a periodic smooth input function.


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  1. 1.
    Ascher, U., Petzold, L.: Computer Methods for Ordinary Differential Equations and Differential–Algebraic Equations. SIAM, Philadelphia (1998)MATHGoogle Scholar
  2. 2.
    Hairer, E., Nørsett, S., Wanner, G.: Solving Ordinary Differential Equations. I. Nonstiff Problems. 2nd edn. Springer–Verlag, Berlin Heidelberg New York (1993)MATHGoogle Scholar
  3. 3.
    Moler, C.: Numerical Computing with MATLAB. SIAM, Philadelphia (2004)MATHGoogle Scholar
  4. 4.
    Bellen, A., Zennaro, M.: Numerical Methods for Delay Differential Equations. Oxford University Press, Oxford, UK (2003)MATHCrossRefGoogle Scholar
  5. 5.
    Walter, W.: Ordinary Differential Equations. Number 182 in Graduate Texts in Mathematics. Springer (1998)Google Scholar
  6. 6.
    Freund, R., Hoppe, R.: Stoer/Bulirsch: Numerische Mathematik 1. 10th edn. Springer–Verlag, Berlin, Heidelberg (2007)MATHGoogle Scholar
  7. 7.
    Hertig, T.: Splineinterpolation bei der numerischen Lösung von gewöhnlichen Differentialgleichungen mit Hilfe expliziter Runge-Kutta-Verfahren. Diploma Thesis, Martin Luther University Halle-Wittenberg (Germany), Department of Mathematics and Computer Science (2005)Google Scholar
  8. 8.
    Pickel, P.: Simulation fahrdynamischer Eigenschaften von Traktoren. Fortschritt-Berichte VDI Reihe 14, Nr. 65. VDI–Verlag, Düsseldorf (1993)Google Scholar
  9. 9.
    Dormand, J., Prince, P.: A family of embedded Runge–Kutta formulae. J. Comp. Appl. Math. 6 (1980) 19–26MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.NWF III – Institute of MathematicsMartin Luther University Halle-WittenbergHalle (Saale)Germany

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