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Tolerance proportionality in ODE codes

Part of the Lecture Notes in Mathematics book series (LNM,volume 1386)

Abstract

Some bounds on the global error of the numerical solution of the initial value problem for a system of ordinary differential equations (ODEs) are developed that show the behavior with respect to the local error tolerance. Problems with solutions that are not smooth at a few points are considered. It is shown how to do local extrapolation with the backward differentiation formulas. An error criterion is proposed that has most of the advantages of both error per step and error per unit step.

This work was partially supported by the Applied Mathematical Sciences program of the Office of Energy Research under DOE grant DE-FG05-86ER25024.

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References

  1. G.D. BYRNE and A.C. HINDMARSH. A polyalgorytm for the numerical solution of ordinary differential equations. ACM TOMS, 1 (1975) 71–96.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. W.H. ENRIGHT. Using a testing package for the automatic assessment of numerical methods for ODE's. pp. 199–217 in L.D. Fosdick, ed., Performance Evaluation of Numerical Software, North-Holland, Amsterdam, 1979.

    Google Scholar 

  3. C.W. GEAR. Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs, NJ, 1971.

    MATH  Google Scholar 

  4. E. HAIRER, S.P. NORSETT and G. WANNER. Solving Ordinary Differential Equations I Nonstiff Problems. Springer, Berlin, 1987.

    CrossRef  MATH  Google Scholar 

  5. L.F. SHAMPINE. Limiting precision in differential equation solvers. Math. Comp., 27 (1974) 141–144.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. _____. Local error control in codes for ordinary differential equations. Appl. Math. Comput., 3 (1977) 189–210.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. _____. The step sizes used by one-step codes for ODEs. Appl. Numer. Math., 1 (1985) 95–106.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. _____ and R.C. ALLEN, Jr., Numerical Computing: An Introduction. W.B.Saunders, Philadelphia, 1973.

    MATH  Google Scholar 

  9. _____ and M.K. GORDON. Computer Solution of Ordinary Differential Equations: the Initial Value Problem. W.H.Freeman, San Francisco, 1975.

    MATH  Google Scholar 

  10. _____ and H.A. WATTS, The art of writing a Runge-Kutta code. II, Appl. Math. Comput., 5 (1979) 93–121.

    CrossRef  MATH  Google Scholar 

  11. __________ and S.M. DAVENPORT, Solving nonstiff ordinary differential equations-the state of the art. SIAM Rev., 18 (1976) 376–411.

    CrossRef  MathSciNet  MATH  Google Scholar 

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© 1989 Springer-Verlag

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Shampine, L.F. (1989). Tolerance proportionality in ODE codes. In: Bellen, A., Gear, C.W., Russo, E. (eds) Numerical Methods for Ordinary Differential Equations. Lecture Notes in Mathematics, vol 1386. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0089235

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  • DOI: https://doi.org/10.1007/BFb0089235

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51478-7

  • Online ISBN: 978-3-540-48144-7

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