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One-step 9-stage Hermite–Birkhoff–Taylor ODE solver of order 10

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Abstract

A one-step 9-stage Hermite–Birkhoff–Taylor method of order 10, denoted by HBT(10)9, is constructed for solving nonstiff systems of first-order differential equations of the form y′=f(x,y), y(x 0)=y 0. The method uses y′ and higher derivatives y (2) to y (4) as in Taylor methods and is combined with a 9-stage Runge–Kutta method. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to Taylor- and Runge–Kutta-type order conditions which are reorganized into Vandermonde-type linear systems whose solutions are the coefficients of the method. The new method has a larger scaled interval of absolute stability than Dormand–Prince DP(8,7)13M. The stepsize is controlled by means of y (2) and y (4). HBT(10)9 is superior to DP(8,7)13M and Taylor method of order 10 in solving several problems often used to test high-order ODE solvers on the basis of the number of steps, CPU time, and maximum global error. These numerical results show the benefits of adding high-order derivatives to Runge–Kutta methods.

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References

  1. Barrio, R.: Sensitivity analysis of ODEs/DAEs using the Taylor series method. SIAM J. Sci. Comput. 27(6), 1929–1947 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Barrio, R., Blesa, F., Lara, M.: VSVO formulation of the Taylor method for the numerical solution of ODEs. Comput. Math. Appl. 50, 93–111 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Berntsen, J., Espelid, T.O.: Error estimation in automatic quadrature routines. ACM Trans. Math. Softw. 17, 233–255 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  4. Butcher, J.C.: On Runge–Kutta processes of high order. J. Aust. Math. Soc. 18, 50–64 (1964)

    MATH  MathSciNet  Google Scholar 

  5. Butcher, J.C.: The Numerical Analysis of Ordinary Differential Equations. Wiley, London (1987)

    MATH  Google Scholar 

  6. Corliss, G.F., Chang, Y.F.: Solving ordinary differential equations using Taylor series. ACM Trans. Math. Softw. 8(2), 114–144 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  7. Davis, P.J., Rabinowitz, P.: Numerical Integration. Blaisdell, Waltham (1967)

    MATH  Google Scholar 

  8. Deprit, A., Zahar, R.M.W.: Numerical integration of an orbit and its concomitant variations. Z. Angew. Math. Phys. 17, 425–430 (1966)

    Article  Google Scholar 

  9. Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I. Nonstiff Problems. Springer, Berlin (1993). Section III.8

    MATH  Google Scholar 

  10. Hajji, M.A., Vaillancourt, R.: Matrix derivation of Gaussian quadratures. Sci. Proc. Riga Techn. Univ. 29(48), 198–213 (2006)

    Google Scholar 

  11. Hoefkens, J., Berz, M., Makino, K.: Computing validated solutions of implicit differential equations. Adv. Comput. Math. 19, 231–253 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Lara, M., Elipe, A., Palacios, M.: Automatic programming of recurrent power series. Math. Comput. Simul. 49, 351–362 (1999)

    Article  MathSciNet  Google Scholar 

  13. Nedialkov, N.S., Jackson, K.R., Corliss, G.F.: Validated solutions of initial value problems for ordinary differential equations. Appl. Math. Comput. 105, 21–68 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  14. Piessens, R., de Doncker-Kapenga, E., Überhuber, C.W., Kahaner, D.K.: QUADPACK. A Subroutine Package for Automatic Integration. Springer Series in Comput. Math., vol. 1. Springer, Berlin (1983)

    MATH  Google Scholar 

  15. Prince, P.J., Dormand, J.R.: High order embedded Runge–Kutta formulae. J. Comput. Appl. Math. 7(1), 67–75 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  16. Rabe, E.: Determination and survey of periodic Trojan orbits in the restricted problem of three bodies. Astron. J. 66(9), 500–513 (1961)

    Article  MathSciNet  Google Scholar 

  17. Sharp, P.W.: Numerical comparison of explicit Runge–Kutta pairs of orders four through eight. ACM Trans. Math. Softw. 17, 387–409 (1991)

    Article  MATH  Google Scholar 

  18. Steffensen, J.F.: On the restricted problem of three bodies. Danske Vid. Selsk., Mat.-fys. Medd. 30(18), 17 (1956)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, K1N 6N5, Canada

    Truong Nguyen-Ba, Vladan Bozic, Emmanuel Kengne & Rémi Vaillancourt

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  1. Truong Nguyen-Ba
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  2. Vladan Bozic
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  3. Emmanuel Kengne
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  4. Rémi Vaillancourt
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Correspondence to Rémi Vaillancourt.

Additional information

This work was supported in part by the Natural Sciences and Engineering Research Council of Canada and the Centre de recherches mathématiques of the Université de Montréal.

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Nguyen-Ba, T., Bozic, V., Kengne, E. et al. One-step 9-stage Hermite–Birkhoff–Taylor ODE solver of order 10. J. Appl. Math. Comput. 31, 335–358 (2009). https://doi.org/10.1007/s12190-008-0216-3

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  • DOI: https://doi.org/10.1007/s12190-008-0216-3

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