Skip to main content
Log in

A new class of efficient one-step contractivity preserving high-order time discretization methods of order 5 to 14

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

A family of one-step, explicit, contractivity preserving, multi-stage, multi-derivative, Hermite–Birkhoff–Taylor methods of order p = 5,6,…,14, that we denote by CPHBTRK4(d,s,p), with nonnegative coefficients are constructed by casting s-stage Runge–Kutta methods of order 4 with Taylor methods of order d. The constructed CPHBTRK4 methods are implemented using efficient variable step control and are compared to other well-known methods on a variety of initial value problems. A selected method: CP 6-stages 9-derivative HBT method of order 12, denoted by CPHBTRK412, has larger region of absolute stability than Dormand–Prince DP(8,7)13M and Taylor method T(12) of order 12. It is superior to DP(8,7)13M and T(12) methods on the basis the number of steps, CPU time, and maximum global error on several problems often used to test higher-order ODE solvers. Also, we show that the contractivity preserving property of CPHBTRK412is very efficient in suppressing the effect of the propagation of discretization errors and the new method compares positively with explicit 17 stages Runge-Kutta-Nyström pair of order 12 by Sharp et al. on a long-term integration of a standard N-body problem. The selected CPHBTRK412is listed in the Appendix.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arenstorf, R. F.: Periodic solutions of the restricted three body problem representing analytic continuations of Keplerian elliptic motions. Amer. J. Math. 163 (2), 525–545 (1963)

    MathSciNet  MATH  Google Scholar 

  2. Barrio, R.: Performance of the taylor series method for ODEs/DAEs. Appl. Math. Comput. 163(2), 525–545 (2005)

    MathSciNet  MATH  Google Scholar 

  3. Barrio, R., Blesa, F., Lara, M.: VSVO Formulation of the Taylor method for the numerical solution of ODEs. Comput. Math. Appl. 50(1), 93–111 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Binney, J., Tremaine, S.: Galactic dynamics. Princeton University Press, Princeton (2011)

    MATH  Google Scholar 

  5. Bozic, V.: Three-stage Hermite–Birkhoff–Taylor ODE solver with a C+ + program. Master’s thesis, University of Ottawa, Canada (2008)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Deprit, A., Zahar, R.: Numerical integration of an orbit and its concomitant variations by recurrent power series. Z. Angew. Math. Phys. 17(3), 425–430 (1966)

    Article  Google Scholar 

  8. Gottlieb, S., Ketcheson, D. I., Shu, C.W.: High order strong stability preserving time discretization. J. Sci. Comput. 38(3), 251–289 (2009). https://doi.org/10.1007/s10915-008-9239-z

    Article  MathSciNet  MATH  Google Scholar 

  9. Gottlieb, S., Ketcheson, D. I., Shu, C. W.: Strong stability preserving Runge–Kutta and multistep time discretizations. World Scientific, Singapore (2011)

    Book  MATH  Google Scholar 

  10. Hairer, E., Nørsett, S., Wanner, G.: Solving ordinary differential equations I: nonstiff problems (Springer Series In Computational Mathematics). Springer, Berlin (2009)

    MATH  Google Scholar 

  11. Hénon, M., Heiles, C.: The applicability of the third integral of motion: some numerical experiments. Astron. J. 69, 73 (1964)

    Article  MathSciNet  Google Scholar 

  12. Hoefkens, J., Berz, M., Makino, K.: Computing valiyeard solutions of implicit differential equations. Adv. Comput. Math. 19(1-3), 231–253 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Huang, C.: Strong stability preserving hybrid methods. Appl. Numer. Math. 59, 891–904 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hull, T., Enright, W., Fellen, B., Sedgwick, A.: Comparing numerical methods for ordinary differential equations. SIAM J. Numer. Anal. 9(4), 603–637 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jorba, À., Zou, M.: A software package for the numerical integration of odes by means of high-order taylor methods. Experiment. Math. 14(1), 99–117 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kennedy, C. A., Carpenter, M. K., Lewis, R. M.: Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations. Appl. Numer. Math. 35, 177–219 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kraaijevanger, J. F. B. M.: Contractivity of Runge–Kutta methods. BIT 31 (3), 482–528 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lambert, J. D.: Numerical methods for ordinary differential systems: the initial value problem. Wiley, Chichester (1991)

    MATH  Google Scholar 

  19. Lara, M., Elipe, A., Palacios, M.: Automatic programming of recurrent power series. Math. Comput. Simul. 49(4), 351–362 (1999)

    Article  MathSciNet  Google Scholar 

  20. Li, G.: Generation of rooted trees and free Trees. Ph.D. thesis, Citeseer (1996)

    Google Scholar 

  21. Li, Y.: Variable-step variable-order 3-stage Hermite–Birkhoff ODE solver of order 5 to 15 with a C+ + program. Master’s thesis, University of Ottawa, Canada (2008)

    Google Scholar 

  22. Nedialkov, N. S., Jackson, K. R., Corliss, G. F.: Valiyeard solutions of initial value problems for ordinary differential equations. Appl. Math. Comput. 105 (1), 21–68 (1999)

    MathSciNet  MATH  Google Scholar 

  23. Nguyen-Ba, T., Desjardins, S. J., Sharp, P. W., Vaillancourt, R.: Contractivity-preserving explicit Hermite–Obrechkoff ODE solver of order 13. Celest. Mech. Dyn. Astr. 117(4), 423–434 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Nguyen-Ba, T., Nguyen-Thu, H., Giordan, T., Vaillancourt, R.: One-step strong–stability–preserving Hermite–Birkhoff–Taylor methods. Scientific J. of Riga Technical University, col 45, 95–104 (2010)

    Google Scholar 

  25. Nguyen-Ba, T., Hao, H., Yagoub, H., Vaillancourt, R.: One-step 5-stage Hermite–Birkhoff–Taylor ODE solver of order 12. Appl. Math. Comput. 211 (2), 313–328 (2009)

    MathSciNet  MATH  Google Scholar 

  26. Nguyen-Ba, T., Hao, H., Yagoub, H., Vaillancourt, R.: One-step 5-stage Hermite–Birkhoff–Taylor ode solver of order 12. Appl. Math. Comput. 211 (2), 313–328 (2009)

    MathSciNet  MATH  Google Scholar 

  27. Nguyen-Ba, T., Karouma, A., Giordano, T., Vaillancourt, R.: Strong-stability-preserving, one-step, 9-stage, Hermite–Birkhoff–Taylor, time-discretization methods combining taylor and RK4 methods. Bound. Field Probl. Comput. Simul. 51, 43–56 (2012)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Reich, S.: Preservation of adiabatic invariants under symplectic discretization. Appl. Numer. Math. 29(1), 45–55 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  30. Shampine, L. F., Gordon, M. K.: Computer solution of ordinary differential equations: the initial value problem. Freeman, San Francisco (1975)

    MATH  Google Scholar 

  31. Sharp, P. W.: N-body simulations: the performance of some integrators. ACM Trans. Math. Softw. (TOMS) 32(3), 375–395 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Sharp, P. W.: Numerical comparisons of some explicit Runge–Kutta pairs of orders 4 through 8. ACM Trans. Math. Softw. 17(3), 387–409 (1991)

    Article  MATH  Google Scholar 

  33. Shu, C. W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  34. Spiteri, R. J., Ruuth, S. J.: A new class of optimal high-order strong-stability-preserving time discretization methods. SIAM J. Numer. Anal. 40 (2), 469–491 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  35. Spiteri, R. J., Ruuth, S. J.: Non-linear evolution using optimal fourth-order strong-stability-preserving Runge–Kutta methods. Comput. Math. Appl. 62(1), 125–135 (2003)

    MathSciNet  MATH  Google Scholar 

  36. Szebehely, V.: Theory of orbits: the restricted problem of three bodies. Tech. rep., DTIC Document (1967)

  37. Sharp, P. W., Qureshi, M. A., Grazier, K. R.: High order explicit Runge–Kutta nyström pairs. Numer Algorithms 62(1), 133–148 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to dedicate this work to Prof. Remi Vaillancourt, may his gentle soul rest in peace. We express our sincere appreciation for his help and advice during this research. Thanks are due to Prof. Martín Lara and Philip Sharp for supplying the authors with their programs and sharing their expertise.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Abdulrahman Karouma.

Additional information

This work was supported in part by the Natural Sciences and Engineering Research Council of Canada.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Karouma, A., Nguyen-Ba, T., Giordano, T. et al. A new class of efficient one-step contractivity preserving high-order time discretization methods of order 5 to 14. Numer Algor 79, 251–280 (2018). https://doi.org/10.1007/s11075-017-0436-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-017-0436-4

Keywords

Mathematics Subject Classification (2010)

Navigation