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

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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.

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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.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Ave, Ottawa, ON, K1N 6N5, Canada

    Abdulrahman Karouma, Truong Nguyen-Ba, Thierry Giordano & Rémi Vaillancourt

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  1. Abdulrahman Karouma
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  2. Truong Nguyen-Ba
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  3. Thierry Giordano
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  4. Rémi Vaillancourt
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Corresponding author

Correspondence to Abdulrahman Karouma.

Additional information

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

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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

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

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