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Contractivity-preserving explicit Hermite–Obrechkoff ODE solver of order 13

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Abstract

A new optimal, explicit, Hermite–Obrechkoff method of order 13, denoted by HO(13), that is contractivity-preserving (CP) and has nonnegative coefficients is constructed for solving nonstiff first-order initial value problems. Based on the CP conditions, the new 9-derivative HO(13) has maximum order 13. The new method usually requires significantly fewer function evaluations and significantly less CPU time than the Taylor method of order 13 and the Runge–Kutta method DP(8,7)13M to achieve the same global error when solving standard \(N\)-body problems.

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Acknowledgments

The authors thank the reviewers for their suggestions, particularly those related to the \(N\)-body problems. The authors also thank Martín Lara for supplying the authors with his programs and sharing his experience with it. This work was supported by the Natural Sciences and Engineering Research Council of Canada.

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Authors and Affiliations

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

    Truong Nguyen-Ba, Steven J. Desjardins & Rémi Vaillancourt

  2. Department of Mathematics, University of Auckland, Auckland, New Zealand

    Philip W. Sharp

Authors
  1. Truong Nguyen-Ba
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  2. Steven J. Desjardins
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  3. Philip W. Sharp
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  4. Rémi Vaillancourt
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Correspondence to Philip W. Sharp.

Appendix: The HO(13) formula

Appendix: The HO(13) formula

For the new HO(13) method \(c(\text {HO(13)})= 0.687\) (3dp), \(c_{\text {eff}}(\text {HO(13)})= 0.0763\) (4dp), the unscaled stability interval is \((-1.83,\ldots , 0)\) (2dp), and

$$\begin{aligned} y_{n+1}&= 4.5537344707090732 \,\text {e-}01 y_{n-1} + 5.4462655292909279 \,\text {e-}01 y_{n} \\&\quad +\,6.6273842336814770 \,\text {e-}01 \Delta t f_{n-1} + 7.9263502370275962 \,\text {e-}01 \Delta t f_{n} \\&\quad +\,3.6304031487037802 \,\text {e-}01 \Delta t^2 y_{n-1}^{(2)} + 5.7201138496231607 \,\text {e-}01 \Delta t^2 y_{n}^{(2)} \\&\quad +\,1.6194516630180894 \,\text {e-}01 \Delta t^3 y_{n-1}^{(3)} + 1.1228817806297973 \,\text {e-}01 \Delta t^3 y_{n}^{(3)} \\&\quad +\,5.1751955213728010 \,\text {e-}02 \Delta t^4 y_{n-1}^{(4)} + 6.1822230586295386 \,\text {e-}02 \Delta t^4 y_{n}^{(4)} \\&\quad +\,1.5800102293137556 \,\text {e-}02 \Delta t^5 y_{n-1}^{(5)} + 0.0 \, \Delta t^5 y_{n}^{(5)} \\&\quad +\,4.5713674152395474 \,\text {e-}03 \Delta t^6 y_{n-1}^{(6)} + 3.4961844977613780 \,\text {e-}03 \Delta t^6 y_{n}^{(6)} \\&\quad +\,9.3883945698842422 \,\text {e-}04 \Delta t^7 y_{n-1}^{(7)} + 3.7179156905590997 \,\text {e-}06 \Delta t^7 y_{n}^{(7)}\\&\quad +\,1.1501709391358726 \,\text {e-}04 \Delta t^8 y_{n-1}^{(8)} + 5.4812853003565492 \,\text {e-}06 \Delta t^8 y_{n}^{(8)}\\&\quad +\,6.6147276073533409 \,\text {e-}06 \Delta t^9 y_{n-1}^{(9)} + 8.4875255409530395 \,\text {e-}06 \Delta t^9 y_{n}^{(9)}. \end{aligned}$$

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Nguyen-Ba, T., Desjardins, S.J., Sharp, P.W. et al. Contractivity-preserving explicit Hermite–Obrechkoff ODE solver of order 13. Celest Mech Dyn Astr 117, 423–434 (2013). https://doi.org/10.1007/s10569-013-9520-9

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