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**New stable closed Newton-Cotes trigonometrically fitted formulae for long-time integration.**
*(English)*
Zbl 1242.65026

Summary: The closed Newton-Cotes differential methods of high algebraic order for small number of function evaluations are unstable. In this work, we propose a new closed Newton-Cotes trigonometrically fitted differential method of high algebraic order which gives much more efficient results than the well-know ones.

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\textit{T. E. Simos}, Abstr. Appl. Anal. 2012, Article ID 182536, 15 p. (2012; Zbl 1242.65026)

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[1] | E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, vol. 31 of Springer Series in Computational Mathematics, Springer, Berlin, Germany, 2nd edition, 2006. · Zbl 1149.65054 |

[2] | J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problem, vol. 7 of Applied Mathematics and Mathematical Computation, Chapman & Hall, London, UK, 1994. · Zbl 0816.65043 |

[3] | W. Zhu, X. Zhao, and Y. Tang, “Numerical methods with a high order of accuracy applied in the quantum system,” Journal of Chemical Physics, vol. 104, no. 6, pp. 2275-2286, 1996. |

[4] | J. C. Chiou and S. D. Wu, “Open Newton-Cotes differential methods as multilayer symplectic integrators,” Journal of Chemical Physics, vol. 107, no. 17, pp. 6894-6898, 1997. |

[5] | T. E. Simos, “A fourth algebraic order exponentially-fitted Runge-Kutta method for the numerical solution of the Schrödinger equation,” IMA Journal of Numerical Analysis, vol. 21, no. 4, pp. 919-931, 2001. · Zbl 0990.65079 |

[6] | T. E. Simos, “Exponentially and trigonometrically fitted methods for the solution of the Schrödinger equation,” Acta Applicandae Mathematicae, vol. 110, no. 3, pp. 1331-1352, 2010. · Zbl 1192.65111 |

[7] | Z. Kalogiratou and T. E. Simos, “Newton-Cotes formulae for long-time integration,” Journal of Computational and Applied Mathematics, vol. 158, no. 1, pp. 75-82, 2003. · Zbl 1041.65104 |

[8] | T. E. Simos, “Closed Newton-Cotes trigonometrically-fitted formulae of high order for long-time integration of orbital problems,” Applied Mathematics Letters, vol. 22, no. 10, pp. 1616-1621, 2009. · Zbl 1171.65449 |

[9] | T. E. Simos, “Exponentially-fitted Runge-Kutta-Nyström method for the numerical solution of initial-value problems with oscillating solutions,” Applied Mathematics Letters, vol. 15, no. 2, pp. 217-225, 2002. · Zbl 1003.65081 |

[10] | Ch. Tsitouras and T. E. Simos, “Optimized Runge-Kutta pairs for problems with oscillating solutions,” Journal of Computational and Applied Mathematics, vol. 147, no. 2, pp. 397-409, 2002. · Zbl 1013.65073 |

[11] | A. Konguetsof and T. E. Simos, “A generator of hybrid symmetric four-step methods for the numerical solution of the Schrödinger equation,” Journal of Computational and Applied Mathematics, vol. 158, no. 1, pp. 93-106, 2003. · Zbl 1027.65094 |

[12] | Z. Kalogiratou, T. Monovasilis, and T. E. Simos, “Symplectic integrators for the numerical solution of the Schrödinger equation,” Journal of Computational and Applied Mathematics, vol. 158, no. 1, pp. 83-92, 2003. · Zbl 1027.65171 |

[13] | G. Psihoyios and T. E. Simos, “Trigonometrically fitted predictor-corrector methods for IVPs with oscillating solutions,” Journal of Computational and Applied Mathematics, vol. 158, no. 1, pp. 135-144, 2003. · Zbl 1027.65095 |

[14] | T. E. Simos, I. T. Famelis, and C. Tsitouras, “Zero dissipative, explicit Numerov-type methods for second order IVPs with oscillating solutions,” Numerical Algorithms, vol. 34, no. 1, pp. 27-40, 2003. · Zbl 1031.65080 |

[15] | T. E. Simos, “Dissipative trigonometrically-fitted methods for linear second-order IVPs with oscillating solution,” Applied Mathematics Letters, vol. 17, no. 5, pp. 601-607, 2004. · Zbl 1062.65075 |

[16] | K. Tselios and T. E. Simos, “Runge-Kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics,” Journal of Computational and Applied Mathematics, vol. 175, no. 1, pp. 173-181, 2005. · Zbl 1063.65113 |

[17] | D. P. Sakas and T. E. Simos, “Multiderivative methods of eighth algebraic order with minimal phase-lag for the numerical solution of the radial Schrödinger equation,” Journal of Computational and Applied Mathematics, vol. 175, no. 1, pp. 161-172, 2005. · Zbl 1063.65067 |

[18] | G. Psihoyios and T. E. Simos, “A fourth algebraic order trigonometrically fitted predictor-corrector scheme for IVPs with oscillating solutions,” Journal of Computational and Applied Mathematics, vol. 175, no. 1, pp. 137-147, 2005. · Zbl 1063.65060 |

[19] | Z. A. Anastassi and T. E. Simos, “An optimized Runge-Kutta method for the solution of orbital problems,” Journal of Computational and Applied Mathematics, vol. 175, no. 1, pp. 1-9, 2005. · Zbl 1063.65059 |

[20] | S. Stavroyiannis and T. E. Simos, “Optimization as a function of the phase-lag order of nonlinear explicit two-step P-stable method for linear periodic IVPs,” Applied Numerical Mathematics. An IMACS Journal, vol. 59, no. 10, pp. 2467-2474, 2009. · Zbl 1169.65324 |

[21] | D. Huybrechs, “Stable high-order quadrature rules with equidistant points,” Journal of Computational and Applied Mathematics, vol. 231, no. 2, pp. 933-947, 2009. · Zbl 1170.65016 |

[22] | http://www.holoborodko.com/pavel/numerical-methods/numerical-integration/stable-newton-cotes-formulas/. |

[23] | G. D. Quinlan and S. Tremaine, “Symmetric multistep methods for the numerical integration of planetary orbits,” The Astronomical Journal, vol. 100, no. 5, pp. 1694-1700, 1990. |

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