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Optimal three-stage implicit exponentially-fitted RKN methods for solving second-order ODEs

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Abstract

In a wide range of applications, second-order ordinary differential equation (ODE) appears frequently. If the second-order ODE is stiff, the implicit Runge–Kutta–Nyström (RKN) method is often used to obtain numerical solutions. In addition, there are often some inherent properties in these problems, such as symmetry, symplecticness and exponentially fitting. Considering these properties, two three-stage implicit modified RKN integrators are obtained in this paper. The new six-order integrators called ISSEFMRKN integrate exactly differential systems whose solutions can be expressed as linear combinations of functions from the set \(\{\exp (\lambda t), \exp (-\lambda t)\mid \lambda \in \mathbb {R}~\text {or}~\lambda \in \mathbb {C}\}\). Furthermore, their final stages are also exact for \(y=t^2\) or \(y\in \{\exp (2\lambda t), \exp (-2\lambda t)\}\). The numerical results show that the new methods are more accurate than some highly accurate codes in the literature.

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References

  1. Abell, M.L., Braselton, J.P.: Modern Differential Equations: Theory, Applications, Technology. Harcourt Brace & Company, San Diego (1996)

    MATH  Google Scholar 

  2. Calvo, M., Franco, J.M., Montijano, J.I., Randez, L.: Sixth-order symmetric and symplectic exponentially fitted modified Runge–Kutta methods of Gauss type. Comput. Phys. Commun. 178, 732–744 (2008)

    Article  MathSciNet  Google Scholar 

  3. Chen, B.Z., Zhai, W.J.: Implicit symmetric and symplectic exponentially fitted modified Runge–Kutta-Nyström methods for solving oscillatory problems. J. Inequal. Appl. 321, 1–17 (2018)

    MATH  Google Scholar 

  4. Coleman, J.P., Ixaru, L. G.R.: P-stability and exponential-fitting methods for \(y^{\prime \prime }=f(x,y)\). IMA J. Numer. Anal 16, 179–199 (1996)

    Article  MathSciNet  Google Scholar 

  5. Cong, N.H.: A stable diagonally implicit Runge–Kutta-Nyström methods for parallel computers. Numer. Algorithms 4(2), 263–281 (1993)

    Article  MathSciNet  Google Scholar 

  6. Franco, J.M., Gomez, I.: Symplectic explicit methods of Runge–Kutta–Nyström type for solving pertubated oscillators. J. Comput. Appl. Math. 260, 482–493 (2014)

    Article  MathSciNet  Google Scholar 

  7. Franco, J.M.: Exponentially fitted symplectic integrators of RKN type for solving oscillatory problems. Comput. Phys. Commun. 177, 479–492 (2007)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  9. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations. Springer, Berlin (2002)

    Book  Google Scholar 

  10. Imoni, S.O., Otunta, F.O., Ramamohan, T.R.: Embedded implicit Runge–Kutta–Nyström method for solving second-order differential equations. Int. J. Comput. Mat. 83(11), 777–784 (2006)

    Article  Google Scholar 

  11. Ismail, F.: Digonally implicit Runge–Kutta–Nyström general method order five for solving second order IVPs. WSEAS Trans. Math. 9(7), 550–560 (2010)

    MathSciNet  Google Scholar 

  12. Ixaru, L.G.: Operations on oscillatory functions. Comput. Phys. Commun. 105, 1–19 (1997)

    Article  MathSciNet  Google Scholar 

  13. Ixaru, L.G.: Exponential and trigonometrical fittings: user-friendly expressions for the coefficients. Numer. Algorithms 82, 1085–1096 (2019)

    Article  MathSciNet  Google Scholar 

  14. Ixaru, L.G., Vanden Berghe, G.: Exponential Fitting. Kluwer Academic Publishers, Dordrecht (2004)

    Book  Google Scholar 

  15. Jator, S.N.: Implicit third derivative Runge–Kutta–Nyström method with trigonometric coefficients. Numer. Algorithms 70(1), 1–18 (2015)

    Article  MathSciNet  Google Scholar 

  16. Kalogiratou, Z.: Diagonally implicit trigonometrically fitted symplectic Runge–Kutta methods. Appl. Math. Comput. 219(14), 7406–7412 (2013)

    MathSciNet  MATH  Google Scholar 

  17. Kalogiratou, Z., Monovasilis, T., Simos, T.E.: A sixth order symmetric and symplectic diagonally implicit Runge–Kutta method. AIP Conf. Proc. 1618, 833–838 (2014)

    Article  Google Scholar 

  18. Li, J., Wang, B., You, X., Wu, X.: Two-step extended RKN methods for oscillatory systems. Comput. Phys. Commun. 182, 2486–2507 (2011)

    Article  MathSciNet  Google Scholar 

  19. Li, J., Deng, S., Wang, X.: Extended explicit pseudo two-step RKN methods for oscillatory systems \(y^{\prime \prime } + My = f(y)\). Numer. Algorithms 8, 673–700 (2018)

    Article  MathSciNet  Google Scholar 

  20. Li, J., Wang, X., Deng, S., Wang, B.: Symmetric trigonometrically-fitted two-step hybrid methods for oscillatory problems. J. Comput. Appl. Math. 344, 115–131 (2018)

    Article  MathSciNet  Google Scholar 

  21. Moo, K.W., Senu, N., Ismail, F., Arifin, N.M.: A zero-dissipative phase-fitted fourth order diagonally implicit Runge–Kutta–Nyström method for solving oscillatory problems. Math. Probl. Eng. 2, 1–8 (2014)

    Article  Google Scholar 

  22. Paternoster, B.: Runge–Kutta(–Nyström) methods for ODEs with periodic solutions based on trigonometric polynomials. Appl. Numer. Math. 28, 401–412 (1998)

    Article  MathSciNet  Google Scholar 

  23. Qin, M.Z., Zhu, W.J.: Canonical Runge–Kutta–Nyström methods for second order ordinary differential equations. Comput. Math. Appl. 22(9), 85–95 (1991)

    Article  MathSciNet  Google Scholar 

  24. Sanz-Serna, J.M., Calvo, M.P.: Numerical Hamiltonian Problems. Chapman and Hall, London (1994)

    Book  Google Scholar 

  25. Senu, N., Suleiman, M., Ismail, F., Othman, M.: A singly diagonally implicit Runge–Kutta–Nyström method with dispersion of high order. IAENG Trans. Eng. Technol. 7, 116–129 (2015)

    MATH  Google Scholar 

  26. Simos, T.E.: An exponentially-fitted Runge–Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions. Comput. Phys. Commun. 115, 1–8 (1998)

    Article  MathSciNet  Google Scholar 

  27. Simos, T.E., Vigo-Aguiar, J.: Exponentially fitted symplectic integrator. Phys. Rev. E. 67, 1–7 (2003)

    Article  MathSciNet  Google Scholar 

  28. Van der Houwen, P.J., Sommeijer, B.P., Cong, N.H.: Parallel diagonally implicit Runge–Kutta–Nyström methods. Appl. Numer. Math. 9(2), 111–131 (1992)

    Article  MathSciNet  Google Scholar 

  29. Wing, M.K., Senu, N., Suleiman, M., Ismail, F., Othman, M.: A five-stage singly diagonally implicit Runge–Kutta–Nyström method with reduced phase-lag. AIP Conf. Proc. 1482, 315–320 (2012)

    Article  Google Scholar 

  30. You, X., Chen, B.Z.: Symmetric and symplectic exponentially fitted Runge–Kutta(–Nyström) methods for Hamiltonian problems. Math. Comput. Simul. 94, 76–95 (2013)

    Article  Google Scholar 

  31. Zhai, W.J., Chen, B.Z.: A fourth order implicit symmetric and symplectic exponentially fitted Runge–Kutta–Nyström method for solving oscillatory problems. Numer. Algebra Control Optim. 9(1), 71–84 (2019)

    Article  MathSciNet  Google Scholar 

  32. Zhai, H.Y., Zhai, W.J., Chen, B.Z.: A class of implicit symmetric symplectic and exponentially fitted Runge–Kutta–Nyström methods for solving oscillatory problems. Adv. Differ. Equ. 463, 1–16 (2018)

    Article  Google Scholar 

  33. Zhao, D., Wang, Z., Dai, Y.: Importance of the first-order derivative formula in the Obrechkoff methods. Comput. Phys. Commun. 167, 65–75 (2005)

    Article  Google Scholar 

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Acknowledgements

The authors are grateful to the anonymous referees for their careful reading of the manuscript and for their invaluable comments and suggestions, which largely help to improve this paper.

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Authors and Affiliations

  1. Institute of Statistics and Big Data, Renmin University of China, Beijing, 100872, China

    Bingzhen Chen

  2. General Education Department, Cangzhou Jiaotong College, Cangzhou, 061100, China

    Wenjuan Zhai

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  1. Bingzhen Chen
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  2. Wenjuan Zhai
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Correspondence to Wenjuan Zhai.

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This work was supported by the Key Program of Haibin College (HB202001002).

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Chen, B., Zhai, W. Optimal three-stage implicit exponentially-fitted RKN methods for solving second-order ODEs. Calcolo 59, 14 (2022). https://doi.org/10.1007/s10092-022-00456-7

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