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01.06.2015

Explicit multi-frequency symmetric extended RKN integrators for solving multi-frequency and multidimensional oscillatory reversible systems

verfasst von: Bin Wang, Xinyuan Wu

Erschienen in: Calcolo | Ausgabe 2/2015

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Abstract

This paper studies explicit multi-frequency symmetric extended Runge–Kutta–Nyström (ERKN) integrators tailored to numerically computing the multi-frequency and multidimensional oscillatory reversible second-order differential equations \(q''(t)+Mq(t)=f\big (q(t)\big )\). We establish the symmetry conditions in a simplified way for multi-frequency ERKN integrators. Five explicit multi-frequency symmetric ERKN integrators are derived based on the simplified symmetry conditions. The arbitrary high-order explicit multi-frequency symmetric ERKN integrators can be achieved by the application of the symmetric composition. The stability and phase properties of the new integrators are discussed. Five numerical experiments are carried out and the numerical results demonstrate the remarkable numerical behavior of the new explicit multi-frequency symmetric integrators when applied to the multi-frequency and multidimensional oscillatory reversible second-order differential equations.

Vorheriger Artikel Lavrentiev-type regularization methods for Hermitian problems

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Chen, Z., You, X., Shi, W., Liu, Z.: Symmetric and symplectic ERNK methods for oscillatory Hamiltonian systems. Comput. Phys. Commun. 183, 86–98 (2012)CrossRefMATHMathSciNet

Cohen, D., Hairer, E., Lubich, C.: Numerical energy conservation for multi-frequency oscillatory differential equations. BIT 45, 287–305 (2005)CrossRefMATHMathSciNet

Fang, Y., Wu, X.: A new pair of explicit ARKN methods for the numerical integration of general perturbed oscillators. Appl. Numer. Math. 57, 166–175 (2007)CrossRefMATHMathSciNet

Franco, J.M.: Runge–Kutta–Nyström methods adapted to the numerical integration of perturbed oscillators. Comput. Phys. Commun. 147, 770–787 (2002)CrossRefMATHMathSciNet

Franco, J.M.: A 5(3) pair of explicit ARKN methods for the numerical integration of perturbed oscillators. J. Comput. Appl. Math. 161, 283–293 (2003)CrossRefMATHMathSciNet

García, A., Martín, P., González, A.B.: New methods for oscillatory problems based on classical codes. Appl. Numer. Math. 42, 141–157 (2002)CrossRefMATHMathSciNet

García-Archilla, B., Sanz-Serna, J.M., Skeel, R.D.: Long-time-step methods for oscillatory differential equations. SIAM J. Sci. Comput. 20, 930–963 (1999)CrossRefMATHMathSciNet

González, A.B., Martín, P., Farto, J.M.: A new family of Runge–Kutta type methods for the numerical integration of perturbed oscillators. Numer. Math. 82, 635–646 (1999)CrossRefMATHMathSciNet

Hairer, E., Lubich, C.: Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J. Numer. Anal. 38, 414–441 (2000)CrossRefMATHMathSciNet

10.

Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer, Berlin, Heidelberg (2006)

11.

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

12.

Hochbruck, M., Lubich, C.: A Gautschi-type method for oscillatory second-order differential equations. Numer. Math. 83, 403–426 (1999)CrossRefMATHMathSciNet

13.

Jimánez, S., Vázquez, L.: Analysis of four numerical schemes for a nonlinear Klein–Gordon equation. Appl. Math. Comput. 35, 61–93 (1990)CrossRefMathSciNet

14.

Sanz-Serna, J.M., Calvo, M.P.: Numerical Hmiltonian Problem. In: Applied Mathematics and Matematical Computation, vol. 7. Chapman & Hall, London (1994)

15.

Stavroyiannis, S., Simos, T.E.: Optimization as a function of the phase-lag order of two-step P-stable method for linear periodic IVPs. App. Numer. Math. 59, 2467–2474 (2009)CrossRefMATHMathSciNet

16.

Tocino, A., Vigo-Aguiar, J.: Symplectic conditions for exponential fitting Runge–Kutta–Nyström methods. Math. Comput. Modell. 42, 873–876 (2005)CrossRefMATHMathSciNet

17.

Van der Houwen, P.J., Sommeijer, B.P.: Explicit Runge–Kutta(–Nyström) methods with reduced phase errors for computing oscillating solutions. SIAM J. Numer. Anal. 24, 595–617 (1987)CrossRefMATHMathSciNet

18.

Van de Vyver, H.: Stability and phase-lag analysis of explicit Runge-Kutta methods with variable coefficients for oscillatory problems. Comput. Phys. Comm. 173, 115–130 (2005)CrossRefMATHMathSciNet

19.

Vigo-Aguiar, J., Simos, T.E., Ferrándiz, J.M.: Controlling the error growth in long-term numerical integration of perturbed oscillations in one or more frequencies. Proc. Roy. Soc. Lond. Ser. A 460, 561–567 (2004)CrossRefMATH

20.

Wang, B., Liu, K., Wu, X.: A Filon-type asymptotic approach to solving highly oscillatory second-order initial value problems. J. Comput. Phys. 243, 210–223 (2013)CrossRefMathSciNet

21.

Wang, B., Wu, X.: A new high precision energy-preserving integrator for system of oscillatory second-order differential equations. Phys. Lett. A 376, 1185–1190 (2012)CrossRefMATHMathSciNet

22.

Wang, B., Wu, X., Xia, J.: Error bounds for explicit ERKN integrators for systems of multi-frequency oscillatory second-order differential equations. Appl. Numer. Math. 74, 17–34 (2013)CrossRefMATHMathSciNet

23.

Wang, B., Wu, X., Zhao, H.: Novel improved multidimensional Strömer–Verlet formulas with applications to four aspects in scientific computation. Math. Comput. Modell. 57, 857–872 (2013)CrossRefMATHMathSciNet

24.

Wu, X.: A note on stability of multidimensional adapted Runge–Kutta–Nyström methods for oscillatory systems. Appl. Math. Modell. 36, 6331–6337 (2012)CrossRef

25.

Wu, X., Wang, B.: Multidimensional adapted Runge–Kutta–Nyström methods for oscillatory systems. Comput. Phys. Commun. 181, 1955–1962 (2010)CrossRefMATH

26.

Wu, X., Wang, B., Shi, W.: Efficient energy-preserving integrators for oscillatory Hamiltonian systems. J. Comput. Phys. 235, 587–605 (2013)CrossRefMATHMathSciNet

27.

Wu, X., Wang, B., Xia, J.: Explicit symplectic multidimensional exponential fitting modified Runge–Kutta–Nyström methods. BIT 52, 773–795 (2012)CrossRefMATHMathSciNet

28.

Wu, X., You, X., Shi, W., Wang, B.: ERKN integrators for systems of oscillatory second-order differential equations. Comput. Phys. Commun. 181, 1873–1887 (2010)CrossRefMATHMathSciNet

29.

Wu, X., You, X., Wang, B.: Structure-Preserving Algorithms for Oscillatory Differential Equations. Springer, Berlin, Heidelberg (2013)CrossRefMATH

Titel: Explicit multi-frequency symmetric extended RKN integrators for solving multi-frequency and multidimensional oscillatory reversible systems
verfasst von: Bin Wang
Xinyuan Wu
Publikationsdatum: 01.06.2015
Verlag: Springer Milan
Erschienen in: Calcolo / Ausgabe 2/2015
Print ISSN: 0008-0624
Elektronische ISSN: 1126-5434
DOI: https://doi.org/10.1007/s10092-014-0114-z

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