87
Views
2
CrossRef citations to date
0
Altmetric
Original Articles

An explicit trigonometrically fitted Runge–Kutta method for stiff and oscillatory problems with two frequencies

ORCID Icon, ORCID Icon & ORCID Icon
Pages 85-94 | Received 29 Aug 2017, Accepted 22 Jan 2018, Published online: 21 Feb 2018

References

  • J.P. Coleman and L.G. Ixaru, P-stability and exponential-fitting methods for y′′=f(x,y), IMA J. Numer. Anal. 16 (1996), pp. 179–199. doi: 10.1093/imanum/16.2.179
  • R. D'Ambrosio, E. Esposito, and B. Paternoster, Exponentially fitted two-step hybrid methods for y′′=f(x,y), J. Comput. Appl. Math. 235 (2011), pp. 4888–4897. doi: 10.1016/j.cam.2011.01.048
  • R. D'Ambrosio, M. Ferro, and B. Paternoster, Trigonometrically fitted two-step hybrid methods for special second order ordinary differential equations, Math. Comput. Simul. 81 (2011), pp. 1068–1084. doi: 10.1016/j.matcom.2010.10.011
  • Y. Fang and Q. Ming, New Runge–Kutta method for stiff oscillatory problems with two frequencies, AIP Conf. Proc. 1168 (2009), pp. 904–907. doi: 10.1063/1.3241628
  • Y. Fang, Y. Song, and X. Wu, Trigonometrically fitted explicit Numerov-type method for periodic IVPs with two frequencies, Comput. Phys. Commun. 179 (2008), pp. 801–811. doi: 10.1016/j.cpc.2008.07.013
  • Y. Fang and X. Wu, A trigonometrically fitted explicit Numerov-type method for second-order initial value problems with oscillating solutions, Appl. Numer. Math. 58 (2008), pp. 341–351. doi: 10.1016/j.apnum.2006.12.003
  • J.M. Franco, Runge–Kutta methods adapted to the numerical integration of oscillatory problems, Appl. Numer. Math. 50 (2004), pp. 427–443. doi: 10.1016/j.apnum.2004.01.005
  • J.M. Franco, Exponentially fitted explicit Runge–Kutta-Nyström methods, J. Comput. Appl. Math. 167 (2004), pp. 1–19. doi: 10.1016/j.cam.2003.09.042
  • J.M. Franco, A class of explicit two-step hybrid methods for second-order IVPs, J. Comput. Appl. Math. 187 (2006), pp. 41–57. doi: 10.1016/j.cam.2005.03.035
  • J.M. Franco and I. Gómez, Accuracy and linear stability of RKN methods for solving second-order stiff problems, Appl. Numer. Math. 59 (2009), pp. 959–975. doi: 10.1016/j.apnum.2008.04.002
  • J.M. Franco, I. Gómez, and L. Rández, SDIRK methods for stiff ODEs with oscillating solutions, J. Comput. Appl. Math. 81 (1997), pp. 197–209. doi: 10.1016/S0377-0427(97)00056-3
  • J.M. Franco, I. Gómez, and L. Rández, Four-stage symplectic and P-stable SDIRKN methods with dispersion of high order, Numer. Algor. 26 (2001), pp. 347–363. doi: 10.1023/A:1016629706668
  • E. Hairer, Ch. Lubich, and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd ed., Springer, Berlin, Heidelberg, 2006.
  • E. Hairer, S.P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems, 2nd ed., Springer, Berlin, Heidelberg, New York, 1993.
  • L. Kramarz, Stability of collocation methods for the numerical solution of y′′=f(t,y), BIT Numer. Math. 20 (1980), pp. 215–222. doi: 10.1007/BF01933194
  • J.D. Lambert and I.A. Watson, Symmetric multistep methods for periodic initial value problems, IMA J. Appl. Math. 18 (1976), pp. 189–202. doi: 10.1093/imamat/18.2.189
  • T. Lyche, Chebyshevian multistep methods for ordinary differential equations, Numer. Math. 19 (1972), pp. 65–75. doi: 10.1007/BF01395931
  • H. Ramos and J. Vigo-Aguiar, On the frequency choice in trigonometrically fitted methods, Appl. Math. Lett. 23 (2010), pp. 1378–1381. doi: 10.1016/j.aml.2010.07.003
  • H. Ramos and J. Vigo-Aguiar, A trigonometrically-fitted method with two frequencies, one for the solution and another one for the derivative, Comput. Phys. Commun. 185 (2014), pp. 1230–1236. doi: 10.1016/j.cpc.2013.12.021
  • T.E. Simos, An exponentially-fitted Runge–Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions, Comput. Phys. Commun. 115 (1998), pp. 1–8. doi: 10.1016/S0010-4655(98)00088-5
  • A. Tocino and J. Vigo-Aguiar, Symplectic conditions for exponential fitting Runge–Kutta–Nyström methods, Math. Comput. Model. 42 (2005), pp. 873–876. doi: 10.1016/j.mcm.2005.09.015
  • H. Van de Vyver, A Runge–Kutta–Nyström pair for the numerical integration of perturbed oscillators, Comput. Phys. Commun. 167 (2005), pp. 129–142. doi: 10.1016/j.cpc.2004.12.011
  • H. Van de Vyver, Comparison of some special optimized fourth-order Runge–Kutta methods for the numerical solution of the Schrödinger equation, Comput. Phys. Commun. 166 (2005), pp. 109–122. doi: 10.1016/j.cpc.2004.11.002
  • H. Van de Vyver, Modified explicit Runge–Kutta methods for the numerical solution of the Schrödinger equation, Appl. Math. Comput. 171 (2005), pp. 1025–1036.
  • H. Van de Vyver, Stability and phase-lag analysis of explicit Runge–Kutta methods with variable coefficients for oscillatory problems, Comput. Phys. Commun. 173 (2005), pp. 115–130. doi: 10.1016/j.cpc.2005.07.007
  • G. Vanden Berghe, H. De Meyer, M. Van Daele, and T. Van Hecke, Exponentially-fitted explicit Runge–Kutta methods, Comput. Phys. Commun. 123 (1999), pp. 7–15. doi: 10.1016/S0010-4655(99)00365-3
  • P.J. Van der Houwen and B.P. Sommeijer, Explicit Runge–Kutta(-Nyström) methods with reduced phase errors for computing oscillating solutions, SIAM. J. Numer. Anal. 24 (1987), pp. 595–617. doi: 10.1137/0724041
  • J. Vigo-Aguiar and J. Martín-Vaquero, Exponential fitting BDF algorithms and their properties, Appl. Math. Comput. 190 (2007), pp. 80–110. doi: 10.1016/j.amc.2007.01.008
  • J. Vigo-Aguiar, J. Martín-Vaquero, and H. Ramos, Exponential fitting BDF-Runge–Kutta algorithms, Comput. Phys. Commun. 178 (2008), pp. 15–34. doi: 10.1016/j.cpc.2007.07.007
  • Z. Wang, Trigonometrically-fitted method for a periodic initial value problem with two frequencies, Comput. Phys. Commun. 175 (2006), pp. 241–249. doi: 10.1016/j.cpc.2006.03.004
  • B. Wang, A. Iserles, and X. Wu, Arbitrary order trigonometric Fourier collocation methods for multi-frequency oscillatory systems, Found. Comput. Math. 16 (2016), pp. 151–181. doi: 10.1007/s10208-014-9241-9
  • B. Wang and G. Li, Bounds on asymptotic-numerical solvers for ordinary differential equations with extrinsic oscillation, Appl. Math. Modell. 39 (2015), pp. 2528–2538. doi: 10.1016/j.apm.2014.10.054
  • B. Wang, F. Meng, and Y. Fang, Efficient implementation of RKN-type Fourier collocation methods for second-order differential equations, Appl. Numer. Math. 119 (2017), pp. 164–178. doi: 10.1016/j.apnum.2017.04.008
  • B. Wang, X. Wu, F. Meng, and Y. Fang, Exponential Fourier collocation methods for solving first-order differential equations, J. Comput. Math. 35 (2017), pp. 711–736. doi: 10.4208/jcm.1611-m2016-0596
  • B. Wang, X. Wu, and F. Meng, Trigonometric collocation methods based on Lagrange basis polynomials for multi-frequency oscillatory second order differential equations, J. Comput. Appl. Math. 313 (2017), pp. 185–201. doi: 10.1016/j.cam.2016.09.017
  • B. Wang, H. Yang, and F. Meng, Sixth order symplectic and symmetric explicit ERKN schemes for solving multifrequency oscillatory nonlinear Hamiltonian equations, Calcolo 54 (2017), pp. 117–140. doi: 10.1007/s10092-016-0179-y
  • X. You and B. Chen, Symmetric and symplectic exponentially fitted Runge–Kutta-Nyström methods for Hamiltonian problems, Math. Comput. Simulat. 94 (2013), pp. 76–95. doi: 10.1016/j.matcom.2013.05.010
  • X. You, Z. Chen, and Y. Fang, New explicit adapted Numerov methods for second-order oscillatory differential equations, Appl. Math. Comput. 219 (2013), pp. 6241–6255.
  • X. You, Y. Zhang, and J. Zhao, Trigonometrically-fitted Scheifele two-step methods for perturbed oscillators, Comput. Phys. Commun. 182 (2011), pp. 1481–1490. doi: 10.1016/j.cpc.2011.04.001
  • X. You, J. Zhao, H. Yang, Y. Fang, and X. Wu, Order conditions for RKN methods solving general second-order oscillatory systems, Numer. Algor. 66 (2014), pp. 147–176. doi: 10.1007/s11075-013-9728-5

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.