Optimized pairs of multidimensional ERKN methods with FSAL property for multi-frequency oscillatory systems

References

• M.M. Chawla, P.S. Rao, A Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems. J. Comput. Appl. Math. 11 (1984), pp. 277–281. https://doi.org/10.1016/0377-0427. doi: 10.1016/0377-0427(84)90002-5
   Web of Science ®Google Scholar
• M.M. Chawla and P.S. Rao, An explicit sixth-order method with phase-lag of order eight for y′′=f(t,y), J. Comput. Appl. Math. 17 (1987), pp. 365–368. doi: 10.1016/0377-0427(87)90113-0
   Google Scholar
• J.R. Dormand, M.E.A. El-Mikkawy, and P.J. Prince, Family of Runge–Kutta–Nyström formulae, IMA J. Numer. Anal. 7 (1987), pp. 235–250. doi: 10.1093/imanum/7.2.235
   Google Scholar
• Y. Fang, Q. Li, and X. Wu, Extended RKN methods with FSAL property for oscillatory systems, Comput. Phys. Commun. 181 (2010), pp. 1538–1548. doi: 10.1016/j.cpc.2010.05.004
   Web of Science ®Google Scholar
• Y.L. Fang, C.Y. Liu, and B. Wang, Efficient energy-preserving methods for general nonlinear oscillatory Hamiltonian systems, Act. Math. Sin. 34 (2018), pp. 1863–1878. doi: 10.1007/s10114-018-6300-1
   Web of Science ®Google Scholar
• J.M. Franco, New methods for oscillatory systems based on ARKN methods, Appl. Numer. Math. 56 (2006), pp. 1040–1053. doi: 10.1016/j.apnum.2005.09.005
   Web of Science ®Google Scholar
• E. Hairer and C. Lubich, Long-time energy conservation of numerical methods for oscillatory differential equations, SIAM J. Numer. Anal. 38 (2000), pp. 414–441. doi: 10.1137/S0036142999353594
   Web of Science ®Google Scholar
• E. Hairer, S.P. Nørsett, and S.P. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems, Springer, Berlin, 1993.
   Google Scholar
• C. Liu and X. Wu, Nonlinear stability and convergence of ERKN integrators for solving nonlinear multi-frequency highly oscillatory second-order ODEs with applications to semi-linear wave equations, Appl. Numer. Math. 153 (2020), pp. 352–380. doi: 10.1016/j.apnum.2020.02.020
   Web of Science ®Google Scholar
• T.E. Simos and T.E. Williams, New insights in the development of Numerov-type methods with minimal phase-lag for the numerical solution of the Schrödinger equation, Comput. Chem. 25 (2001), pp. 77–82. doi: 10.1016/S0097-8485(00)00090-5
   PubMedGoogle Scholar
• E. Stiefel and D.G. Bettis, Stabilization of Cowell's method, Numer. Math. 13 (1969), pp. 154–175. doi: 10.1007/BF02163234
   Web of Science ®Google Scholar
• H. Van de Vyver, An embedded exponentially fitted Runge–Kutta–Nyström method for the numerical solution of orbital problems, New Astron. 11 (2006), pp. 577–587. doi: 10.1016/j.newast.2006.03.001
   Web of Science ®Google Scholar
• P.J. van der Houwen and B.P. Sommeijer, Explicit Runge–Kutta(–Nyström) methods with reduced phase errors for computing oscillating solution, SIAM J. Numer. Anal. 24 (1987), pp. 595–617. doi: 10.1137/0724041
   Web of Science ®Google Scholar
• B. Wang, X. Wu, and J. Xia, Error bounds for explicit ERKN integrators for systmes of multi-frequency oscillatory second-order differential equations, Appl. Numer. Math. 74 (2013), pp. 17–34. doi: 10.1016/j.apnum.2013.08.002
   Web of Science ®Google Scholar
• B. Wang, X.Y. Wu, and H. Zhao, Novel improved multidimensional Strömer–Verlet formulas with applications to four aspects in scientific computation, Math. Comput. Model. 57 (2013), pp. 857–872. doi: 10.1016/j.mcm.2012.09.014
   Google Scholar
• X. Wu, K. Liu, and W. Shi, Structure-Preserving Algorithms for Oscillatory Differential Equations II, Springer, Science Press, Beijing, 2015.
   Google Scholar
• X.Y. Wu and B. Wang, Multidimensional adapted Runge–Kutta–Nyström methods for oscillatory systems, Comput. Phys. Commun. 181 (2010), pp. 1955–1962. doi: 10.1016/j.cpc.2010.09.006
   Web of Science ®Google Scholar
• X. Wu, B. Wang, and L. Mei, Oscillation-preserving algorithms for efficiently solving highly oscillatory second-order ODEs, Numer. Algor. Available at https://doi.org/10.1007/s11075-020-00908-7.
   Google Scholar
• X.Y. Wu, B. Wang, and J. Xia, Explicit symplectic multidimensional exponential fitting modified Runge–Kutta–Nyström methods, BIT Numer. Math. 52 (2012), pp. 773–795. doi: 10.1007/s10543-012-0379-z
   Web of Science ®Google Scholar
• X. Wu, X. You, W. Shi, and B. Wang, ERKN integrators for systems of oscillatory second-order differential equations, Comput. Phys. Commun. 181 (2010), pp. 1873–1887. doi: 10.1016/j.cpc.2010.07.046
   Web of Science ®Google Scholar
• X. Wu, X. You, and B. Wang, Structure-Preserving Algorithms for Oscillatory Differential Equations, Springer, Science Press, Beijing, 2013.
   Google Scholar
• X.Y. Wu, X. You, and J.L. Xia, Order conditions for ARKN methods solving oscillatory systems, Comput. Phys. Commun. 180 (2009), pp. 2250–2257. doi: 10.1016/j.cpc.2009.07.011
   Web of Science ®Google Scholar
• X. You, Y. Fang, and J. Zhao, Special extended Nyström tree theory for ERKN methods, J. Comput. Appl. Math. 263 (2014), pp. 478–499. doi: 10.1016/j.cam.2013.12.043
   Web of Science ®Google Scholar