Abstract
In this paper, we consider highly oscillatory second-order differential equations \(\ddot{x}(t)+\Omega ^2x(t)=g(x(t))\) with a single frequency confined to the linear part, and \(\Omega \) is singular. It is known that the asymptotic-numerical solvers are an effective approach to numerically solve the highly oscillatory problems. Unfortunately, however, the existing asymptotic-numerical solvers fail to apply to the highly oscillatory second-order differential equations when \(\Omega \) is singular. We propose an efficient improvement on the existing asymptotic-numerical solvers, so that the asymptotic-numerical solvers can be able to solve this class of highly oscillatory ordinary differential equations. The error estimation of the asymptotic-numerical solver is analyzed and nearly conservation of the energy in the Hamiltonian case is proved. Two numerical examples including the Fermi–Pasta–Ulam problem are implemented to show the efficiency of our proposed methods.
Similar content being viewed by others
References
Bambusi D, Giorgilli A (1994) Exponential stability of states close to resonance in infinite dimensional Hamiltonian systems. J Stat Phys 74:569–606
Cardone A, Conte D, D’Ambrosio R, Paternoster B (2017) Multivalue approximation of second order differential problems: a review. Int J Circ Syst Sign Proc 11:319–327
Castella F, Chartier P, Faou E (2009) An averaging technique for highly oscillatory Hamiltonian problems. SIAM J Numer Anal 47:2808–2837
Citro V, D’Ambrosio R (2020) Long-term analysis of stochastic \(\theta \)-methods for damped stochastic oscillators. Appl Numer Math 150:18–26
Cohen D (2012) On the numerical discretisation of stochastic oscillators. Math Comput Simul 82:1478–1495
de la Cruz H, Jimenez JC, Zubelli JP (2017) Locally linearized methods for the simulation of stochastic oscillators driven by random forces. BIT Numer Math 57:123–151
Condon M, Deaño A, Gao J, Iserles A (2014) Asymptotic solvers for second-order differential equation systems with multiple frequencies. Calcolo 51:109–139
Condon M, Deaño A, Gao J, Iserles A (2015) Asymptotic solvers for ordinary differential equations with multiple frequencies. Sci Chin Math 58:2279–2300
Condon M, Deaño A, Iserles A (2011) Asymptotic solvers for oscillatory systems of differential equations. \(S\vec{e}MA\) J 53:79–101
Condon M, Deaño A, Iserles A, Maczynski K, Xu T (2009) On numerical methods for highly oscillatory problems in circuit simulation. Int J Comput 28:1607–1618
Condon M, Iserles A, Nørsett S P (2014) Differential equations with general highly oscillatory forcing terms. Proc R Soc Lond Ser A Math Phys Eng Sci 470(2161):20130490
D’Ambrosio R, Moccaldi M, Paternoster B (2017) Adapted numerical methods for advection–reaction–diffusion problems generating periodic wavefronts. Comput Math Appl 74(5):1029–1042
D’Ambrosio R, Scalone C (2021) Asymptotic quadrature based numerical integration of stochastic damped oscillators. Lect Notes Comput Sci in press
D’Ambrosio R, Scalone C (2021) Filon quadrature for stochastic oscillators driven by time-varying forces. Appl Numer Math 169:21–31
Garcia-Archilla B, Sanz-Serna JM, Skeel RD (1998) Long-time-step methods for oscillatory differential equations. SIAM J Sci Comput 20:930–963
Garrett C, Munk W (1979) Internal waves in the ocean. Ann Rev Fluid Mech 14
Grimm V, Hochbruck M (2006) Error analysis of exponential integrators for oscillatory second-order differential equations. J Phys A Math Gen 39:5495–5507
Hairer E, Lubich C (2001) Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J Numer Anal 38:414–441
Hairer E, Lubich C (2009) Oscillations over long times in numerical Hamiltonian systems. In: Engquist B, Fokas A, Hairer E, Iserles A (eds), Highly oscillatory problems, London Mathematical Society Lecture Note Series, vol 366, Cambridge Univ. Press, Cambridge
Hairer E, Lubich C, Wanner G (2002) Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations, Springer Series in Computational Mathematics, No 31, Springer, Berlin
Hochbruck M, Lubich C (1999) A Gautschi-type method for oscillatory second-order differential equations. Numer Math 83:403–426
Khanamiryan M (2008) Quadrature methods for highly oscillatory linear and nonlinear systems of ordinary differential equations: Part I. BIT 48:743–761
Kopell N (1985) Invariant manifolds and the initialization problem for some atmospheric equations. Physica D 14:203–215
Liu Z, Tian H, You X (2017) Adiabatic Filon-type methods for highly oscillatory second-order ordinary differential equations. J Comput Appl Math 320:1–14
Liu Z, Tian T, Tian H (2019) Asymptotic-numerical solvers for highly oscillatory second-order differential equations. Appl Numer Math 137:184–202
Lorenz K, Jahnke T, Lubich Ch (2005) Adiabatic integrators for highly oscillatory second-order linear differential equations with time-varying eigen decomposition. BIT 45:91–115
Miranker WL, van Veldhuizen M (1978) The method of envelopes. Math Comp 32:453–496
Petzold LR, Jay LO, Yen J (1997) Numerical solution of highly oscillatory ordinary differential equations. Acta Numer 7:437–483
Sanz-Serna JM (2008) Mollified impulse methods for highly oscillatory differential equations. SIAM J Numer Anal 46:1040–1059
Sanz-Serna JM (2009) Modulated Fourier expansions and heterogeneous multiscale methods. IMA J Numer Anal 29:595–605
Senosiain MJ, Tocino A (2015) A review on numerical schemes for solving a linear stochastic oscillator. BIT Numer Math 55:515–529
Tidblad J, Graedel TE (1996) Gildes model studies of aqueous chemistry. III. Initial SO2-induced atmospheric corrosion of copper. Corros Sci 38:2201–2224
Wang B, Iserles A, Wu X (2016) Arbitrary order trigonometric Fourier collocation methods for second-order ODEs. Found Comput Math 16:151–181
Wang B, Liu K, Wu X (2013) A Filon-type asymptotic approach to solving highly oscillatory second-order initial value problems. J Comput Phys 243:210–223
Wang B, Wu X (2021) A long-term numerical energy-preserving analysis of symmetric and/or symplectic extended RKN integrators for efficiently solving highly oscillatory Hamiltonian systems. BIT Numer Math. https://doi.org/10.1007/s10543-021-00846-3
Wang B, Zhao X, Error estimates of some splitting schemes for charged-particle dynamics under strong magnetic field. To appear in SIAM J Numer Anal
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Valeria Neves Domingos.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Hongjiong Tian: The work of this author is supported in part by the National Natural Science Foundation of China under Grant Nos. 11671266 and 11871343, Science and Technology Innovation Plan of Shanghai under Grant No. 20JC1414200 and E-Institutes of Shanghai Municipal Education Commission under Grant No. E03004.
Rights and permissions
About this article
Cite this article
Liu, Z., Sa, X. & Tian, H. Asymptotic-numerical solvers for highly oscillatory ordinary differential equations and Hamiltonian systems. Comp. Appl. Math. 40, 291 (2021). https://doi.org/10.1007/s40314-021-01675-4
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-021-01675-4
Keywords
- Highly oscillatory problem
- Hamiltonian system
- Fermi–Pasta–Ulam problem
- Asymptotic expansion
- Modulated Fourier series