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Asymptotic-numerical solvers for highly oscillatory ordinary differential equations and Hamiltonian systems

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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.

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References

  • Bambusi D, Giorgilli A (1994) Exponential stability of states close to resonance in infinite dimensional Hamiltonian systems. J Stat Phys 74:569–606

    MathSciNet  MATH  Google Scholar 

  • 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

    Google Scholar 

  • Castella F, Chartier P, Faou E (2009) An averaging technique for highly oscillatory Hamiltonian problems. SIAM J Numer Anal 47:2808–2837

    Article  MathSciNet  Google Scholar 

  • Citro V, D’Ambrosio R (2020) Long-term analysis of stochastic \(\theta \)-methods for damped stochastic oscillators. Appl Numer Math 150:18–26

    Article  MathSciNet  Google Scholar 

  • Cohen D (2012) On the numerical discretisation of stochastic oscillators. Math Comput Simul 82:1478–1495

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • Condon M, Deaño A, Gao J, Iserles A (2014) Asymptotic solvers for second-order differential equation systems with multiple frequencies. Calcolo 51:109–139

    Article  MathSciNet  Google Scholar 

  • Condon M, Deaño A, Gao J, Iserles A (2015) Asymptotic solvers for ordinary differential equations with multiple frequencies. Sci Chin Math 58:2279–2300

    Article  MathSciNet  Google Scholar 

  • 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

    MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • Garcia-Archilla B, Sanz-Serna JM, Skeel RD (1998) Long-time-step methods for oscillatory differential equations. SIAM J Sci Comput 20:930–963

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • Hairer E, Lubich C (2001) Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J Numer Anal 38:414–441

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • Khanamiryan M (2008) Quadrature methods for highly oscillatory linear and nonlinear systems of ordinary differential equations: Part I. BIT 48:743–761

    Article  MathSciNet  Google Scholar 

  • Kopell N (1985) Invariant manifolds and the initialization problem for some atmospheric equations. Physica D 14:203–215

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • Liu Z, Tian T, Tian H (2019) Asymptotic-numerical solvers for highly oscillatory second-order differential equations. Appl Numer Math 137:184–202

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • Miranker WL, van Veldhuizen M (1978) The method of envelopes. Math Comp 32:453–496

    Article  MathSciNet  Google Scholar 

  • Petzold LR, Jay LO, Yen J (1997) Numerical solution of highly oscillatory ordinary differential equations. Acta Numer 7:437–483

    Article  MathSciNet  Google Scholar 

  • Sanz-Serna JM (2008) Mollified impulse methods for highly oscillatory differential equations. SIAM J Numer Anal 46:1040–1059

    Article  MathSciNet  Google Scholar 

  • Sanz-Serna JM (2009) Modulated Fourier expansions and heterogeneous multiscale methods. IMA J Numer Anal 29:595–605

    Article  MathSciNet  Google Scholar 

  • Senosiain MJ, Tocino A (2015) A review on numerical schemes for solving a linear stochastic oscillator. BIT Numer Math 55:515–529

    Article  MathSciNet  Google Scholar 

  • Tidblad J, Graedel TE (1996) Gildes model studies of aqueous chemistry. III. Initial SO2-induced atmospheric corrosion of copper. Corros Sci 38:2201–2224

    Article  Google Scholar 

  • Wang B, Iserles A, Wu X (2016) Arbitrary order trigonometric Fourier collocation methods for second-order ODEs. Found Comput Math 16:151–181

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • 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

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Correspondence to Hongjiong Tian.

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Communicated by Valeria Neves Domingos.

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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.

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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

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  • DOI: https://doi.org/10.1007/s40314-021-01675-4

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