Long-time momentum and actions behaviour of energy-preserving methods for semi-linear wave equations via spatial spectral semi-discretisations

  • Bin WangEmail author
  • Xinyuan Wu


It is known that wave equations have physically very important properties which should be respected by numerical schemes in order to predict correctly the solution over a long time period. In this paper, the long-time behaviour of momentum and actions for energy-preserving methods is analysed for semi-linear wave equations. A full discretisation of wave equations is derived and analysed by firstly using a spectral semi-discretisation in space and then by applying the adopted average vector field (AAVF) method in time. This numerical scheme can exactly preserve the energy of the semi-discrete system. The main theme of this paper is to analyse another important physical property of the scheme. It is shown that this scheme yields near conservation of a modified momentum and modified actions over long times. The results are rigorously proved based on the technique of modulated Fourier expansions in two stages. First, a multi-frequency modulated Fourier expansion of the AAVF method is constructed, and then two almost-invariants of the modulation system are derived.


Semi-linear wave equations Energy-preserving methods Multi-frequency modulated Fourier expansion Momentum and actions conservation 

Mathematics Subject Classification (2010)

35L70 65M70 65M15 


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The authors sincerely thank the two anonymous reviewers for their valuable suggestions, which helped improve this paper significantly. The authors are grateful to Professor Christian Lubich for his helpful comments and discussions on the topic of modulated Fourier expansions. We also thank him for drawing our attention to the long-term analysis of energy-preserving methods, which motives this paper.

Funding information

The research of the first author is financially supported in part by the Alexander von Humboldt Foundation and by the Natural Science Foundation of Shandong Province (Outstanding Youth Foundation) under Grant ZR2017JL003. The research of the second author is financially supported in part by the National Natural Science Foundation of China under Grant 11671200.


  1. 1.
    Bambusi, D.: Birkhoff normal form for some nonlinear PDEs. Comm. Math. Phys. 234, 253–285 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brugnano, L., Frasca Caccia, G., Iavernaro, F.: Energy conservation issues in the numerical solution of the semilinear wave equation. Appl. Math. Comput. 270, 842–870 (2015)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Cano, B.: Conservation of invariants by symmetric multistep cosine methods for second-order partial differential equations. BIT 53, 29–56 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cano, B.: Conserved quantities of some Hamiltonian wave equations after full discretization. Numer. Math. 103, 197–223 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cano, B., Moreta, M.J.: Multistep cosine methods for second-order partial differential systems. IMA J. Numer. Anal. 30, 431–461 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Celledoni, E., Grimm, V., McLachlan, R.I., McLaren, D.I., O’Neale, D., Owren, B., Quispel, G.R.W.: Preserving energy resp. dissipation in numerical PDEs using the “Average Vector Field” method. J. Comput. Phys. 231, 6770–6789 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cohen, D., Gauckler, L.: One-stage exponential integrators for nonlinear schrödinger equations over long times. BIT 52, 877–903 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cohen, D., Hairer, E., Lubich, C.: Conservation of energy, momentum and actions in numerical discretizations of nonlinear wave equations. Numer. Math. 110, 113–143 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cohen, D., Hairer, E., Lubich, C.: Long-time analysis of nonlinearly perturbed wave equations via modulated Fourier expansions. Arch. Ration. Mech. Anal. 187, 341–368 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Feng, K., Qin, M.: The symplectic methods for the computation of Hamiltonian equations, Numerical Methods for Partial Differential Equations, pp 1–37. Springer, Berlin (2006)Google Scholar
  11. 11.
    Gauckler, L.: Error analysis of trigonometric integrators for semilinear wave equations. SIAM J. Numer. Anal. 53, 1082–1106 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gauckler, L.: Numerical long-time energy conservation for the nonlinear schrödinger equation. IMA J. Numer. Anal. 37, 2067–2090 (2017)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Gauckler, L., Hairer, E., Lubich, C.: Long-term analysis of semilinear wave equations with slowly varying wave speed. Comm. Part. Diff. Equa. 41, 1934–1959 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gauckler, L., Hairer, E., Lubich, C., Weiss, D.: Metastable energy strata in weakly nonlinear wave equations. Comm. Part. Diff. Equa. 37, 1391–1413 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gauckler, L., Lubich, C.: Nonlinear schrödinger equations and their spectral semi-discretizations over long times. Found. Comput. Math. 10, 141–169 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gauckler, L., Lubich, C.: Splitting integrators for nonlinear schrödinger equations over long times. Found. Comput. Math. 10, 275–302 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gauckler, L., Weiss, D.: Metastable energy strata in numerical discretizations of weakly nonlinear wave equations. Disc. Contin. Dyn. Syst. 37, 3721–3747 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gauckler, L., Lu, J., Marzuola, J., Rousset, F., Schratz K.: Trigonometric integrators for quasilinear wave equations. Math. Comput. (2018)
  19. 19.
    Grimm, V.: On the use of the Gautschi-type exponential integrator for wave equations. In: Numerical Mathematics and Advanced Applications. pp. 557–563, Springer, Berlin (2006)Google Scholar
  20. 20.
    Hairer, E.: Energy-preserving variant of collocation methods. J. Numer. Anal. Ind. Appl. Math. 5, 73–84 (2010)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Hairer, E., Lubich, C.: Long-term analysis of a variational integrator for charged-particle dynamics in a strong magnetic field, Preprint, (2018)
  22. 22.
    Hairer, E., Lubich, C.: Long-term analysis of the störmer-verlet method for Hamiltonian systems with a solution-dependent high frequency. Numer. Math. 134, 119–138 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hairer, E., Lubich, C.: Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J. Numer. Anal. 38, 414–441 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Hairer, E., Lubich, C.: Spectral semi-discretisations of weakly nonlinear wave equations over long times. Found. Comput. Math. 8, 319–334 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hairer, E., Lubich, C., Wanner G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  26. 26.
    Li, Y.W., Wu, X.: Exponential integrators preserving first integrals or Lyapunov functions for conservative or dissipative systems. SIAM J. Sci. Comput. 38, 1876–1895 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Li, Y.W., Wu, X.: General local energy-preserving integrators for solving multi-symplectic Hamiltonian PDEs. J. Comput. Phys. 301, 141–166 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Liu, C., Iserles, A., Wu, X.: Symmetric and arbitrarily high-order Birkhoff–Hermite time integrators and their long-time behaviour for solving nonlinear Klein–Gordon equations. J. Comput. Phys. 356, 1–30 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Liu, C., Wu, X.: An energy-preserving and symmetric scheme for nonlinear Hamiltonian wave equations. J. Math. Anal. Appl. 440, 167–182 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Liu, K., Wu, X., Shi, W: A linearly-fitted conservative (dissipative) scheme for efficiently solving conservative (dissipative) nonlinear wave PDEs. J. Comput. Math. 35, 780–800 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    McLachlan, R.I., Stern, A.: Modified trigonometric integrators. SIAM J. Numer. Anal. 52, 1378–1397 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Mei, L., Liu, C., Wu, X.: An essential extension of the finite-energy condition for extended Runge–Kutta–Nyström integrators when applied to nonlinear wave equations. Commun. Comput. Phys. 22, 742–764 (2017)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Quispel, G.R.W., McLaren, D.I.: A new class of energy-preserving numerical integration methods. J. Phys. A 41(045206), 7 (2008)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Sanz-Serna, J.M.: Modulated Fourier expansions and heterogeneous multiscale methods. IMA J. Numer. Anal. 29, 595–605 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Wang, B., Iserles, A., Wu, X.: Arbitrary-order trigonometric Fourier collocation methods for multi-frequency oscillatory systems. Found. Comput. Math. 16, 151–181 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Wang, B., Wu, X.: Global error bounds of one-stage extended RKN integrators for semilinear wave equations. Numer. Algo. 81, 1203–1218. (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Wang, B., Wu, X.: The formulation and analysis of energy-preserving schemes for solving high-dimensional nonlinear Klein-Gordon equations. IMA. J. Numer. Anal. (2018)
  39. 39.
    Wu, X., Wang, B.: Recent developments in Structure-Preserving algorithms for oscillatory differential equations. Springer Nature Singapore Pte Ltd (2018)Google Scholar
  40. 40.
    Wu, X., Wang, B., Shi, W.: Efficient energy preserving integrators for oscillatory Hamiltonian systems. J. Comput Phys. 235, 587–605 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXi’an Jiaotong UniversityXi’anPeople’s Republic of China
  2. 2.Mathematisches InstitutUniversity of TübingenTübingenGermany
  3. 3.Department of MathematicsNanjing UniversityNanjingPeople’s Republic of China
  4. 4.School of Mathematical SciencesQufu Normal UniversityQufuPeople’s Republic of China

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