Open Access
Volume 57, Number 2, March-April 2023
Page(s) 899 - 919
Published online 30 March 2023
  1. W. Bao and L. Yang, Efficient and accurate numerical methods for the Klein–Gordon–Schrödinger equations. J. Comput. Phys. 225 (2007) 1863–1893. [CrossRef] [MathSciNet] [Google Scholar]
  2. W. Bao, Y. Feng and C. Su, Uniform error bounds of time-splitting spectral methods for the long-time dynamics of the nonlinear Klein-Gordon equation with weak nonlinearity. Math. Comp. 91 (2022) 811–842. [Google Scholar]
  3. A. Barone, F. Esposito and C.J. Magee, Theory and applications of the sine-gordon equation. La Rivista del Nuovo Cimento 1 (1971) 227–267. [Google Scholar]
  4. Y. Bruned and K. Schratz, Resonance based schemes for dispersive equations via decorated trees, in Forum of Mathematics, Pi. Vol. 10. Cambridge University Press (2022) E2. [CrossRef] [Google Scholar]
  5. S. Buchholz, B. Dörich and M. Hochbruck, On averaged exponential integrators for semilinear Klein-Gordon equations with solutions of low-regularity. SN Part. Differ. Equ. Appl. 2 (2021) 2662–2963. [Google Scholar]
  6. W. Cao, D. Li and Z. Zhang, Unconditionally optimal convergence of an energy-conserving and linearly implicit scheme for nonlinear Klein-Gordon equations. Sci. China Math. 65 (2021) 1731–1748. [Google Scholar]
  7. C. Chen, J. Hong, C. Sim and K. Sonwu, Energy and quadratic invariants preserving (EQUIP) multi-symplectic methods for Hamiltonian Klein-Gordon equations. J. Comput. Phys. 418 (2020) 10959. [Google Scholar]
  8. D. Cohen, E. Hairer and C. Lubich, Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equations. Numer. Math. 110 (2008) 113–143. [Google Scholar]
  9. P. Deuflhard, A study of extrapolation methods based on multistep schemes without parasitic solutions. Z. Angew. Math. Phys. 30 (1979) 177–189. [Google Scholar]
  10. B. García-Archilla, J.M. Sanz-Serna and R.D. Skeel, Long-time-step methods for oscillatory differential equations. SIAM J. Sci. Comput. 20 (1998) 930–963. [Google Scholar]
  11. L. Gauckler, Error analysis of trigonometric integrators for semilinear Klein-Gordon equations. SIAM J. Numer. Anal. 53 (2015) 1082–1106. [Google Scholar]
  12. R. Glowinski and A. Quaini, On the numerical solution to a nonlinear wave equation associated with the first Painlevé equation: an operator-splitting approach, in Partial Differential Equations: Theory, Control and Approximation, Springer, Dordrecht (2014) 243–264. [Google Scholar]
  13. V. Grimm and M. Hochbruck, Error analysis of exponential integrators for oscillatory second-order differential equations. J. Phys. A 39 (2006) 5495–5507. [Google Scholar]
  14. E. Hairer and C. Lubich, Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J. Numer. Anal. 38 (2000) 414–441. [Google Scholar]
  15. E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edition. Springer (2006). [Google Scholar]
  16. E. Hansen and A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numer. Math. 56 (2016) 1303–1316. [CrossRef] [Google Scholar]
  17. M. Hochbruck and J. Leibold, An implicit-explicit time discretization scheme for second-order semilinear Klein-Gordon equations with application to dynamic boundary conditions. Numer. Math. 147 (2021) 869–899. [Google Scholar]
  18. M. Hochbruck and C. Lubich, A Gautschi-type method for oscillatory second-order differential equations. Numer. Math. 83 (1999) 403–426. [Google Scholar]
  19. M. Hofmanová and K. Schratz, An exponential-type integrator for the KdV equation. Numer. Math. 136 (2017) 1117–1137. [Google Scholar]
  20. W. Layton, Y. Li and C. Trenchea, Recent developments in IMEX methods with time filters for systems of evolution equations. J. Comput. Appl. Math. 299 (2016) 50–67. [CrossRef] [MathSciNet] [Google Scholar]
  21. D. Li and W. Sun, Linearly implicit and high-order energy-conserving schemes for nonlinear Klein-Gordon equations. J. Sci. Comput. 83 (2020) 65. [Google Scholar]
  22. J. Li and M.R. Visbal, High-order compact schemes for nonlinear dispersive waves. J. Sci. Comput. 26 (2006) 1–23. [Google Scholar]
  23. Y. Li, Y. Wu and F. Yao, Convergence of an embedded exponential-type low-regularity integrators for the KdV equation without loss of regularity. Ann. Appl. Math. 37 (2021) 1–21. [CrossRef] [MathSciNet] [Google Scholar]
  24. B. Li, S. Ma and K. Schratz, A semi-implicit low-regularity integrator for Navier-Stokes equations. SIAM J. Numer. Anal. 60 (2022) 2273–2292. [Google Scholar]
  25. D. Murai and T. Koto, Stability and convergence of staggered Runge-Kutta schemes for semilinear Klein-Gordon equations. J. Comput. Appl. Math. 235 (2011) 4251–4264. [CrossRef] [MathSciNet] [Google Scholar]
  26. A. Ostermann and K. Schratz, Low regularity exponential-type integrators for semilinear Schrödinger equations. Found. Comput. Math. 18 (2018) 731–755. [CrossRef] [MathSciNet] [Google Scholar]
  27. A. Ostermann, F. Rousset and K. Schratz, Fourier integrator for periodic NLS: low regularity estimates via discrete Bourgain spaces. J. Eur. Math. Soc. (2022). DOI: 10.4171/jems/1275. [Google Scholar]
  28. R. Qi and X. Wang, Error estimates of finite element method for semilinear stochastic strongly damped Klein-Gordon equation. IMA J. Numer. Anal. 39 (2019) 1594–1626. [CrossRef] [MathSciNet] [Google Scholar]
  29. A. Quaini and R. Glowinski, Splitting methods for some nonlinear wave problems, in Splitting Methods in Communication, Imaging, Science, and Engineering, Scientific Computation Series. Springer, Cham (2016) 643–676. [Google Scholar]
  30. Z. Rong and C. Xu, Numerical approximation of acoustic waves by spectral element methods. Appl. Numer. Math. 58 (2008) 999–1016. [CrossRef] [MathSciNet] [Google Scholar]
  31. F. Rousset and K. Schratz, A general framework of low-regularity integrators. SIAM J. Numer. Anal. 59 (2021) 1735–1768. [Google Scholar]
  32. G. Strang, On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5 (1968) 506–517. [Google Scholar]
  33. B. Wang and X. Wu, The formulation and analysis of energy-preserving schemes for solving high-dimensional nonlinear Klein-Gordon equations. IMA J. Numer. ANal. 39 (2019) 2016–2044. [CrossRef] [MathSciNet] [Google Scholar]
  34. B. Wang and X. Wu, Global error bounds of one-stage extended RKN integrators for semilinear Klein-Gordon equations. Numer. Algorithm 81 (2019) 1203–1218. [Google Scholar]
  35. Y. Wang and X. Zhao, A symmetric low-regularity integrator for nonlinear Klein-Gordon equation. Math. Comp. 91 (2022) 2215–2245. [Google Scholar]
  36. Y. Wu and X. Zhao, Optimal convergence of a first order low-regularity integrator for the KdV equation. IMA J. Numer. Anal. 42 (2022) 3499–3528. [CrossRef] [MathSciNet] [Google Scholar]
  37. Y. Wu and X. Zhao, Embedded exponential-type low-regularity integrators for KdV equation under rough data. BIT Numer. Math. 62 (2022) 1049–1090. [CrossRef] [Google Scholar]

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