EDP Sciences logo
Subscriber Authentication Point
Search
Menu
All issues Volume 57 / No 2 (March-April 2023) ESAIM: M2AN, 57 2 (2023) 899-919 References
Open Access
Issue
ESAIM: M2AN
Volume 57, Number 2, March-April 2023
Page(s) 899 - 919
DOI https://doi.org/10.1051/m2an/2022096
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. [CrossRef] [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. [CrossRef] [MathSciNet] [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. [CrossRef] [MathSciNet] [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. [CrossRef] [Google Scholar]
  11. L. Gauckler, Error analysis of trigonometric integrators for semilinear Klein-Gordon equations. SIAM J. Numer. Anal. 53 (2015) 1082–1106. [CrossRef] [MathSciNet] [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. [CrossRef] [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. [CrossRef] [MathSciNet] [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. [CrossRef] [MathSciNet] [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. [CrossRef] [MathSciNet] [Google Scholar]
  18. M. Hochbruck and C. Lubich, A Gautschi-type method for oscillatory second-order differential equations. Numer. Math. 83 (1999) 403–426. [CrossRef] [MathSciNet] [Google Scholar]
  19. M. Hofmanová and K. Schratz, An exponential-type integrator for the KdV equation. Numer. Math. 136 (2017) 1117–1137. [CrossRef] [MathSciNet] [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. [CrossRef] [Google Scholar]
  22. J. Li and M.R. Visbal, High-order compact schemes for nonlinear dispersive waves. J. Sci. Comput. 26 (2006) 1–23. [CrossRef] [MathSciNet] [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. [CrossRef] [MathSciNet] [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. [CrossRef] [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. [CrossRef] [MathSciNet] [Google Scholar]
  32. G. Strang, On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5 (1968) 506–517. [CrossRef] [MathSciNet] [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. [CrossRef] [Google Scholar]
  35. Y. Wang and X. Zhao, A symmetric low-regularity integrator for nonlinear Klein-Gordon equation. Math. Comp. 91 (2022) 2215–2245. [CrossRef] [MathSciNet] [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]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you