Free Access
Issue
ESAIM: M2AN
Volume 54, Number 5, September-October 2020
Page(s) 1491 - 1508
DOI https://doi.org/10.1051/m2an/2020006
Published online 26 June 2020
  1. R.A. Adams and J.J. Fournier, Sobolev Spaces. Elsevier (2003). [Google Scholar]
  2. W. Bao and X. Dong, Analysis and comparison of numerical methods for the Klein-Gordon equation in the nonrelativistic limit regime. Numer. Math. 120 (2012) 189–229. [Google Scholar]
  3. W. Bao and C. Su, Uniform error bounds of a finite difference method for the Zakharov system in the subsonic limit regime via an asymptotic consistent formulation. Multiscale Model. Simul. 15 (2017) 977–1002. [Google Scholar]
  4. W. Bao and C. Su, Uniform error bounds of a finite difference method for the Klein–Gordon–Zakharov system in the subsonic limit regime. Math. Comput. 87 (2018) 2133–2158. [Google Scholar]
  5. W. Bao and C. Su, Uniformly and optimally accurate methods for the Zakharov system in the subsonic limit regime. SIAM J. Sci. Comput. 40 (2018) A929–A953. [Google Scholar]
  6. W. Bao and X. Zhao, A uniformly accurate multiscale time integrator spectral method for the Klein–Gordon–Zakharov system in the high-plasma-frequency limit regime. J. Comput. Phys. 327 (2016) 270–293. [Google Scholar]
  7. W. Bao, Y. Cai and X. Zhao, A uniformly accurate multiscale time integrator pseudospectral method for the Klein-Gordon equation in the nonrelativistic limit regime. SIAM J. Numer. Anal. 52 (2014) 2488–2511. [Google Scholar]
  8. S. Baumstark and K. Schratz, Uniformly accurate oscillatory integrators for the Klein–Gordon–Zakharov system from low to high-plasma frequency regimes. SIAM J. Numer. Anal. 57 (2019) 429–457. [Google Scholar]
  9. S. Baumstark, E. Faou and K. Schratz, Uniformly accurate exponential-type integrators for Klein-Gordon equations with asymptotic convergence to the classical NLS splitting. Math. Comput. 87 (2018) 1227–1254. [Google Scholar]
  10. M. Berti and A. Maspero, Long time dynamics of Schrödinger and wave equations on flat tori. J. Differ. Equ. 267 (2019) 1167–1200. [Google Scholar]
  11. C. Besse, B. Bidégaray and S. Descombes, Order estimates in time of splitting methods for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 40 (2002) 26–40. [Google Scholar]
  12. J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations. Geom. Funct. Anal. 3 (1993) 107–156. [CrossRef] [MathSciNet] [Google Scholar]
  13. J. Bourgain, Exponential sums and nonlinear Schrödinger equations. Geom. Funct. Anal. 3 (1993) 157–178. [CrossRef] [Google Scholar]
  14. J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations. In Vol. 46 of Colloquium Publications. American Mathematical Society, Providence, RI (1999). [CrossRef] [Google Scholar]
  15. J. Bourgain, Growth of Sobolev norms in linear Schrödinger operators with quasi-periodic potential. Commun. Math. Phys. 204 (1999) 207–247. [CrossRef] [Google Scholar]
  16. S. Buchholz, L. Gauckler, V. Grimm, M. Hochbruck and T. Jahnke, Closing the gap between trigonometric integrators and splitting methods for highly oscillatory differential equations. IMA J. Numer. Anal. 38 (2017) 57–74. [CrossRef] [Google Scholar]
  17. R. Carles, Nonlinear Schrodinger equation with time dependent potential. Commun. Math. Sci. 9 (2011) 937–964. [Google Scholar]
  18. R. Carles and C. Gallo, On Fourier time splitting methods for NLS equations in the semi-classical limit II. Analytic regularity. Numer. Math. 136 (2017) 315–342. [Google Scholar]
  19. Ph. Chartier, N. Crouseilles, M. Lemou and F. Méhats, Uniformly accurate numerical schemes for highly oscillatory Klein-Gordon and nonlinear Schrödinger equations. Numer. Math. 129 (2015) 211–250. [Google Scholar]
  20. Ph. Chartier, N.J. Mauser, F. Méhats and Y. Zhang, Solving highly-oscillatory NLS with SAM: numerical efficiency and geometric properties. Disc. Contin. Dyn. Syst. Ser. S 9 (2016) 1327–1349. [CrossRef] [Google Scholar]
  21. Ph. Chartier, F. Méhats, M. Thalhammer and Y. Zhang, Improved error estimates for splitting methods applied to highly-oscillatory nonlinear Schrödinger equations. Math. Comput. 85 (2016) 2863–2885. [Google Scholar]
  22. Ph. Chartier, M. Lemou, F. Méhats and G. Vilmart, A new class of uniformly accurate numerical schemes for highly oscillatory evolution equations. Found. Comput. Math. 20 (2020) 1–33. [CrossRef] [Google Scholar]
  23. N. Crouseilles, S.A. Hirstoaga and X. Zhao, Multiscale Particle-In-Cell methods and comparisons for long time two-dimensional Vlasov-Poisson equation with strong magnetic field. Comput. Phys. Commun. 222 (2018) 136–151. [Google Scholar]
  24. N. Crouseilles, M. Lemou, F. Méhats and X. Zhao, Uniformly accurate Particle-in-Cell method for the long time two-dimensional Vlasov-Poisson equation with uniform strong magnetic field. J. Comput. Phys. 346 (2017) 172–190. [Google Scholar]
  25. D. Fang and Q. Zhang, On growth of Sobolev norms in linear Schrödinger equations with time dependent Gevrey potential. J. Dyn. Differ. Equ. 24 (2012) 151–180. [Google Scholar]
  26. E. Faou, Geometric Numerical Integration and Schrödinger Equations. European Mathematical Society (2012). [Google Scholar]
  27. L. Gauckler, Convergence of a split-step Hermite method for the Gross-Pitaevskii equation. IMA J. Numer. Anal. 31 (2011) 396–415. [CrossRef] [MathSciNet] [Google Scholar]
  28. E. Hairer, Ch. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer, Berlin (2006). [Google Scholar]
  29. S. Herr, D. Tataru and N. Tzvetkov, Global well-posedness of the energy critical nonlinear Schrödinger equation with small initial data in H1(𝕋3). Duke Math. J. 159 (2011) 329–349. [CrossRef] [Google Scholar]
  30. A.D. Ionescu and B. Pausader, The energy-critical defocusing NLS on (𝕋3). Duke Math. J. 161 (2012) 1581–1612. [CrossRef] [Google Scholar]
  31. A. Iserles, K. Kropielnicka and P. Singh, Solving Schrödinger equation in semiclassical regime with highly oscillatory time-dependent potentials. J. Comput. Phys. 376 (2019) 564–584. [Google Scholar]
  32. Ch. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations. Math. Comput. 77 (2008) 2141–2153. [Google Scholar]
  33. Y. Ma and C. Su, A uniformly and optimally accurate multiscale time integrator method for the Klein–Gordon–Zakharov system in the subsonic limit regime. Comput. Math. Appl. 76 (2018) 602–619. [Google Scholar]
  34. N. Masmoudi and K. Nakanishi, From the Klein–Gordon–Zakharov system to the nonlinear Schrödinger equation. J. Hyperbol. Differ. Equ. 2 (2005) 975–1008. [CrossRef] [Google Scholar]
  35. N. Mauser, Y. Zhang and X. Zhao, On the rotating nonlinear Klein-Gordon equation: non-relativistic limit and numerical methods. Preprint hal-01956352 (2018). [Google Scholar]
  36. J. Shen, T. Tang and L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer, Berlin (2011). [Google Scholar]
  37. J. Shen and Z.Q. Wang, Error analysis of the Strang time-splitting Laguerre–Hermite/Hermite collocation methods for the Gross–Pitaevskii equation. Found. Comput. Math. 13 (2013) 99–137. [CrossRef] [Google Scholar]
  38. M. Thalhammer, Convergence analysis of high-order time-splitting pseudo-spectral methods for nonlinear Schrödinger equations. SIAM J. Numer. Anal. 50 (2012) 3231–3258. [Google Scholar]
  39. W.M. Wang, Logarithmic bounds on Sobolev norms for time dependent linear Schrödinger equations. Commun. Part. Differ. Equ. 33 (2008) 2164–2179. [CrossRef] [Google Scholar]
  40. Y. Wang and X. Zhao, Symmetric high order Gautschi-type exponential wave integrators pseudospectral method for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime. Int. J. Numer. Anal. Model. 15 (2017) 405–427. [Google Scholar]
  41. J.A.C. Weideman and B.M. Herbst, Split-step methods for the solution of the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 23 (1986) 485–507. [Google Scholar]
  42. C. Xiong, M. Good, Y. Guo, X. Liu and K. Huang, Relativistic superfluidity and vorticity from the nonlinear Klein-Gordon equation. Phys. Rev. D 90 (2014) 125019. [Google Scholar]
  43. H. Yoshida, Construction of higher order symplectic integrators. Phys. Lett. A 150 (1990) 262–268. [Google Scholar]
  44. X. Zhao, On error estimates of an exponential wave integrator sine pseudospectral method for the Klein–Gordon–Zakharov system. Numer. Methods Part. Differ. Equ. 32 (2016) 266–291. [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