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 Journal on Numerical Analysis, vol.52, issue.5, pp.2488-2511, 2014.
DOI : 10.1137/130950665

W. Bao, X. Dong, and X. Zhao, Uniformly accurate multiscale time integrators for highly oscillatory second order differential equations, J. Math. Study, vol.47, pp.111-150, 2014.

W. Bao and X. Dong, Analysis and comparison of numerical methods for the Klein???Gordon equation in the nonrelativistic limit regime, Numerische Mathematik, vol.8, issue.2, pp.189-229, 2012.
DOI : 10.1007/s00211-011-0411-2

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geometric and Functional Analysis, vol.34, issue.1, pp.209-262, 1993.
DOI : 10.1007/BF01895688

P. Chartier, N. Crouseilles, M. Lemou, and F. Méhats, Uniformly accurate numerical schemes for highly oscillatory Klein???Gordon and nonlinear Schr??dinger equations, Numerische Mathematik, vol.150, issue.5???7, pp.211-250, 2015.
DOI : 10.1007/s00211-014-0638-9

URL : http://arxiv.org/abs/1308.0507

D. Cohen, E. Hairer, and C. Lubich, Modulated Fourier Expansions of Highly Oscillatory Differential Equations, Foundations of Computational Mathematics, vol.3, issue.4, pp.327-345, 2003.
DOI : 10.1007/s10208-002-0062-x

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.545.7596

B. Engquist, A. Fokas, E. Hairer, and A. , Iserles, Highly Oscillatory Problems, 2009.

E. Faou, Geometric numerical integration and Schrödinger equations, 2012.

E. Faou and K. Schratz, Asymptotic preserving schemes for the Klein???Gordon equation in the non-relativistic limit regime, Numerische Mathematik, vol.8, issue.3, pp.441-469, 2014.
DOI : 10.1007/s00211-013-0567-z

URL : https://hal.archives-ouvertes.fr/hal-00762161

L. Gauckler, Error Analysis of Trigonometric Integrators for Semilinear Wave Equations, SIAM Journal on Numerical Analysis, vol.53, issue.2, pp.1082-1106, 2015.
DOI : 10.1137/140977217

E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, 2006.
DOI : 10.4171/owr/2006/14

URL : https://hal.archives-ouvertes.fr/hal-01403326

M. Hochbruck and C. Lubich, A Gautschi-type method for oscillatory second-order differential equations, Numerische Mathematik, vol.83, issue.3, pp.403-426, 1999.
DOI : 10.1007/s002110050456

M. Hochbruck and A. Ostermann, Exponential integrators, Acta Numerica, vol.19, pp.209-286, 2010.
DOI : 10.1017/S0962492910000048

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.187.6794

M. Hofmanova and K. Schratz, An exponential-type integrator for the KdV equation. arxiv.org:1601, p.5311, 2016.

J. D. Lawson, Generalized Runge-Kutta Processes for Stable Systems with Large Lipschitz Constants, SIAM Journal on Numerical Analysis, vol.4, issue.3
DOI : 10.1137/0704033

K. Lorenz, T. Jahnke, and C. Lubich, Adiabatic Integrators for Highly Oscillatory Second-Order Linear Differential Equations with Time-Varying Eigendecomposition, BIT Numerical Mathematics, vol.7, issue.1, pp.91-115, 2005.
DOI : 10.1007/s10543-005-2637-9

C. Lubich, On splitting methods for Schr??dinger-Poisson and cubic nonlinear Schr??dinger equations, Mathematics of Computation, vol.77, issue.264, pp.2141-2153, 2008.
DOI : 10.1090/S0025-5718-08-02101-7

N. Masmoudi and K. Nakanishi, From nonlinear Klein-Gordon equation to a system of coupled nonlinear Schr??dinger equations, Mathematische Annalen, vol.324, issue.2, pp.359-389, 2002.
DOI : 10.1007/s00208-002-0342-4

A. Ostermann and K. Schratz, Low regularity exponential-type integrators for semilinear Schrödinger equations in the energy space, arxiv.org:1603, p.7746, 2016.

W. Strauss and L. Vazquez, Numerical solution of a nonlinear Klein-Gordon equation, Journal of Computational Physics, vol.28, issue.2, pp.271-278, 1978.
DOI : 10.1016/0021-9991(78)90038-4

S. Baumstark and . Fakultät-für-mathematik, Karlsruhe Institute of Technology, Englerstr. 2, 76131 Karlsruhe, Germany ? E-mail : simon.baumstark@kit