This paper is concerned with the numerical solution of nonlinear Hamiltonian highly oscillatory systems of second-order differential equations of a special form. We present numerical methods of high asymptotic as well as time stepping order based on the modulated Fourier expansion of the exact solution. In particular we obtain time stepping orders higher than with only a finite energy assumption on the initial values of the problem. In addition, the stepsize of these new numerical integrators is not restricted by the high frequency of the problem. Furthermore, numerical experiments on the modified Fermi–Pasta–Ulam problem as well as on a one dimensional model of a diatomic gas with short-range interaction forces support our investigations.
DOI : 10.1051/m2an/2014056
Mots-clés : Highly oscillatory differential equations, multiple time scales, Fermi–Pasta–Ulam problem, modulated Fourier expansions, high order numerical schemes, adiabatic invariants
@article{M2AN_2015__49_3_695_0,
author = {Cohen, David and Schweitzer, Julia},
title = {High order numerical methods for highly oscillatory problems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {695--711},
publisher = {EDP-Sciences},
volume = {49},
number = {3},
year = {2015},
doi = {10.1051/m2an/2014056},
zbl = {1317.34137},
mrnumber = {3342224},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2014056/}
}
TY - JOUR AU - Cohen, David AU - Schweitzer, Julia TI - High order numerical methods for highly oscillatory problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2015 SP - 695 EP - 711 VL - 49 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2014056/ DO - 10.1051/m2an/2014056 LA - en ID - M2AN_2015__49_3_695_0 ER -
%0 Journal Article %A Cohen, David %A Schweitzer, Julia %T High order numerical methods for highly oscillatory problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2015 %P 695-711 %V 49 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2014056/ %R 10.1051/m2an/2014056 %G en %F M2AN_2015__49_3_695_0
Cohen, David; Schweitzer, Julia. High order numerical methods for highly oscillatory problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 3, pp. 695-711. doi : 10.1051/m2an/2014056. http://www.numdam.org/articles/10.1051/m2an/2014056/
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