High order numerical methods for highly oscillatory problems

¹ Matematik och matematisk statistik, Umeå universitet, 90187 Umeå, Sweden.
david.cohen@math.umu.se
² Institut für Angewandte und Numerische Mathematik, Karlsruher Institut für Technologie, 76128 Karlsruhe, Germany.
julia.schweitzer@kit.edu

Received: 4 March 2014
Revised: 9 November 2014

Abstract

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 2 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.