High order numerical methods for highly oscillatory problems
1 Matematik och matematisk statistik,
Umeå universitet, 90187
2 Institut für Angewandte und Numerische Mathematik, Karlsruher Institut für Technologie, 76128 Karlsruhe, Germany.
Revised: 9 November 2014
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.
Mathematics Subject Classification: 34E05 / 34E13 / 65L20 / 65P10
Key words: Highly oscillatory differential equations / multiple time scales / Fermi–Pasta–Ulam problem / modulated Fourier expansions / high order numerical schemes / adiabatic invariants
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