Section: New Results

Higher-order averaging, formal series and numerical integration

Participant : Philippe Chartier.

In [17] , we show how B-series may be used to derive in a systematic way the analytical expressions of the high-order stroboscopic averaged equations that approximate the slow dynamics of highly oscillatory systems. For first order systems we give explicitly the form of the averaged systems with -errors, j = 1, 2, 3 (2 denotes the period of the fast oscillations). For second order systems with large forces, we also give the explicit form of the averaged systems. The Fermi-Pasta-Ulam model and the inverted Kapitsa pendulum are used as illustrations. For the former it is shown that our approach establishes the adiabatic invariance of the oscillatory energy. Finally we use B-series to analyze multiscale numerical integrators that implement the method of averaging. We construct integrators that are able to approximate not only the simplest, lowest order averaged equation but also its high-order counterparts.