Numerical stroboscopic averaging for ODEs and DAEs

The stroboscopic averaging method (SAM) is a technique for the integration of highly oscillatory differential systems d y / d t = f ( y , t ) with a single high frequency. The method may be seen as a purely numerical way of implementing the analytical technique of stroboscopic averaging which constructs an averaged differential system d Y / d t = F ( Y ) whose solutions Y interpolate the sought highly oscillatory solutions y. SAM integrates numerically the averaged system without using the analytic expression of F; all information on F required by the algorithm is gathered on the fly by numerically integrating the originally given system in small time windows. SAM may be easily implemented in combination with standard software and may be applied with variable step-sizes. Furthermore it may also be used successfully to integrate oscillatory DAEs. The paper provides an analytic and experimental study of SAM and two related techniques: the LIPS algorithm of Kirchgraber and multirevolution methods. An error analysis is provided that indicates that the efficiency of all these techniques increases even further when combined with splitting integrators.