Mollified Impulse Methods for Highly Oscillatory Differential Equations

Abstract

We introduce a family of impulselike methods for the integration of highly oscillatory second-order differential equations whose forces can be split into a fast part and a slow part. Methods of this family are specified by two weight functions $\phi$, $\psi$; one is used to average positions and the other to mollify the force. When the fast forces are conservative and $\phi=\psi$, the methods here coincide with the mollified impulse methods introduced by Garcia-Archilla, Sanz-Serna, and Skeel. On the other hand, the methods here extend to nonlinear situations a well-known class of exponential integrators introduced by Hairer and Lubich for cases of linear fast forces. A convergence analysis is presented that provides insight into the role played by the processes of mollification and averaging in avoiding order reduction. A simple condition on the weight functions is shown to be both necessary and sufficient to avoid order reduction.