MOLLIFIED IMPULSE METHODS FOR HIGHLY OSCILLATORYDIFFERENTIAL EQUATIONS

摘要

We introduce a family of impulselike methods for the integration of highly oscillatorysecond-order differential equations whose forces can be split into a fast part and a slow part. Methodsof this family are specified by two weight functions (1), ,L; one is used to average positions and theother to mollify the force. When the fast forces are conservative and = t/), the methods herecoincide 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 exponentialintegrators introduced by Hairer and Lubich for cases of linear fast forces. A convergence analysis ispresented that provides insight into the role played by the processes of mollification and averaging inavoiding order reduction. A simple condition on the weight functions is shown to be both necessaryand sufficient to avoid order reduction.