Oscillatory Differential Equations with Constant High Frequencies

Abstract

This chapter deals with numerical methods for second-order differential equations with oscillatory solutions. These methods are designed to require a new complete function evaluation only after a time step over one or many periods of the fastest oscillations in the system. Various such methods have been proposed in the literature – some of them decades ago, some very recently, motivated by problems from molecular dynamics, astrophysics and nonlinear wave equations. For these methods it is not obvious what implications geometric properties like symplecticity or reversibility have on the long-time behaviour, e.g., on energy conservation.