On the numerical analysis of rapid oscillation

This paper surveys recent advances in the allied challenges of discretizing highly oscillatory ordinary differential equations and computing numerical quadrature of highly oscillatory integrals, and attempts to sketch the mathematical foundations of a general approach to these issues. Having described the Magnus, Cayley and Neumann methods for highly oscillatory ordinary differential equations and analysed the Filon quadrature of highly oscillatory integrals, its overall conclusion is that, once numerical methods are properly crafted, their accuracy increases in the presence of high oscillation.