  We propose a third-order numerical integrator based on the Neumann series and
the Filon quadrature, designed mainly for highly oscillatory partial
differential equations. The method can be applied to equations that exhibit
small or moderate oscillations; however, counter-intuitively, large
oscillations increase the accuracy of the scheme. With the proposed approach,
the convergence order of the method can be easily improved. Error analysis of
the method is also performed. We consider linear evolution equations involving
first- and second-time derivatives that feature elliptic differential
operators, such as the heat equation or the wave equation. Numerical
experiments consider the case in which the space dimension is greater than one
and confirm the theoretical study.
