Taylor-Fourier approximation

In this paper, we introduce an algorithm that provides approximate solutions to semi-linear ordinary differential equations with highly oscillatory solutions, which, after an appropriate change of variables, can be rewritten as non-autonomous systems with a $(2\pi/\omega)$-periodic dependence on $t$. The proposed approximate solutions are given in closed form as functions $X(\omega t,t)$, where $X(\theta,t)$ is (i) a truncated Fourier series in $\theta$ for fixed $t$ and (ii) a truncated Taylor series in $t$ for fixed $\theta$, which motivates the name of the method. These approximations are uniformly accurate in $\omega$, meaning that their accuracy does not degrade as $\omega \to \infty$. In addition, Taylor-Fourier approximations enable the computation of high-order averaging equations for the original semi-linear system, as well as related maps that are particularly useful in the highly oscillatory regime (i.e., for sufficiently large $\omega$). The main goal of this paper is to develop an efficient procedure for computing such approximations by combining truncated power series arithmetic with the Fast Fourier Transform (FFT). We present numerical experiments that illustrate the effectiveness of the proposed method, including applications to the nonlinear Schr\"odinger equation with non-smooth initial data and a perturbed Kepler problem from satellite orbit dynamics.