Taylor-Fourier approximation

TL;DR

The paper tackles highly oscillatory semi-linear ODEs of the form $\dot{\mathbf{x}}=\mathcal{A}\mathbf{x}+\mathbf{g}(\mathbf{x})$ with purely imaginary spectra by transforming to a slow variable $\mathbf{y}$ via $\mathbf{x}=e^{t\omega A}\mathbf{y}$, yielding a $(2\pi)$-periodic right-hand side $\mathbf{f}(\omega t,\mathbf{y})$. It introduces $(M,d)$-Taylor-Fourier approximations, representing solutions as $\mathbf{X}(\omega t,t)=e^{t\omega A}\sum_{k=-M}^{M} e^{ik\omega t}\sum_{j=0}^{d} t^{j} \mathbf{y}_{k,j}^{[M,d]}$, and develops an efficient FFT-based procedure to compute these coefficients with uniform accuracy in $\omega$ over time intervals independent of $\omega$. The work connects these approximations to high-order stroboscopic averaging, extends the framework to quasi-periodic and complex-domain problems, and demonstrates the method on the cubic nonlinear Schrödinger equation and a perturbed Kepler problem, achieving high-precision results and robust long-time behavior. Overall, the Taylor-Fourier approach provides reliable reference solutions and benchmarks for highly oscillatory dynamics, while offering a practical computational pathway via truncated power-series arithmetic and FFTs.

Abstract

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 $(2π/ω)$-periodic dependence on $t$. The proposed approximate solutions are given in closed form as functions $X(ωt,t)$, where $X(θ,t)$ is (i) a truncated Fourier series in $θ$ for fixed $t$ and (ii) a truncated Taylor series in $t$ for fixed $θ$, which motivates the name of the method. These approximations are uniformly accurate in $ω$, meaning that their accuracy does not degrade as $ω\to \infty$. 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 $ω$). 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 Schrödinger equation with non-smooth initial data and a perturbed Kepler problem from satellite orbit dynamics.

Figures (6)

• Figure 1: Numerical solution of \ref{['eq:NLS']}-\ref{['NLS_ic']} at $t=0.3$ (top left), $t=0.31$ (top right), $t=0.314$ (bottom left) and $t=\pi/10$ (bottom right) computed with $J=2^9$, and the $(2^{11},3)$-Taylor-Fourier approximation. Real part is plotted with blue solid line and imaginary part with brown dashed line.
• Figure 2: $(M,d)$-Taylor-Fourier approximations to the solution of \ref{['eq:NLS']}-\ref{['NLS_ic']} at $t=\pi/10$ for $J=2^9$, computed with $M=4J$, $d=1$ (left), $M=J$, $d=3$ (center) and $M=J$, $d=1$ (right). Real part is plotted with blue solid line and imaginary part with brown dashed line.
• Figure 3: Errors of the $(2^{11},7)$-Taylor-Fourier approximation at $t=\epsilon^{-2} \pi/10$ for $\epsilon=2^{-m}$, $m=1,2,3,4$ with $J=2^6$ and initial data (\ref{['eq:initial_data_NLS']}).
• Figure 4: Relative errors in position (top) and absolute errors in physical time (bottom) versus fictitious time $\tau$ (scaled by $P=2\pi/\omega$) of the $(8,8)$-Taylor-Fourier approximation.
• Figure 5: Energy error of the averaging approximated solution for $(M,d)=(8,8)$ (top) and the position error of the averaged approximation with respect the $(8,8)$-Taylor-Fourier approximation (bottom) in 600 revolutions.