arXiv subjects

J. Makazaga

Publications and source records attributed to J. Makazaga.

3 recordsLinked to original sources

arXiv2025

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 $(2π/ω)$-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.

math.NA↗

arXiv2022

An implicit symplectic solver for high-precision long term integrations of the Solar System

Compared to other symplectic integrators (the Wisdom and Holman map and its higher order generalizations) that also take advantage of the hierarchical nature of the motion of the planets around the central star, our methods require solving implicit equations at each time-step. We claim that, despite this disadvantage, FCIRK16 is more efficient than explicit symplectic integrators for high precision simulations thanks to: (i) its high order of precision, (ii) its easy parallelization, and (iii) its efficient mixed-precision implementation which reduces the effect of round-off errors. In addition, unlike typical explicit symplectic integrators for near Keplerian problems, FCIRK16 is able to integrate problems with arbitrary perturbations (non necessarily split as a sum of integrable parts). We present a novel analysis of the effect of close encounters in the leading term of the local discretization errors of our integrator. Based on that analysis, a mechanism to detect and refine integration steps that involve close encounters is incorporated in our code. That mechanism allows FCIRK16 to accurately resolve close encounters of arbitrary bodies. We illustrate our treatment of close encounters with the application of FCIRK16 to a point mass Newtonian 15-body model of the Solar System (with the Sun, the eight planets, Pluto, and five main asteroids) and a 16-body model treating the Moon as a separate body. We also present some numerical comparisons of FCIRK16 with a state-of-the-art high order explicit symplectic scheme for 16-body model that demonstrate the superiority of our integrator when very high precision is required.

physics.comp-ph↗

arXiv2020

Global time-renormalization of the gravitational $N$-body problem

This work considers the {\em gravitational} $N$-body problem and introduces global time-renormalization {\em functions} that allow the efficient numerical integration with fixed time-steps. First, a lower bound of the radius of convergence of the solution to the original equations is derived, which suggests an appropriate time-renormalization. In the new fictitious time $τ$, it is then proved that any solution exists for all $τ\in \mathbb{R}$, and that it is uniquely extended as a holomorphic function to a strip of fixed width. As a by-product, a global power series representation of the solutions of the $N$-body problem is obtained. Noteworthy, our global time-renormalizations remain valid in the limit when one of the masses vanishes. Finally, numerical experiments show the efficiency of the new time-renormalization functions for some $N$-body problems with close encounters.

math.DS↗