arXiv subjects

A. Murua

Publications and source records attributed to A. Murua.

7 recordsLinked to original sources

arXiv2024

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.

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↗

arXiv2017

Hopf algebra techniques to handle dynamical systems and numerical integrators

In a series of papers the present authors and their coworkers have developed a family of algebraic techniques to solve a number of problems in the theory of discrete or continuous dynamical systems and to analyze numerical integrators. Given a specific problem, those techniques construct an abstract, {\em universal} version of it which is solved algebraically; then, the results are tranferred to the original problem with the help of a suitable morphism. In earlier contributions, the abstract problem is formulated either in the dual of the shuffle Hopf algebra or in the dual of the Connes-Kreimer Hopf algebra. In the present contribution we extend these techniques to more general Hopf algebras, which in some cases lead to more efficient computations.

math.DS↗

arXiv2015

Averaging and computing normal forms with word series algorithms

In the first part of the present work we consider periodically or quasiperiodically forced systems of the form $(d/dt)x = \epsilon f(x,t \omega )$, where $\epsilon\ll 1$, $\omega\in\mathbb{R}^d$ is a nonresonant vector of frequencies and $f(x,\theta)$ is $2\pi$-periodic in each of the $d$ components of $\theta$ (i.e.\ $\theta\in\mathbb{T}^d$). We describe in detail a technique for explicitly finding a change of variables $x = u(X,\theta;\epsilon)$ and an (autonomous) averaged system $(d/dt) X = \epsilon F(X;\epsilon)$ so that, formally, the solutions of the given system may be expressed in terms of the solutions of the averaged system by means of the relation $x(t) = u(X(t),t\omega;\epsilon)$. Here $u$ and $F$ are found as series whose terms consist of vector-valued maps weighted by suitable scalar coefficients. The maps are easily written down by combining the Fourier coefficients of $f$ and the coefficients are found with the help of simple recursions. Furthermore these coefficients are {\em universal} in the sense that they do not depend on the particular $f$ under consideration. In the second part of the contribution, we study problems of the form $(d/dt) x = g(x)+f(x)$, where one knows how to integrate the "unperturbed" problem $(d/dt)x = g(x)$ and $f$ is a perturbation satisfying appropriate hypotheses. It is shown how to explicitly rewrite the system in the "normal form" $(d/dt) x = \bar g(x)+\bar f(x)$, where $\bar g$ and $\bar f$ are {\em commuting} vector fields and the flow of $(d/dt) x = \bar g(x)$ is conjugate to that of the unperturbed $(d/dt)x = g(x)$. In Hamiltonian problems the normal form directly leads to the explicit construction of formal invariants of motion. Again, $\bar g$, $\bar f$ and the invariants are written as series consisting of known vector-valued maps and universal scalar coefficients that may be found recursively.

math.DS↗

arXiv2015

Computing normal forms and formal invariants of dynamical systems by means of word series

We show how to use extended word series in the reduction of continuous and discrete dynamical systems to normal form and in the computation of formal invariants of motion in Hamiltonian systems. The manipulations required involve complex numbers rather than vector fields or diffeomorphisms. More precisely we construct a group G and a Lie algebra g in such a way that the elements of G and g are families of complex numbers; the operations to be performed involve the multiplication F in G and the bracket of g and result in universal coefficients that are then applied to write the normal form or the invariants of motion of the specific problem under consideration.

math.DS↗

arXiv2015

An efficient algorithm based on splitting for the time integration of the Schr\"odinger equation

We present a practical algorithm based on symplectic splitting methods to integrate numerically in time the Schr\"odinger equation. When discretized in space, the Schr\"odinger equation can be recast as a classical Hamiltonian system corresponding to a generalized high-dimensional separable harmonic oscillator. The particular structure of this system combined with previously obtained stability and error analyses allows us to construct a set of highly efficient symplectic integrators with sharp error bounds and optimized for different tolerances and time integration intervals. They can be considered, in this setting, as polynomial approximations to the matrix exponential in a similar way as methods based on Chebyshev and Taylor polynomials. The theoretical analysis, supported by numerical experiments, indicates that the new methods are more efficient than schemes based on Chebyshev polynomials for all tolerances and time intervals. The algorithm we present incorporates the new splitting methods and automatically selects the most efficient scheme given a tolerance, a time integration interval and an estimate on the spectral radius of the Hamiltonian.

math.NA↗