10 research outputs found
An implicit symplectic solver for high-precision long term integrations of the Solar System
We present FCIRK16, a 16th-order implicit symplectic integrator for long-term high precision Solar System simulations. Our integrator takes advantage of the near-Keplerian motion of the planets around the Sun by alternating Keplerian motions with corrections accounting for the planetary interactions. 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.Consolidated Research Group MATHMODE (IT1294-19
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
New families of symplectic splitting methods for numerical integration in dynamical astronomy
We present new splitting methods designed for the numerical integration of
near-integrable Hamiltonian systems, and in particular for planetary N-body
problems, when one is interested in very accurate results over a large time
span. We derive in a systematic way an independent set of necessary and
sufficient conditions to be satisfied by the coefficients of splitting methods
to achieve a prescribed order of accuracy. Splitting methods satisfying such
(generalized) order conditions are appropriate in particular for the numerical
simulation of the Solar System described in Jacobi coordinates. We show that,
when using Poincar\'e Heliocentric coordinates, the same order of accuracy may
be obtained by imposing an additional polynomial equation on the coefficients
of the splitting method. We construct several splitting methods appropriate for
each of the two sets of coordinates by solving the corresponding systems of
polynomial equations and finding the optimal solutions. The experiments
reported here indicate that the efficiency of our new schemes is clearly
superior to previous integrators when high accuracy is required.Comment: 24 pages, 2 figures. Revised version, accepted for publication in
Applied Numerical Mathematic
Global time-regularization of the gravitational N -body problem
This work considers the gravitational N-body problem and introduces time-reparametrization functions that allow to define globally solutions of the N-body equations. First, a lower bound of the radius of convergence of the solution to the original equations is derived, which suggests an appropriate time-reparametrization. In the new fictitious time τ , it is then proved that any solution exists for all τ ∈ 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-regularization remain valid in the limit when one of the masses vanishes. Finally, numerical experiments show the efficiency of the new time-regularization functions for some N-problems with close encounters
Global Time-Renormalization of the Gravitational N-body Problem
This work considers the gravitational N-body problem and introduces global time-renormalization 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 \tau , it is then proved that any solution exists for all \tau \in 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. Notably, 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 the numerical integration of some N-body problems with close encounters
An Algorithm based on continuation techniques for minimization problems with highly non linear equality constraints
We present an algorithm based on continuation techniques that can be applied to solve numerically minimization problems with equality constraints. We focus on problems with a great number of local minima which are hard to obtain by local minimization algorithms with random starting guesses. We are particularly interested in the computation of minimal norm solutions of underdetermined systems of polynomial equations. Such systems arise, for instance, in the context of the construction of high order optimized differential equation solvers. By applying our algorithm, we are able to obtain 10th order time-symmetric composition integrators with smaller 1-norm than any other integrator found in the literature up to now