High order symplectic integrators for perturbed Hamiltonian systems

By Jacques Laskar, P. Robutel. 2000

263 citations recorded by the source corpus.

A family of symplectic integrators adapted for the integration of perturbed Hamiltonian systems of the form H = A + εB was given in (McLachlan, 1995). We give here a constructive proof that for all integer p, such integrator exists, with only positive steps, and with a remainder of order O(τpε + τ2ε2), where τ is the stepsize of the integrator. Moreover, we compute the analytical expressions of the leading terms of the remainders at all orders. We show also that for a large class of systems, a corrector step can be performed such that the remainder becomes O(τpε +τ4ε2). The performances of these integrators are compared for the simple pendulum and the planetary three-body problem of Sun–Jupiter–Saturn.

Subfields: Control and Systems Engineering, Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/75c4c60154f4907911ede04ba35b1bd1a32050d0

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Control and Systems Engineering

In Numerical Analysis

In Statistical and Nonlinear Physics

Citations in the corpus

Geometric integrators for ODEs (2006, unknown): “One can easily build, for example, a 2-stage method of order O(ε2τ 4 + ε3τ 3), a 3-stage method of order O(ε2τ 6 + ε3τ 4), and so on [10, 69, 81].”
La2010: a new orbital solution for the long-term motion of the Earth (2011, methodology): “As in La2004, the numerical integration was performed with the symplectic integrator scheme SABAC4 of Laskar & Robutel (2001), including a correction step for the integration of the Moon.”
The HARPS search for southern extra-solar planets - XXVIII. Up to seven planets orbiting HD 10180: probing the architecture of low-mass planetary systems (2010, methodology): “We tested the stability ofthe system by performing a numerical integration of the orbits of the 6 outermost planets in Table 7 over 150 Myr using the symplectic integrator SABAC4 of Laskar & Robutel (2001) with a step size of 10−3 years, including general relativity.”
REBOUNDx: a library for adding conservative and dissipative forces to otherwise symplectic N-body integrations (2019, background): “This means any higher-order splitting scheme that tries to correct for higher and higher order terms in the BCH expansion (e.g., Wisdom et al. 1996; Laskar & Robutel 2001) will always have such a geometrical suppression in its leading error term.”
Long term evolution and chaotic diffusion of the insolation quantities of Mars (2004, unknown): “In the ﬁrst section, we will present the derivation of the orbital solution for Mars over 250 Myr using the new symplectic integrator developed in our group (Laskar and Robutel 2001).”
A dynamically-packed planetary system around GJ 667C with three super-Earths in its habitable zone (2013, methodology): “Therefore we tested the stability of each configuration by a separate numerical integration using the symplectic integrator SABA2 of Laskar & Robutel (2001) with a step sizeτ = 0.0625 days.”
Yarkovsky/YORP chronology of asteroid families (2006, methodology): “, in Marzari et al. (1995). relation (2) is no longer tailored to the model yet it can still be formally used to define the density function D(C) of the formal parameter C.”
A long-term numerical solution for the insolation quantities of the Earth (2004, unknown): “After a detailed analysis of the main limiting factors for long term integrations (Laskar, 1999), and the design of new symplectic integrators (Laskar and Robutel, 2001), we have obtained a new numerical solution for both orbital and rotational motion of the Earth that can be used over about 50 Myr ”