New families of symplectic splitting methods for numerical integration in dynamical astronomy

Blanes, Sergio; Casas, Fernando; Farrés, Ariadna; Laskar, Jacques; Makazaga, Joseba; Murua, Ander

doi:10.1016/j.apnum.2013.01.003

Cited by 98 publications

(134 citation statements)

References 23 publications

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“…The evolution of the e, inclination and j z is presented in Figure 1 using the accurate, direct N-body (inverse square law) integration. The Nbody algorithm applies a Wisdom-Holman (Wisdom & Holman 1991) operator splitting with a high order (8-6-4) coefficient set taken from Blanes et al (2012), and for more details see descriptions in . The time is shown in units of the secular (Kozai) time scale…”

Section: The Double-averaging Approximation Breaks Down Over Long Timmentioning

confidence: 99%

Double-averaging can fail to characterize the long-term evolution of Lidov–Kozai Cycles and derivation of an analytical correction

Luo

Katz

Dong

2016

Mon. Not. R. Astron. Soc.

127

153

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The double-averaging (DA) approximation is widely employed as the standard technique in studying the secular evolution of the hierarchical three-body system. We show that effects stemmed from the short-timescale oscillations ignored by DA can accumulate over long timescales and lead to significant errors in the long-term evolution of the Lidov-Kozai cycles. In particular, the conditions for having an orbital flip, where the inner orbit switches between prograde and retrograde with respect to the outer orbit and the associated extremely high eccentricities during the switch, can be modified significantly. The failure of DA can arise for a relatively strong perturber where the mass of the tertiary is considerable compared to the total mass of the inner binary. This issue can be relevant for astrophysical systems such as stellar triples, planets in stellar binaries, stellar-mass binaries orbiting massive black holes and moons of the planets perturbed by the Sun. We derive analytical equations for the short-term oscillations of the inner orbit to the leading order for all inclinations, eccentricities and mass ratios. Under the test particle approximation, we derive the "corrected double-averaging" (CDA) equations by incorporating the effects of short-term oscillations into the DA. By comparing to N-body integrations, we show that the CDA equations successfully correct most of the errors of the long-term evolution under the DA approximation for a large range of initial conditions. We provide an implementation of CDA that can be directly added to codes employing DA equations.

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Section: The Double-averaging Approximation Breaks Down Over Long Timmentioning

confidence: 99%

Double-averaging can fail to characterize the long-term evolution of Lidov–Kozai Cycles and derivation of an analytical correction

Luo

Katz

Dong

2016

Mon. Not. R. Astron. Soc.

127

153

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“…The particular values of the coefficients appearing in the expression of the ABA864 SI are given in Table 3 of [28].…”

Section: The Formal Solution Of the Equations Of Motion For Initial mentioning

confidence: 99%

“…2−2 1/3 , and the ABA864 method introduced in [28] having 15 steps. The particular values of the coefficients appearing in the expression of the ABA864 SI are given in Table 3 of [28].…”

Section: The Formal Solution Of the Equations Of Motion For Initial mentioning

confidence: 99%

On the symplectic integration of the discrete nonlinear Schrödinger equation with disorder

Gerlach

Meichsner

Skokos

2016

Eur. Phys. J. Spec. Top.

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Abstract. We present several methods, which utilize symplectic integration techniques based on two and three part operator splitting, for numerically solving the equations of motion of the disordered, discrete nonlinear Schrödinger (DDNLS) equation, and compare their efficiency. Our results suggest that the most suitable methods for the very long time integration of this one-dimensional Hamiltonian lattice model with many degrees of freedom (of the order of a few hundreds) are the ones based on three part splits of the system's Hamiltonian. Two part split techniques can be preferred for relatively small lattices having up to N ≈ 70 sites. An advantage of the latter methods is the better conservation of the system's second integral, i.e. the wave packet's norm.

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“…This problem can be considered as a perturbed problem and we study the performance of the splitting methods (4,2) and (8,4) for the splitting given by (23) and (24), which are tailored for this class of problems. We solve separately the dominant part of the system, given by the equations  …”

Section: Numerical Examplesmentioning

confidence: 99%

High order structure preserving explicit methods for solving linear-quadratic optimal control problems

Blanes

2014

Numer Algor

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We consider the numerical integration of linear-quadratic optimal control problems. This problem requires the solution of a boundary value problem: a non-autonomous matrix Riccati differential equation (RDE) with final conditions coupled with the state vector equation with initial conditions. The RDE has positive definite matrix solution and to numerically preserve this qualitative property we propose first to integrate this equation backward in time with a sufficiently accurate scheme. Then, this problem turns into an initial value problem, and we analyse splitting and Magnus integrators for the forward time integration which preserve the positive definite matrix solutions for the RDE. Duplicating the time as two new coordinates and using appropriate splitting methods, high order methods preserving the desired property can be obtained. The schemes make sequential computations and do not require the storrage of intermediate results, so the storage requirements are minimal. The proposed methods are also adapted for solving linear-quadratic N -player differential games. The performance of the splitting methods can be considerably improved if the system is a perturbation of an exactly solvable problem and the system is properly split. Some numerical examples illustrate the performance of the proposed methods.

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New families of symplectic splitting methods for numerical integration in dynamical astronomy

Cited by 98 publications

References 23 publications

Double-averaging can fail to characterize the long-term evolution of Lidov–Kozai Cycles and derivation of an analytical correction

Double-averaging can fail to characterize the long-term evolution of Lidov–Kozai Cycles and derivation of an analytical correction

On the symplectic integration of the discrete nonlinear Schrödinger equation with disorder

High order structure preserving explicit methods for solving linear-quadratic optimal control problems

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