Derivation of symmetric composition constants for symmetric integrators

Sofroniou, Mark; Spaletta, Giulia

doi:10.1080/10556780500140664

Cited by 51 publications

(68 citation statements)

References 24 publications

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“…The coefficients used in those integrators are the ones found by McLachlan (1995) and McLachlan et al (2002). The 6th order integrators implemented in AMUSE are of 11 and 13 stages (M11 and M13), with the coefficients found by Sofroniou & Spaletta (2005). In the simulations performed here, we used the Rotating Bridge with a 6th order integrator.…”

Section: A1 High Order Integratorsmentioning

confidence: 99%

Radial migration of the Sun in the Milky Way: a statistical study

Martínez-Barbosa

Brown

Zwart

2014

Monthly Notices of the Royal Astronomical Society

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The determination of the birth radius of the Sun is important to understand the evolution and consequent disruption of the Sun's birth cluster in the Galaxy. Motivated by this fact, we study the motion of the Sun in the Milky Way during the last 4.6 Gyr in order to find its birth radius. We carried out orbit integrations backward in time using an analytical model of the Galaxy which includes the contribution of spiral arms and a central bar. We took into account the uncertainty in the parameters of the Milky Way potential as well as the uncertainty in the present day position and velocity of the Sun. We find that in general the Sun has not migrated from its birth place to its current position in the Galaxy (R ). However, significant radial migration of the Sun is possible when: 1) The 2 : 1 Outer Lindblad resonance of the bar is separated from the corrotation resonance of spiral arms by a distance ∼ 1 kpc. 2) When these two resonances are at the same Galactocentric position and further than the solar radius. In both cases the migration of the Sun is from outer regions of the Galactic disk to R , placing the Sun's birth radius at around 11 kpc. We find that in general it is unlikely that the Sun has migrated significantly from the inner regions of the Galactic disk to R .

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Section: A1 High Order Integratorsmentioning

confidence: 99%

Radial migration of the Sun in the Milky Way: a statistical study

Martínez-Barbosa

Brown

Zwart

2014

Monthly Notices of the Royal Astronomical Society

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“…However, I recommend an implementation following that of Bader and Deuflhard [1,5]. Using the Mathematica's NDSolve framework developed by Sofroniou et al [18,19], the current state of the art implementation of each of the methods presented in this paper requires only a few lines of Mathematica code. This author has shown that such implementation may solve all the ODE problems in the Test Set of IVP Solvers at Bari, Italy (formerly at CWI, Amsterdam) [15], including those derived from partial differential equations (PDE) by the method of lines.…”

Section: Appendix A: Practical Implementation Of Non-linear One-step mentioning

confidence: 99%

Stability by order stars for non-linear theta-methods based on means

Villatoro

2010

International Journal of Computer Mathematics

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The linear stability analysis of non-linear one-step methods based on means is studied by means of the concept of stability regions and order stars. Concretely, non-linear θ -methods based on harmonic, contraharmonic, quadratic, geometric, Heronian, centroidal and logarithmic means are considered. Their stability diagrams and order stars show their A-stability for θ ≥ 1/2, and L-stability in some cases. Order stars in the Riemann surface are a requirement for non-linear one-step methods. The advantages and disadvantages of this technique are presented.

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“…3, we compare some well known higher order integrators with the PV-based, minimal extrapolated integrators (17). KL8 and SS10 are eighth and tenth order symplectic integrators by Kahan and Li (1997) and Sofroniou and Spaletta (2005) with 17 and 35 force evaluations, respectively. Both are recommended by Hairer et al (2002).…”

Section: Higher Order Comparisonsmentioning

confidence: 99%

Multi-product splitting and Runge-Kutta-Nyström integrators

Chin

2010

Celest Mech Dyn Astr

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The splitting of e h(A+B) into a single product of e h A and e h B results in symplectic integrators when A and B are classical Lie operators. However, at high orders, a single product splitting, with exponentially growing number of operators, is very difficult to derive. This work shows that, if the splitting is generalized to a sum of products, then a simple choice of the basis product reduces the problem to that of extrapolation, with analytically known coefficients and only quadratically growing number of operators. When a multi-product splitting is applied to classical Hamiltonian systems, the resulting algorithm is no longer symplectic but is of the Runge-Kutta-Nyström (RKN) type. Multi-product splitting, in conjunction with a special force-reduction process, explains why at orders p = 4 and 6, RKN integrators only need p − 1 force evaluations.

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Derivation of symmetric composition constants for symmetric integrators

Cited by 51 publications

References 24 publications

Radial migration of the Sun in the Milky Way: a statistical study

Radial migration of the Sun in the Milky Way: a statistical study

Stability by order stars for non-linear theta-methods based on means

Multi-product splitting and Runge-Kutta-Nyström integrators

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