MPFR

Fousse, Laurent; Hanrot, Guillaume; Lefèvre, Vincent; Pélissier, Patrick; Zimmermann, Paul

doi:10.1145/1236463.1236468

Cited by 672 publications

(116 citation statements)

References 15 publications

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“…Second, we made sure every constant used in IAS15 is as accurate as possible. All hard coded constants have been precalculated in extended floating-point precision using the GNU MP library (Fousse et al 2007). In addition, multiplications of a number x with a rational number p/q are implemented as p · x/q.…”

Section: Random Errorsmentioning

confidence: 99%

ias15: a fast, adaptive, high-order integrator for gravitational dynamics, accurate to machine precision over a billion orbits

Rein

Spiegel

2014

Monthly Notices of the Royal Astronomical Society

635

428

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We present IAS15, a 15th-order integrator to simulate gravitational dynamics. The integrator is based on a Gauß-Radau quadrature and can handle conservative as well as non-conservative forces. We develop a step-size control that can automatically choose an optimal timestep. The algorithm can handle close encounters and high-eccentricity orbits. The systematic errors are kept well below machine precision and long-term orbit integrations over 10 9 orbits show that IAS15 is optimal in the sense that it follows Brouwer's law, i.e. the energy error behaves like a random walk. Our tests show that IAS15 is superior to a mixed-variable symplectic integrator (MVS) and other popular integrators, including high-order ones, in both speed and accuracy. In fact, IAS15 preserves the symplecticity of Hamiltonian systems better than the commonly-used nominally symplectic integrators to which we compared it.We provide an open-source implementation of IAS15. The package comes with several easy-to-extend examples involving resonant planetary systems, Kozai-Lidov cycles, close encounters, radiation pressure, quadrupole moment, and generic damping functions that can, among other things, be used to simulate planet-disc interactions. Other non-conservative forces can be added easily.

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Section: Random Errorsmentioning

confidence: 99%

ias15: a fast, adaptive, high-order integrator for gravitational dynamics, accurate to machine precision over a billion orbits

Rein

Spiegel

2014

Monthly Notices of the Royal Astronomical Society

635

428

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“…The MCT provides wide range of numerical analysis routines implemented with arbitrary precision support ranging from elementary arithmetic operations [45,46] to advanced solvers for ill-conditioned linear systems and eigenvalue problems [47]. The MCT is heavily optimized for modern multi-core parallel architectures and orders of magnitude faster than Maple, Mathematica, and MATLABs Variable-Precision Arithmetic packages [44].…”

Section: Implementation Of the Galperin-zheng Methodsmentioning

confidence: 99%

Radial basis functions: achievements and challenges

Kansa¹

2015

WIT Transactions on Modelling and Simulation

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Traditionally, the various forms of compactly supported polynomial based schemes such as finite difference, element, and volume methods were used in the numerical solution of ordinary and partial differential equations (ODEs and PDEs) as well as integral equations (IEs). The primitive computers had limited memory and processing speed. As time progressed, spectral and pseudo-spectral methods that possess exponential or spectral convergence that use tensor products of global expansions of one-dimensional orthonormal functions were used to solve ODEs, PDEs, and IEs. In the past 25 years, radial basis functions (RBFs) that can be either compactly supported or global gained in importance. However, those global RBFs that are C ∞ with shape parameters enjoy exponential convergence. The solution accuracy can be increased by either the spatial resolution or increasing the shape parameters. The performance of C ∞ RBFs has been demonstrated in a wide variety of linear and nonlinear elliptic, hyperbolic and parabolic PDEs and IE applications. Both well posed and ill-posed problems can be solved as well as with defined arithmetic and fuzzy arithmetic, problems with fractional derivatives, etc. Computers with the standard 32 bit chips limit the inherent power of the global RBFs by giving rise to potentially severely ill-conditioned systems. However, procedures using preconditioning and domain decomposition can be used to overcome this limitation. Recently, fast extended precision software enables the use of large shape parameters outperforming compactly supported finite elements and other methods. Also, the Galperin-Zheng weak formulation allows treatment of both well posed and ill-posed problems because it looks for solutions with global, rather than local minima. Presently, meshfree C ∞ RBFs appear to have the potential of solving many important scientific problems in higher dimensions.

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“…Additionally, due to the exponential scaling of uniaxial deformation with strain, double precision arithmetic was insufficient for performing simulations at large strain. Following Hunt, 33 the arbitrary precision libraries GMP 37 and MPFR 38 were used for the computation of the box shape and the LLL algorithm, 39 as implemented in the fplll package, 40 was used for lattice reduction. In order to mitigate a known momentum instability in the SLLOD equations of motion under extension, the center of mass was periodically reset to zero every 100 timesteps.…”

Section: Simulation Detailsmentioning

confidence: 99%

Molecular simulation of flow-enhanced nucleation in n-eicosane melts under steady shear and uniaxial extension

Nicholson

Rutledge

2016

The Journal of Chemical Physics

118

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Non-equilibrium molecular dynamics is used to study crystal nucleation of n-eicosane under planar shear and, for the first time, uniaxial extension. A method of analysis based on the mean first-passage time is applied to the simulation results in order to determine the effect of the applied flow field type and strain rate on the steady-state nucleation rate and a characteristic growth rate, as well as the effects on kinetic parameters associated with nucleation: the free energy barrier, critical nucleus size, and monomer attachment pre-factor. The onset of flow-enhanced nucleation (FEN) occurs at a smaller critical strain rate in extension as compared to shear. For strain rates larger than the critical rate, a rapid increase in the nucleation rate is accompanied by decreases in the free energy barrier and critical nucleus size, as well as an increase in chain extension. These observations accord with a mechanism in which FEN is caused by an increase in the driving force for crystallization due to flow-induced entropy reduction. At high applied strain rates, the free energy barrier, critical nucleus size, and degree of stretching saturate, while the monomer attachment pre-factor and degree of orientational order increase steadily. This trend is indicative of a significant diffusive contribution to the nucleation rate under intense flows that is correlated with the degree of global orientational order in a nucleating system. Both flow fields give similar results for all kinetic quantities with respect to the reduced strain rate, which we define as the ratio of the applied strain rate to the critical rate. The characteristic growth rate increases with increasing strain rate, and shows a correspondence with the nucleation rate that does not depend on the type of flow field applied. Additionally, a structural analysis of the crystalline clusters indicates that the flow field suppresses the compaction and crystalline ordering of clusters, leading to the formation of large articulated clusters under strong flow fields, and compact well-ordered clusters under weak flow fields. Published by AIP Publishing. [http://dx

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MPFR

Cited by 672 publications

References 15 publications

ias15: a fast, adaptive, high-order integrator for gravitational dynamics, accurate to machine precision over a billion orbits

ias15: a fast, adaptive, high-order integrator for gravitational dynamics, accurate to machine precision over a billion orbits

Radial basis functions: achievements and challenges

Molecular simulation of flow-enhanced nucleation in n-eicosane melts under steady shear and uniaxial extension

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