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Parallel/Vector Integration Methods for Dynamical Astronomy

  • Toshio Fukushima (a1)

Abstract

This paper reviews three recent works on the numerical methods to integrate ordinary differential equations (ODE), which are specially designed for parallel, vector, and/or multi-processor-unit (PU) computers. The first is the Picard-Chebyshev method (Fukushima, 1997a). It obtains a global solution of ODE in the form of Chebyshev polynomial of large (> 1000) degree by applying the Picard iteration repeatedly. The iteration converges for smooth problems and/or perturbed dynamics. The method runs around 100-1000 times faster in the vector mode than in the scalar mode of a certain computer with vector processors (Fukushima, 1997b). The second is a parallelization of a symplectic integrator (Saha et al., 1997). It regards the implicit midpoint rules covering thousands of timesteps as large-scale nonlinear equations and solves them by the fixed-point iteration. The method is applicable to Hamiltonian systems and is expected to lead an acceleration factor of around 50 in parallel computers with more than 1000 PUs. The last is a parallelization of the extrapolation method (Ito and Fukushima, 1997). It performs trial integrations in parallel. Also the trial integrations are further accelerated by balancing computational load among PUs by the technique of folding. The method is all-purpose and achieves an acceleration factor of around 3.5 by using several PUs. Finally, we give a perspective on the parallelization of some implicit integrators which require multiple corrections in solving implicit formulas like the implicit Hermitian integrators (Makino and Aarseth, 1992), (Hut et al., 1995) or the implicit symmetric multistep methods (Fukushima, 1998), (Fukushima, 1999).

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References

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Fukushima, T.: 1997 a, ‘Picard Iteration Method, Chebyshev Polynomial Approximation, and Global Numerical Integration of Dynamical Motions’, Astron. J., 113, 19091914.
Fukushima, T.: 1997b, ‘Vector Integration of Dynamical Motions by the Picard-Chebyshev Method’, Astron. J., 113, 23252328.
Fukushima, T.: 1998, ‘Symmetric Multistep Methods Revisited’, in Proc. 30th Symp. on Cele. Mech. (Fukushima, et al. eds), 229247.
Fukushima, T.: 1999, ‘Symmetric Multistep Methods Revisited II: Numerical Experiments’, in Proc. IAU Coll. No. 173, to be printed.
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Ito, T., and Fukushima, T.: 1997, ‘Parallelized Extrapolation Method and Its Application to the Orbital Dynamics’, Astron. J., 114, 12601267.
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Saha, P., Stadel, J., and Tremaine, S.: 1997, ‘A Parallel Integration Method for Solar System Dynamics’, Astron. J., 114, 409415.
Wisdom, J., and Holman, M.: 1991, ‘Symplectic Maps for the N-body Problem’, Astron. J., 102, 15281538.

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