Parallel/vector integration methods for dynamical astronomy

This paper reviews three recent works on the numerical methods to integrate ordinary differential equations (ODE), which are specially designed for pa rallel, vector, and/or multi-processor-unit(PU) computers. The first is the Picard-Chebyshev method (Fukushima, 1997a). It obtains a global solution o f ODE in the form of Chebyshev polynomial of large (> 1000) degree by apply ing the Picard iteration repeatedly. The iteration converges for smooth pro blems and/or perturbed dynamics. The method runs around 100-1000 times fast er in the vector mode than in the scalar mode of a certain computer with ve ctor processors (Fukushima, 1997b). The second is a parallelization of a sy mplectic integrator (Saha et al., 1997). It regards the implicit midpoint r ules covering thousands of timesteps as large-scale nonlinear equations and solves them by the fixed-point iteration. The method is applicable to Hami ltonian systems and is expected to lead an acceleration factor of around 50 in parallel computers with more than 1000 PUs. The last is a parallelizati on of the extrapolation method (Ito and Fukushima, 1997). It performs trial integrations in parallel. Also the trial integrations are further accelera ted 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 parallelizatio n of some implicit integrators which require multiple corrections in solvin g implicit formulas like the implicit Hermitian integrators (Makino and Aar seth, 1992), (Hut et al., 1995) or the implicit symmetric multistep methods (Fukushima, 1998), (Fukushima, 1999).