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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
High precision parallel algorithms of numerical
integration of celestial mechanics problems
N. V. Batkhina, A. B. Batkhin
Volzhsky Institute of Humanities, Volgograd State University, Volzhsky, Russia
We investigate different scenarios of dynamical chaos origin in Hamiltonian
systems generated by some celestial mechanics problems. Numerical investiga­
tions were carried out of phenomena as follows:
ffl period doubling cascades [1];
ffl transversal intersections of hyperbolic point invariant manifolds [2].
Both above--mentioned phenomena require a big amount of processor time using
high precision numerical methods of integration of systems of ordinary differential
equations. There are some reasons for necessity of using such numerical methods.
First of all, we use Poincar'e mapping for searching, continuation and bifurcation
analysis of periodic solutions of Hamiltonian systems. This method require a big
amount of calculation at the first stage of its using. The second reason is that
both phenomena require computation of orbits for a lot of revolutions.
Traditional PC­based computer complex has essential restriction on hard­
ware bit grid (no more than 18 decimal digits) and hence we can define the
coordinates of points on Poincar'e section for long period orbits with accuracy no
more than 10 \Gamma15 and therefore we can obtain the other features of motion (such
as multiplicators, Feigenbaum and scaling constants, homoclinic invariants, etc.)
with essentially less accuracy.
We took as a base for constructing a high precision numerical method two
very popular algorithms:
ffl variable--order extrapolation method of Gragg--Burlirsch--StЁoer with variable--
step described in [3];
ffl fixed--order variable--step implicit Runge--Kutta method by Everhart (up to
27 order) [4].
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We compiled these algorithms to C programming language with free multiple
precision library GMP (GNU MP library) [5]. This library supports high effec­
tive floating point arithmetic operations with arbitrary precision. Testing of high
precision integrator for the planar Hill problem [1] showed that we could calcu­
late periodic orbits with number of revolutions 1024 and more with accuracy no
less than 10 \Gamma25 and obtain the Feigenbaum and scaling constants with accuracy
10 \Gamma10 .
For the reason of intensification of the computation process the programs
were carried on a parallel cluster which was built with the set of PC computers
under LINUX operating system with LAM/MPI software [6]. We created the
parallel version of numerical algorithm for calculation of Poincar'e map in MPI
environment and obtained the reduction of computing time at the small cluster
with 4 processors.
References
1. Batkhin A. B., Batkhina N. V., Sumarokov S. I. Doubling period bifurcations
in the Hill's problem. Vestnik of Volgograd State University, Ser. Mathemat­
ics, Physics, 2000, 5, 6--11 (in Russian).
2. Ivanov A. V. Study of the double mathematical pendulum -- I. Numerical
investigation of homoclinic transversal intersections. Regular and haotic Dy­
namics, 1999, 4, 104--116.
3. Hairer E., NЬrsett S. P., Wanner G. Solving ordinary differential equations
I. Berlin: Springer--Verlag, 1987.
4. Bordovitsyna T. V. The modern numerical methods in celestial mechanic
problems. M.: Nauka, 1984 (in Russian).
5. GNU MP library, http://www.swox.com/gmp.
6. LAM/MPI version 6.5.6, http://www.lam­mpi.org.
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