N-body simulations: The performance of some integrators

ABSTRACT

We describe four challenging N-body test problems involving the Sun and planets and use them to compare the performance of nine nonsymplectic and two symplectic integrators. Each problem has a long interval of integration and two have non-Newtonian gravitational interactions. The emphasis in our comparison is on the accuracy of the solution, including the phase information produced by nonsympletic methods; the symplectic methods have been included to provide a contrast. Long intervals of integration necessitate small local error tolerances for the nonsymplectic integrators.Among variable-stepsize integrators, RKNINT requires the least CPU time on the two problems with Newtonian interactions and DIVA the least CPU time on the other two problems for the intervals of integration we used. We find that the error growth on some integrations is noticeably slower than predicted by asymptotic analysis of the truncation and round-off error. Our comparisons suggest that the numerical solutions near the end of a billion year simulation in double precision with variable-stepsize nonsymplectic methods would have poor accuracy.

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INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.7 Ordinary Differential Equations

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
        G.1.10 Applications

General Terms:
Algorithms, Performance

Keywords:
N-body test problems, accurate integration, solar system