Two FORTRAN packages for assessing initial value methods
ACM Transactions on Mathematical Software (TOMS)
Algorithm 670: a Runge-Kutta-Nyström code
ACM Transactions on Mathematical Software (TOMS)
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
The development of variable-step symplectic integrators with application to the two-body problem
SIAM Journal on Scientific Computing
High-order symplectic Runge-Kutta-Nystro¨m methods
SIAM Journal on Scientific Computing
ACM Transactions on Mathematical Software (TOMS)
The stability of planetesimal niches in the outer solar system: a numerical investigation
The stability of planetesimal niches in the outer solar system: a numerical investigation
Explicit variable step-size and time-reversible integration
Applied Numerical Mathematics
The performance of phase-lag enhanced explicit Runge-Kutta Nyström pairs on N-body problems
Journal of Computational and Applied Mathematics
Algorithm 924: TIDES, a Taylor Series Integrator for Differential EquationS
ACM Transactions on Mathematical Software (TOMS)
High order explicit Runge---Kutta Nyström pairs
Numerical Algorithms
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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.