The error growth of some symplectic explicit Runge-Kutta Nystrom methods on long N-body simulations

Abstract

At one extreme, the global error for symplectic explicit Runge-Kutta Nystrom (SERKN) methods consists entirely of truncation error and grows as t. At the other extreme, the global error consists entirely of random round-off error and grows stochastically as t^1.5. We use numerical testing to investigate how the global error grows for stepsizes between these two extremes. The testing is of representative SERKN methods of orders four to seven on three long N-body simulations of the Solar System. The work also provides an opportunity to introduce two new test problems for symplectic methods and to present comparisons of the efficiency of SERKN methods.