Symmetric multistep methods for the numerical integration of planetary orbits
The limit to the numerical accuracy of integrations of planetary orbits is set by the accumulation of round-off and truncation error. For the usual Stoermer multistep methods, even if steps are taken to reduce round-off error, truncation error still results in an energy error that grows linearly with time, which leads to a longitude error that grows quadratically with time. Here, 'symmetric' multistep methods are developed for which truncation leads to a longitude error that grows only linearly with time. The superiority of the symmetric methods over the Stoermer methods is illustrated by numerical examples. The optimum choice of the order and the coefficients of a symmetric multistep method for planetary integrations are discussed.