Performance of Gauss implicit Runge-Kutta methods on separable Hamiltonian systems

We consider implementations of a variable step size (and, separately, constant step size), fourth-order symplectic Gauss implict Runge-Kutta method for the solution of Hamiltonian systems. We test our implementations on Keplerʹs problem with the aim of judging the algorithmsʹ qualitative behavior and efficiency. In particular, we introduce compensated summation as a method of controlling roundoff accumulation. Also, we show how the variable step size Gauss implicit Runge-Kutta method performs on Keplerʹs problem with solution orbits of high eccentricity, and compare its performance with that of two Runge-Kutta-Nyström codes. Finally, we discuss the calculation of efficient starting values for the associated iterations, measure the cost in iterations of our various predictors, and comment on the strategies for terminating the iteration.