Runge-Kutta-Nyström Pairs of Orders 8(6) with Coefficients Trained to Perform Best on Classical Orbits

Abstract

In this study, we consider eight stages per step family of explicit Runge-Kutta-Nyström pai·rs of orders eight and six. The pairs from this family effectively use eight stages for each step. The coefficients p·rovided by such a method are much less than the number of non linear o·rder conditions requi·red to be solved. Thus, we t·raditionally apply various simplified assumptions in o·rder to address this drawback. The assumptions taken in the family we consider he·re deliver a subsystem where all the coefficients are evaluated successively and explicitly with respect to five f·ree parameters. We t·rain (adjust) these f·ree parameters in o·rder to derive a certain pair that outperforms other similar pai·rs of orders 8(6) in Keplerian type o·rbits, e.g., Kepler, pe·rturbed Kepler, A·renstorf o·rbit, or Pleiades. Diffe·rential evolution technique is used for the training. The pair that we finally present offe·rs about an additional digit of accu·racy in a va·riety of orbits.