Controlling the Wrapping Effect in the Solution of ODEs for Asteroids

During the last decade, substantial progress has been made in fighting the wrapping effect in self-validated integrations of linear systems. However, it is still the main problem limiting the applicability of such methods to the long-term integration of non-linear systems. Here we show how high-order self-validated methods can successfully overcome this obstacle.We study and compare the validated integration of a Kepler problem with conventional and high-order methods represented by AWA and Taylor models, respectively. We show that this simple model problem exhibits significant wrapping that is particularly difficult to control for conventional first-order methods. It will become clear that utilizing high-order methods with shrink wrapping allows the system to be analyzed in a fully validated context over large integration times. By comparing high-order Taylor model integrations with Taylor model methods subjected to an artificial wrapping effect, we show that utilizing high-order methods to propagate initial conditions is indeed the foremost reason for the successful suppression of the wrapping effect.To further demonstrate that high-order Taylor model methods can be used for the integration of complicated non-linear systems, we summarize results obtained from a fully verified and self-validated orbit integration of the near earth asteroid 1997 XF11. Since this asteroid will have several close encounters with Earth, its analysis is an important application of reliable computations.