On Reusing the Stages of a Rejected Runge-Kutta Step

Runge-Kutta (RK) pairs are amongst the most popular methods for numerically solving Initial Value Problems. While using an RK pair, we may experience rejection of some steps through interval integration. Traditionally, all evaluations then dropped, and proceed with a completely new round computations. In this work, propose avoiding waste continuing by reusing rejected stages. We focus especially on pair orders six five. After step rejection, reuse previously evaluated stages only add three evaluating output smaller step. By technique, manage to significantly reduce cost in set problems that known pose difficulties algorithms changing sizes.