Obtaining Accurate Derivatives Across Adaptive-Step Integration

Obtaining derivatives through the integration of a set of ordinary differential equations is of interest to a number of fields of research. Approaches for doing so that are available in the literature, however, do not account for details associated with variable step integrator that cause inaccuracies in the total derivatives. In this paper, we explore the causes of the accumulation of derivative error in adaptive-step Runge-Kutta integration schemes and propose a solution to eliminate them. The new algorithm proposed is a transformation of each step of a Runge-Kutta integration into a set of residual equations. This work demonstrates the improvement in accuracy of total derivatives when the new algorithm is utilized.