Reducing rounding errors and achieving Brouwerʼs law with Taylor Series Method

The integration of problems derived from dynamical systems is an important topic, both in mathematics and physics. In many publications, specific algorithms for each problem are proposed to obtain high accuracy in the integration. In this paper, we study the performance of the Taylor Series Method and the ways to obtain optimal accuracy in the integration of differential equations. We present different sources of rounding errors and how to reduce them. All the different strategies are compared to show their efficiency.