The employ of an adapted numerical scheme within the integration of differential equations shows benefits in terms of accuracy and stability. In particular, we focus on differential equations modeling chemical phenomena with an oscillatory dynamics. In this work, the adaptation can be performed thanks to the information arising from existing theoretical studies and especially the observation of time series. Such information is properly merged into the exponential fitting technique, which is specially suitable to follow the a-priori known qualitative behavior of the solution. Some numerical experiments will be provided to exhibit the effectiveness of this approach.