Error Estimation and Adaptivity for Differential Equations with Multiple Scales in Time

We consider systems of ordinary differential equations with multiple scales in time. In general, we are interested the long time horizon a slow variable that is coupled to solution components act on fast scale. Although scale variables essential for dynamics problem, they often no interest themselves. Recently have proposed temporal multiscale approach fits into framework heterogeneous method and allows efficient simulations significant speedups. Fast decoupled by introducing local averages replacing contributions localized periodic-in-time problems. Here, generalize this larger class problems but particular, derive an posteriori error estimator based dual weighted residual splitting averaging error, demonstrate accuracy also its use adaptive control numerical scheme.