  We consider initial value problems where we are interested in a quantity of
interest (QoI) that is the integral in time of a functional of the solution of
the IVP. For these, we look into local error based time adaptivity. We derive a
goal oriented error estimate and timestep controller, based on error
contribution to the error in the QoI, for which we prove convergence of the
error in the QoI for tolerance to zero under weak assumptions. We analyze
global error propagation of this method and derive guidelines to predict
performance of the method. In numerical tests we verify convergence results and
guidelines on method performance. Additionally, we compare with the
dual-weighted residual method (DWR) and classical local error based
time-adaptivity. The local error based methods show better performance than DWR
and the goal oriented method shows good results in most examples, with
significant speedups in some cases.
