Goal oriented time adaptivity using local error estimates
For computational scientists solving IVPs with specific quantities of interest, this provides a more efficient time-adaptivity method, though it is an incremental improvement over existing local error-based approaches.
The paper derives a goal-oriented error estimate and timestep controller for initial value problems where the quantity of interest is a time integral of a functional, proving convergence under weak assumptions. Numerical tests show the method outperforms dual-weighted residual and classical local error-based adaptivity, with significant speedups in some cases.
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.