Goal oriented time adaptivity using local error estimates

TL;DR

This paper develops a goal-oriented local error estimation and adaptive timestep control for initial value problems, improving efficiency in computing quantities of interest with proven convergence and better performance than existing methods.

Contribution

It introduces a new goal-oriented error estimate and timestep controller for time adaptivity, with convergence proof and performance analysis, outperforming traditional methods in numerical tests.

Findings

The method achieves convergence of the QoI error as tolerance approaches zero.

Numerical tests confirm the effectiveness and efficiency of the proposed approach.

The approach outperforms the dual-weighted residual method and classical local error methods in most cases.

Abstract

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.