Goal Oriented Time Adaptivity Using Local Error Estimates

Philipp Birken,Peter Meisrimel

doi:10.3390/a13050113

Abstract

We consider initial value problems (IVPs) where we are interested in a quantity of interest (QoI) that is the integral in time of a functional of the solution. For these, we analyze goal oriented time adaptive methods that use only local error estimates. A local error estimate and timestep controller for step-wise contributions to the QoI are derived. We prove convergence of the error in the QoI for tolerance to zero under a controllability assumption. By analyzing global error propagation with respect to the QoI, we can identify possible issues and make performance predictions. Numerical tests verify these results. We compare performance with classical local error based time-adaptivity and a posteriori based adaptivity using the dual-weighted residual (DWR) method. For dissipative problems, 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.

Highlights

A typical situation in numerical simulations based on differential equations is that one is not interested in the solution per se but in a Quantity of Interest (QoI) that is given as a functional of the solution
The purpose of this article is to present a formal derivation and analysis of such a goal oriented adaptive method based on local error estimates, allowing us to answer this question in detail
We compare the performance of the dual-weighted residual (DWR) method and the local error based adaptive methods

Summary

Introduction

A typical situation in numerical simulations based on differential equations is that one is not interested in the solution per se but in a Quantity of Interest (QoI) that is given as a functional of the solution. The QoI would be the lift coefficient divided by the drag coefficient. In simulations of the Greenland ice sheet, it could be the net amount of ice loss over a year [1]. The amount of energy produced during a certain time period is more important than the actual flow solution. One may want to optimize blade shape [2] or determine optimal placement of, for example, tidal turbines [3], for maximal energy output

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Discussion

Conclusion