Abstract
AbstractWe consider adaptive time discretization methods for ordinary differential equations where one aims to control the error in a quantity of interest of the form , with j : ℝd → ℝ. In this setting we propose a new timestep controller based on local error estimates of the quantity of interest. The new method converges when the tolerance goes to zero.We experimentally compare the new scheme with the classic norm‐based time‐adaptivity based on local error estimates as well as the dual‐weighted residual (DWR) method. The results show significantly lower efficiency for the DWR method. The local error based schemes are similarly efficient, with the new scheme showing significant improvement in some cases. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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
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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