Error estimation for the time to a threshold value in evolutionary partial differential equations
Abstract
We develop an \textit{a posteriori} error analysis for a numerical estimate of the time at which a functional of the solution to a partial differential equation (PDE) first achieves a threshold value on a given time interval. This quantity of interest (QoI) differs from classical QoIs which are modeled as bounded linear (or nonlinear) functionals {of the solution}. Taylor's theorem and an adjoint-based \textit{a posteriori} analysis is used to derive computable and accurate error estimates in the case of semi-linear parabolic and hyperbolic PDEs. The accuracy of the error estimates is demonstrated through numerical solutions of the one-dimensional heat equation and linearized shallow water equations (SWE), representing parabolic and hyperbolic cases, respectively.
Cite
@article{arxiv.2111.09834,
title = {Error estimation for the time to a threshold value in evolutionary partial differential equations},
author = {Jehanzeb Chaudhry and Don Estep and Trevor Giannini and Zachary Stevens and Simon Tavener},
journal= {arXiv preprint arXiv:2111.09834},
year = {2022}
}