Error estimation for the time to a threshold value in evolutionary partial differential equations

TL;DR

This paper introduces an  posteriori error analysis method to accurately estimate the time at which a functional of a PDE solution first reaches a threshold, applicable to semi-linear parabolic and hyperbolic equations.

Contribution

It develops a novel  posteriori error estimation technique for the first hitting time of a PDE solution functional, extending classical approaches.

Findings

Error estimates are accurate for heat and shallow water equations.

Method is applicable to both parabolic and hyperbolic PDEs.

Numerical validation confirms the effectiveness of the approach.

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.