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arXiv · 2111.09834

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

Jehanzeb Chaudhry·Don Estep·Trevor Giannini·Zachary Stevens·Simon Tavener
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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.

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Jehanzeb Chaudhry, Don Estep, Trevor Giannini, Zachary Stevens, Simon Tavener. 2021-11-18. Error estimation for the time to a threshold value in evolutionary partial differential equations. https://arxiv.org/abs/2111.09834

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