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

We develop an 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 bo...

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Bibliographic Details
Published in:BIT 2023-03, Vol.63 (1), Article 12
Main Authors: Chaudhry, Jehanzeb H., Estep, Donald, Giannini, Trevor, Stevens, Zachary, Tavener, Simon J.
Format: Article
Language:English
Subjects:
Computational Mathematics and Numerical Analysis
Error analysis
Estimates
Hyperbolic differential equations
Mathematics
Mathematics and Statistics
Numeric Computing
Partial differential equations
Shallow water equations
Thermodynamics
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Summary:We develop an 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 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.
ISSN:0006-3835
1572-9125
DOI:10.1007/s10543-023-00947-1