We develop an \textit{a posteriori} error analysis for the time of the first occurrence of an event, specifically, 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 novel quantity of interest (QoI) differs from classical QoIs which are modeled as bounded linear (or nonlinear) functionals. Taylor's theorem and an adjoint-based \textit{a posteriori} analysis is used to derive computable and accurate error estimates for 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.

翻译：我们为首次发生事件时的错误分析,具体来说,是部分差分方程(PDE)解决方案的功能首次在特定时间间隔内达到临界值的时间。这种新颖的利息量(QoI)不同于典型的典型的QI,这些利益量以捆绑线性(或非线性)功能为模型。Taylor的理论和基于联合的\textit{a postiori}分析用于为半线性抛物线和双线性PDE得出可比较和准确的错误估计值。错误估计的准确性通过单维热方程和线性浅水方程(SWE)的数字解决方案分别代表单面和双向情况。