We develop an a posteriori error analysis for a novel quantity of interest (QoI) evolutionary partial differential equations (PDEs). Specifically, the QoI is the first time at which a functional of the solution to the PDE achieves a threshold value signifying a particular event, and differs from classical QoIs which are modeled as bounded linear functionals. We use Taylor's theorem and adjoint based analysis to derive computable and accurate error estimates for linear parabolic and hyperbolic PDEs. Specifically, the heat equation and linearized shallow water equations (SWE) are used for the parabolic and hyperbolic cases, respectively. Numerical examples illustrate the accuracy of the error estimates.

翻译：我们为新数量的利息(QoI)的进化部分差异方程式(PDEs)开发了事后误差分析。 具体地说, QoI是PDE解决方案的功能首次达到代表特定事件的临界值,并且不同于典型的、以捆绑线性功能为模型的二次误差分析。 我们使用泰勒的理论和基于联合的分析得出线性抛物线和双曲性PDEs的可计算和准确误差估计值。 具体地说,热方程式和线性浅水方程分别用于parbolic和双曲型案例。 数字示例显示了错误估计的准确性。