One dimensional troubled cell indicator, which is based on the extreme point of approximated polynomial, was proposed by Zhu and Qiu (SIAM J. Sci. Comput. 39:A1089-A1113, 2017). This troubled cell indicator performs well, but still has two shortcomings. First, it calculates the extreme point of high-degree polynomial, however the calculation is not simple. The reason is that, the explicit formula of extreme point of quartic polynomial has been complex, the one of higher degree polynomial becomes more complex or even is hard to be obtained, so has to be calculated by using numerical method. Second, a complex or without explicit formula of extreme point will bring tremendous difficulty to analyze why the troubled cell indicator can capture the major character of solution. In order to overcome these shortcomings, we first improve the troubled cell indicator, then use it to design the hybrid WENO-AO method for solving one dimensional hyperbolic conservation laws.

Zhu和Qiu（SIAM J.Sci.Comput.39：A1089-A1113，2017）提出了基于近似多项式极值的一维故障单元指标。该故障电池指示器性能良好，但仍存在两个缺点。首先，它计算高次多项式的极点，但是计算并不简单。原因在于，四次多项式的极值的显式公式比较复杂，高阶多项式之一甚至更复杂甚至难以获得，因此必须采用数值方法进行计算。其次，复杂的或没有明确的极端公式，将给分析有问题的电池指示器为何能够捕获解决方案的主要特征带来巨大的困难。为了克服这些缺点，