Hybrid algorithms are an efficient and popular choice for computing the solutions of hyperbolic conservation laws. In general, hybrid algorithms involve low-cost high-order accurate schemes in smooth regions and non-oscillatory shock-capturing schemes in the vicinity of discontinuities. Troubled-cell indicators which measure the smoothness of the solution play a significant role in the efficiency of hybrid algorithms. This article proposes a new troubled-cell indicator utilising the smoothness indicators of WENO schemes for hyperbolic conservation laws. The proposed troubled-cell indicators are simple, efficient, effective, and are used to construct three new adaptive WENO algorithms of high-order accuracy. The hybrid algorithms developed are independent of the order and type of the WENO reconstruction. For demonstration, we have considered the fifth and seventh order WENO-Z reconstruction. The first two algorithms have comparable accuracy and resolution of the solution across discontinuities to that of the WENO-Z scheme but at a less computational cost. The third algorithm ensures the convergence of the proposed scheme to the correct entropy solution when applied to a hyperbolic conservation law with non-convex flux for which the WENO schemes fail. We have performed several 1D and 2D numerical experiments to demonstrate the efficiency of the proposed algorithms and their performance compared with the WENO-Z schemes. The proposed algorithms are efficient and take 30%–75% less computational time than the WENO-Z schemes while retaining the advantages of WENO-Z schemes.

中文翻译：

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基于 WENO 平滑度指标的双曲守恒定律问题单元指标

混合算法是计算双曲守恒定律解的有效且流行的选择。一般来说，混合算法涉及平滑区域中的低成本高阶精确方案和不连续性附近的非振荡冲击捕获方案。衡量解决方案平滑度的问题单元指标在混合算法的效率中发挥着重要作用。本文提出了一种新的问题单元指标，利用双曲守恒定律的 WENO 方案的平滑度指标。所提出的问题单元指标简单、高效、有效，并用于构造三种新的高阶精度自适应WENO算法。开发的混合算法独立于 WENO 重建的顺序和类型。为了演示，我们考虑了五阶和七阶 WENO-Z 重建。前两种算法在不连续性上具有与 WENO-Z 方案相当的精度和分辨率，但计算成本较低。第三种算法确保当应用于具有非凸通量的双曲守恒定律（WENO 方案失败）时，所提出的方案收敛到正确的熵解。我们进行了多次一维和二维数值实验，以证明所提出算法的效率及其与 WENO-Z 方案相比的性能。所提出的算法非常高效，并且比 WENO-Z 方案减少 30%–75% 的计算时间，同时保留了 WENO-Z 方案的优点。