This short note introduces a novel diagnostic tool for evaluating the convection boundedness properties of numerical schemes across discontinuities. The proposed method is based on the convection boundedness criterion and the normalised variable diagram. By utilising this tool, we can determine the CFL conditions for numerical schemes to satisfy the convection boundedness criterion, identify the locations of over- and under-shoots, optimize the free parameters in the schemes, and develop strategies to prevent numerical oscillations across the discontinuity. We apply the diagnostic tool to assess representative discontinuity-capturing schemes, including THINC, fifth-order WENO, and fifth-order TENO, and validate the conclusions drawn through numerical tests. We further demonstrate the application of the proposed method by formulating a new THINC scheme with less stringent CFL conditions.

翻译：本短文介绍了一种新颖的诊断工具，用于评估数值格式在间断处的对流有界性。所提出的方法基于对流有界性准则和归一化变量图。利用该工具，我们可以确定数值格式满足对流有界性准则的CFL条件，识别过冲与欠冲的位置，优化格式中的自由参数，并制定防止跨越间断处数值振荡的策略。我们应用该诊断工具评估了具有代表性的间断捕捉格式，包括THINC、五阶WENO和五阶TENO，并通过数值试验验证了所得结论。我们进一步通过构建一个具有更宽松CFL条件的新型THINC格式，展示了所提方法的应用。