The numerical entropy production (NEP) for shallow water equations (SWE) is discussed and implemented as a smoothness indicator. We consider SWE in three different dimensions, namely, one-dimensional, one-and-a-half-dimensional, and two-dimensional SWE. An existing numerical entropy scheme is reviewed and an alternative scheme is provided. We prove the properties of these two numerical entropy schemes relating to the entropy steady state and consistency with the entropy equality on smooth regions. Simulation results show that both schemes produce NEP with the same behaviour for detecting discontinuities of solutions and perform similarly as smoothness indicators. An implementation of the NEP for an adaptive numerical method is also demonstrated.

中文翻译：

---

数值熵产生作为浅水方程的光滑度指标

浅水方程 (SWE) 的数值熵产生 (NEP) 被讨论并实现为平滑度指标。我们从三个不同的维度考虑 SWE，即一维、一维半和二维 SWE。回顾了现有的数值熵方案并提供了替代方案。我们证明了这两种数值熵方案与熵稳态相关的性质以及与平滑区域上的熵等式的一致性。仿真结果表明，两种方案都产生了具有相同行为的 NEP，用于检测解的不连续性，并且作为平滑度指标的表现相似。还演示了用于自适应数值方法的 NEP 的实现。