一种稳健且稳定的深度平均欧拉系统数值方案
A robust and stable numerical scheme for a depth-averaged Euler system
AI总结：
本文提出了一种高效的数值方案，用于求解非静力圣维南型模型，采用动能解释和投影修正方法，证明了方案的正性、平衡性和熵不等式性质，并在水深趋近于零时保持稳定性。
摘要：
我们提出了一种高效的数值方案，用于非静力圣维南型模型的求解。该模型是不可压缩欧拉系统的浅水近似，具有自由表面且与格林-纳赫迪模型略有不同。数值近似基于模型的动能解释和投影修正型方案。系统的双曲部分使用基于动能的有限体积求解器进行近似，修正步骤涉及求解包含非静力压力部分的椭圆问题。我们证明了该数值方案满足正性、平衡和完全离散熵不等式性质。该数值方案与各种时间依赖的解析解进行对比。值得注意的是，当水深趋近于零时，数值过程保持稳定。
英文摘要：
We propose an efficient numerical scheme for the resolution of a non-hydrostatic Saint-Venant type model. The model is a shallow water type approximation of the incompressbile Euler system with free surface and slightly differs from the Green-Naghdi model. The numerical approximation relies on a kinetic interpretation of the model and a projection-correction type scheme. The hyperbolic part of the system is approximated using a kinetic based finite volume solver and the correction step implies to solve an elliptic problem involving the non-hydrostatic part of the pressure. We prove the numerical scheme satisfies properties such as positivity, well-balancing and a fully discrete entropy inequality. The numerical scheme is confronted with various time-dependent analytical solutions. Notice that the numerical procedure remains stable when the water depth tends to zero.
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