SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page B1136-B1172, January 2020.
A new well-balanced asymptotic preserving scheme for two-dimensional low Froude number shallow water flows over irregular bottom is developed in this study. The bed-slope terms in the low Froude number regime are nontrivial since their stiffness has the same order as the gravity waves, they change the flow behavior in the low Froude number regime, and thus require special treatment when developing a numerical scheme to ensure such terms will not introduce high order numerical diffusion and spurious waves. To this end, the governing system is reformulated to obtain the well-balanced property. Since the system is stiff in the low Froude number flow regime, conventional explicit numerical schemes are extremely inefficient and often impractical. In order to overcome such difficulties, an asymptotic preserving scheme is developed by splitting the flux into a slow nonlinear part and fast linear part first, then approximating the slow dynamics explicitly using an explicit shock capturing scheme while estimating the fast dynamics implicitly. Using in space the linear piecewise reconstruction with minmod limiter for the shock explicit capturing scheme and central difference method for implicit derivatives, and in time the second order implicit-explicit Runge--Kutta methods, the second order accuracy of the proposed scheme is achieved. It is proved that the proposed numerical schemes are asymptotically consistent and stable uniformly with respect to small Froude number. Several numerical experiments are conducted to demonstrate the performance of the proposed asymptotic preserving numerical methods.

中文翻译：

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不规则底形上二维浅水方程组的一个平衡的渐近保存格式

SIAM科学计算杂志，第42卷，第5期，第B1136-B1172页，2020年1月。
本文研究了二维低弗鲁德数浅水流向不规则底部的一种新的平衡良好的渐近保存方案。低弗洛德数形式的床坡项是不平凡的，因为它们的刚度与重力波具有相同的阶数，它们改变了低弗洛德数形式的流动特性，因此在制定数值方案以确保这种情况时需要进行特殊处理。这些术语不会引入高阶数值扩散和杂散波。为此，重新制定了治理制度以获得平衡的财产。由于系统在低弗洛德数流态下是刚性的，因此常规的显式数值方案效率极低，而且通常不切实际。为了克服这些困难，通过先将磁通分成慢速非线性部分和快速线性部分，然后使用显式激振捕获方案显式近似慢速动力学，同时隐式估计快速动力学，从而开发了一种渐近的保存方案。在空间中使用具有minmod限制器的线性分段重构进行震荡显式捕获，并使用隐式导数的中心差分方法，并及时采用二阶隐式-显式Runge-Kutta方法，从而实现了该方案的二阶精度。实践证明，所提出的数值格式对于较小的弗洛德数是渐近一致的，并且是一致稳定的。进行了几个数值实验，以证明所提出的渐近保持数值方法的性能。