In this paper, we develop a robust and efficient numerical method for shallow water equations with moving bottom topography. The model consists of the Saint-Venant system governing the water flow coupled with the Exner equation for the sediment transport. One of the main difficulties in designing good numerical methods for such models is related to the fact that the speed of water surface gravity waves is typically much faster than the speed at which the changes in the bottom topography occur. This imposes a severe stability restriction on the size of time steps, which, in turn, leads to excessive numerical diffusion that affects the computed bottom structure. In order to overcome this difficulty, we develop an operator splitting approach for the underlying coupled system, which allows one to treat slow and fast waves in a different manner and using different time steps. Our method is based on the application of a finite-volume central-upwind scheme introduced in [A. Kurganov and G. Petrova, Commun. Math. Sci., 5:133–160, 2007], and incorporates a staggered grid strategy needed for a proper approximation of the bottom topography function. A number of oneand two-dimensional numerical examples are presented to demonstrate the performance of the proposed method.

中文翻译：

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具有移动底部地形的浅水方程基于算子分裂的中央迎风方案

在本文中，我们为具有移动底部地形的浅水方程开发了一种稳健有效的数值方法。该模型由控制水流的 Saint-Venant 系统和用于沉积物迁移的 Exner 方程组成。为此类模型设计良好的数值方法的主要困难之一与以下事实有关：水面重力波的速度通常比底部地形发生变化的速度快得多。这对时间步长的大小施加了严重的稳定性限制，进而导致影响计算的底部结构的过度数值扩散。为了克服这个困难，我们为底层耦合系统开发了一种算子分裂方法，这允许人们以不同的方式和使用不同的时间步长处理慢波和快波。我们的方法基于在 [A. Kurganov 和 G. Petrova，Commun。数学。Sci., 5:133–160, 2007]，并结合了适当逼近底部地形函数所需的交错网格策略。提供了许多一维和二维数值例子来证明所提出方法的性能。