Shallow-water equations are widely used to model water flow in rivers, lakes, reservoirs, coastal areas, and other situations in which the water depth is much smaller than the horizontal length scale of motion. The classical shallow-water equations, the Saint-Venant system, were originally proposed about 150 years ago and still are used in a variety of applications. For many practical purposes, it is extremely important to have an accurate, efficient and robust numerical solver for the Saint-Venant system and related models. As their solutions are typically non-smooth and even discontinuous, finite-volume schemes are among the most popular tools. In this paper, we review such schemes and focus on one of the simplest (yet highly accurate and robust) methods: central-upwind schemes. These schemes belong to the family of Godunov-type Riemann-problem-solver-free central schemes, but incorporate some upwinding information about the local speeds of propagation, which helps to reduce an excessive amount of numerical diffusion typically present in classical (staggered) non-oscillatory central schemes. Besides the classical one- and two-dimensional Saint-Venant systems, we will consider the shallow-water equations with friction terms, models with moving bottom topography, the two-layer shallow-water system as well as general non-conservative hyperbolic systems.

中文翻译：

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浅水方程的有限体积格式

浅水方程广泛用于模拟河流、湖泊、水库、沿海地区和其他水深远小于水平运动长度尺度的水流。经典的浅水方程 Saint-Venant 系统最初是在大约 150 年前提出的，至今仍在各种应用中使用。对于许多实际目的，为 Saint-Venant 系统和相关模型提供准确、高效和稳健的数值求解器非常重要。由于他们的解决方案通常是不平滑的，甚至是不连续的，有限体积方案是最流行的工具之一。在本文中，我们回顾了这些方案，并专注于最简单（但高度准确和稳健）的方法之一：中央迎风方案。这些方案属于 Godunov 型 Riemann-problem-solver-free 中心方案家族，但包含一些关于局部传播速度的逆向信息，这有助于减少经典（交错）非求解中通常存在的过多数值扩散。 - 振荡中心方案。除了经典的一维和二维 Saint-Venant 系统外，我们还将考虑具有摩擦项的浅水方程、具有移动底部地形的模型、两层浅水系统以及一般的非保守双曲线系统。