This work provides a new formulation of the one-dimensional Shallow Water Equations system for open channels and rivers with arbitrarily shaped cross sections, suitable for numerical integration when discontinuous geometry is encountered. The additional variable considered can be the bottom elevation, a reference width, a shape coefficient, or a vector containing these or other geometric parameters. The appropriate numerical method, which is well suited to coupling with the mathematical one, is a path-conservative method, capable of reconstructing the behaviour of physical and geometrical variables at the cell boundaries, where the discrete solution of hyperbolic systems of equations is discontinuous. A nonlinear path suitable for the shallow water context is adopted. The resulting model is shown to be well-balanced and accurate to the second order and is further validated against analytical solutions related to channels with power-law cross-sections, specifically for dam break patterns over a variable-width channel and the run-up dynamics of long water waves over sloping bays.

中文翻译：

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任意截面渠道的一维增广浅水方程组


这项工作为具有任意形状横截面的明渠和河流提供了一维浅水方程组的新公式，适用于遇到不连续几何形状时的数值积分。所考虑的附加变量可以是底部标高、参考宽度、形状系数或包含这些或其他几何参数的向量。适当的数值方法非常适合与数学方法耦合，是一种路径保守方法，能够重建单元边界处的物理和几何变量的行为，其中双曲方程组的离散解是不连续的。采用适合浅水环境的非线性路径。结果表明，所得模型具有良好的平衡性和二阶精度，并根据与幂律横截面的渠道相关的分析解决方案进行了进一步验证，特别是针对可变宽度渠道上的溃坝模式和爬升倾斜海湾上长水波的动力学。