Riemann problems at geometric discontinuities are a classic and fascinating topic of hydraulics. In the present paper, the exact solution to the Riemann problem of the one-dimensional (1-d) Shallow water Equations at monotonic width discontinuities is completely determined for any initial condition. This solution is based on the assumption that the relationship between the states immediately to the left and to the right of the discontinuity is a stationary weak solution of the 1-d variable-width Shallow water Equations. Under this hypothesis, it is demonstrated that the solution to the Riemann problem always exists, although there are cases where the solution is triple. This proves that it is possible to define width-jump interior boundary conditions of Saint Venant models that lead to well-posed problems, and that additional physical information is required to pick the relevant Riemann wave configuration among the alternatives when the solution is multiple. The analysis of an existing Finite Volume numerical scheme from the literature, based on flow variables reconstruction with preservation of specific energy and discharge, shows that the algorithm captures the solution with supercritical flow through the width discontinuity when multiple solutions are possible. This suggests that it is possible to change the algorithm accordingly to the structure of the physically relevant solution. Interestingly, the 1-d variable-width Shallow water Equations are formally identical to the 1-d Porous Shallow water Equations, implying that the exact solutions provided in the present paper are relevant for two-dimensional (2-d) Porous Shallow water numerical models aiming at urban flooding simulations.

The procedure presented in this paper may be used not only to construct new challenging benchmarks for numerical schemes, but also as a guide for the construction of new algorithms, and for the interpretation of on-field and laboratory data related to transients at rapid channel width variations. The appearance of multiple solutions, which is connected to the hydraulic hysteresis phenomenon observed in the case of supercritical flows impinging a cross-section contraction, requires a criterion for the disambiguation of solutions. This will be the object of future research.

几何不连续处的黎曼问题是水力学的一个经典而引人入胜的话题。在本文中，对于任何初始条件完全确定了单调宽度不连续处的一维 (1-d) 浅水方程的黎曼问题的精确解。该解基于以下假设：紧邻不连续点左侧和右侧的状态之间的关系是一维变宽浅水方程的平稳弱解。在这个假设下，证明了黎曼问题的解总是存在的，尽管也有解是三重的。这证明可以定义导致适定问题的 Saint Venant 模型的宽度跳跃内部边界条件，并且当解决方案是多个时，需要额外的物理信息来在备选方案中选择相关的黎曼波配置。对文献中现有的有限体积数值方案的分析，基于流量变量重构，保留比能量和流量，表明当多个解是可能的时，该算法通过宽度不连续性捕获具有超临界流的解。这表明可以根据物理相关解决方案的结构相应地更改算法。有趣的是，一维变宽浅水方程在形式上与一维多孔浅水方程相同，

本文中介绍的程序不仅可用于为数值方案构建新的具有挑战性的基准，还可用作构建新算法的指南，以及解释与快速通道宽度下的瞬态相关的现场和实验室数据变化。多个解的出现与在超临界流冲击横截面收缩的情况下观察到的水力滞后现象有关，需要一个消除解歧义的标准。这将是未来研究的对象。