This work presents an advancement in solving the Shallow Water Equations (SWE) in one-dimensional (1D) networks of channels using the Junction Riemann Problem (JRP). The necessity for robust solvers for junctions in networks is evident from the extensive literature and the variety of proposed methods. While multidimensional coupled approaches that model junctions as two-dimensional spaces have shown success, they lack the computational efficiency of pure 1D methods that treat junctions as singular points. In this context, existing JRP-based methods have primarily been limited to subcritical flow regimes. For the first time, this paper demonstrates that the Junction Riemann Problem can be effectively used as an internal boundary condition across all flow regimes representing a junction of channels. The proposed JRP solution is both simple and robust, accommodating various flow regimes and an arbitrary number of channels without requiring additional information. Furthermore, it is shown that the JRP can be solved efficiently at internal boundaries and integrated with a standard first-order Godunov scheme to yield accurate results. The validation of this method is confirmed through a series of test cases, highlighting its effectiveness in modeling free-surface flows.

中文翻译：

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一维浅水河道中的结黎曼问题，包括超临界流条件


这项工作提出了使用结黎曼问题 （JRP） 在一维 （1D） 通道网络中求解浅水方程 （SWE） 的进展。从大量文献和各种建议的方法中可以明显看出，网络中的交汇点需要稳健的求解器。虽然将交汇点建模为二维空间的多维耦合方法取得了成功，但它们缺乏将交汇点视为奇异点的纯 1D 方法的计算效率。在这种情况下，现有的基于 JRP 的方法主要局限于亚临界流态。本文首次证明，结黎曼问题可以有效地用作代表通道结点的所有流态的内部边界条件。所提出的 JRP 解决方案既简单又稳健，适用于各种流态和任意数量的通道，而无需其他信息。此外，结果表明 JRP 可以在内部边界处有效求解，并与标准的一阶 Godunov 方案集成以产生准确的结果。通过一系列测试用例证实了该方法的验证，突出了它在模拟自由表面流方面的有效性。