The Junction Riemann Problem in 1D shallow water channels including supercritical flow conditions

Abstract

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.