A steady-state-preserving numerical model is developed for the shallow water equations in channels in the current study. The proposed numerical model is based on the finite volume method and capable to preserve both the moving-water and still-water steady states. In order to preserve the general steady states, the source terms in the momentum equation are incorporated into a global flux term; and the new momentum equation with global flux term is then relaxed in order to avoid the non-linearity in the computation. One can thus develop an upwind/modified-HLL hybrid Riemann solver to estimate the fluxes in different flow regimes. The developed numerical model can preserve the general steady states, and one avoids: (1) non-trivial root-finding for point values of wetted cross-sectional area A at cell interfaces; (2) complex discretization of geometric source terms incorporating artificial conservation corrections. The developed numerical model has second order accuracy in both space and time. Numerical solutions are presented to demonstrate that the proposed numerical model is capable to exactly preserve both moving-water and still-water steady states.

在本研究中，为河道中的浅水方程建立了一个保持稳态的数值模型。所提出的数值模型基于有限体积法，并且能够同时保留运动水和静止水的稳态。为了保持一般的稳态，动量方程中的源项被合并到全局通量项中。然后放宽具有全局通量项的新动量方程，以避免计算中的非线性。因此，可以开发出迎风/改进型HLL混合Riemann解算器，以估算不同流态下的通量。所建立的数值模型可以保留一般的稳态，并且可以避免：（1）对润湿截面积A的点值进行非平凡的根查找在单元界面；（2）包含人工守恒校正的几何源项的复杂离散化。所开发的数值模型在空间和时间上均具有二阶精度。提出了数值解法，以证明所提出的数值模型能够精确地保持运动水和静止水的稳态。