SIAM Journal on Scientific Computing, Volume 42, Issue 4, Page A2206-A2229, January 2020.
We develop a new moving-water equilibria preserving numerical scheme for the Saint-Venant system. The new scheme is designed in two major steps. First, the geometric source term is incorporated into the discharge flux, which results in a hyperbolic system with a global flux. Second, the discharge equation is relaxed so that the nonlinearity is moved into the stiff right-hand side of the added auxiliary equation. The main advantages of the new scheme are that (i) no special treatment of the geometric source term is required, and (ii) no nonlinear (cubic) equations should be solved to obtain the point values of the water depth out of the reconstructed equilibrium variables, as it must be done in the existing alternative methods. We also develop a hybrid numerical flux, which helps to handle various flow regimes in a stable manner. Several numerical experiments are performed to verify that the proposed scheme is capable of exactly preserving general moving-water steady states and accurately capturing their small perturbations.

中文翻译：

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Saint-Venant系统的动水平衡保持部分松弛方案

SIAM科学计算杂志，第42卷，第4期，第A2206-A2229页，2020年1月。
我们为Saint-Venant系统开发了一种新的保水平衡数值方案。新方案的设计分为两个主要步骤。首先，将几何源项合并到放电通量中，从而产生具有整体通量的双曲线系统。第二，放宽放电方程，使非线性移动到添加的辅助方程的右手边。新方案的主要优点是：（i）不需要对几何源项进行特殊处理，并且（ii）无需求解任何非线性（三次）方程，即可获得重建平衡以外的水深点值变量，因为它必须在现有的替代方法中完成。我们还开发了一种混合数值通量，有助于稳定地处理各种流动状态。