In this paper, we investigate the shallow water (SW) flow over the erodible layer using a fully coupled mathematical model in two-dimensional (2D) space. A well-balanced discontinuous Galerkin (DG) scheme is proposed for solving the SW equations with sediment transport and bed evolution. To achieve the well-balanced property of the numerical scheme easily, the nonlinear SW equations are first reformulated into a new form by introducing an auxiliary variable. Then the DG method is used to discretize the model, in which the FORCE flux is used. By choosing an appropriate value of the auxiliary variable, we can prove that the numerical method can accurately maintain the steady solution in still water, so it is indeed an equilibrium preserving scheme. The well-balanced property can be extended to any numerical flux which satisfies the consistency. Moreover, we investigate the impact on the numerical results if the appropriate value of auxiliary variable is slightly modified. The effectiveness of the numerical method is finally verified by numerical experiments.

在本文中，我们使用二维 (2D) 空间中的完全耦合数学模型研究了可蚀层上的浅水 (SW) 流动。提出了一种平衡良好的不连续 Galerkin (DG) 方案，用于求解具有泥沙输运和河床演化的 SW 方程。为了容易地实现数值格式的良好平衡特性，首先通过引入辅助变量将非线性SW方程重新表述为新形式。然后使用DG方法对模型进行离散化，其中使用了FORCE通量。通过选择适当的辅助变量值，我们可以证明数值方法可以准确地保持静止水中的稳态解，因此它确实是一个平衡保持方案。良好平衡的性质可以扩展到任何满足一致性的数值通量。此外，我们研究了如果辅助变量的适当值稍作修改，对数值结果的影响。最后通过数值实验验证了数值方法的有效性。