One important issue in the approach with shallow water equations, which is not restricted to discontinuous Galerkin formulation, is the limitation to step geometries (discontinuous bathymetry), due to the hydrostatic assumption employed for the derivation of shallow water equations from the Navier-Stokes equations. In addition, the explicit Runge-Kutta time-stepping schemes do not come without any drawbacks even though the majority of discontinuous Galerkin applications have employed explicit ones due to simplicity. In this study, the recently developed implicit discontinuous Galerkin scheme is combined with the surface gradient method for steps (SGMS). The developed scheme is verified with flows over discontinuous bathymetry, i.e., vertical steps and weirs. For one-dimensional problems, the flows over a step and over a rectangular weir are simulated. As for two-dimensional problems, the flow over a weir and the dam-break flow over a step followed by the L-shaped and 45°-bend channels are simulated. The numerical solutions are compared with the experimental data. In all cases, good agreements were observed and the effectiveness of the developed scheme was verified.

浅水方程方法中的一个重要问题（不限于不连续的Galerkin公式）是阶跃几何形状（不连续测深法）的限制，这是由于从Navier-Stokes方程推导浅水方程所采用的静水假设。此外，即使大多数不连续的Galerkin应用程序由于简单性而采用了显式的应用程序，显式的Runge-Kutta时间步移方案也没有任何缺点。在这项研究中，最近开发的隐式不连续Galerkin方案与阶梯表面梯度法（SGMS）相结合。所开发的方案通过不连续的测深法（即垂直台阶和堰）上的流量进行了验证。对于一维问题，模拟了台阶上和矩形堰上的流动。对于二维问题，模拟了堰上的水流和台阶上的堰坝水流，以及随后的L形和45°弯曲通道。将数值解与实验数据进行了比较。在所有情况下，都可以观察到良好的协议，并且可以验证所开发方案的有效性。