The nonlinear shallow water equations (SWEs) are widely used to model the unsteady water flows in rivers and coastal areas, with extensive applications in ocean and hydraulic engineering. In this work, we propose entropy stable, well-balanced and positivity-preserving discontinuous Galerkin (DG) methods, under arbitrary choices of quadrature rules, for the SWEs with a non-flat bottom topography. In Chan (J Comput Phys 362:346–374, 2018), a SBP-like differentiation operator was introduced to construct the discretely entropy conservative DG methods. We extend this idea to the SWEs and establish an entropy stable scheme by adding additional dissipative terms. Careful approximation of the source term is included to ensure the well-balanced property of the resulting method. A simple positivity-preserving limiter, compatible with the entropy stable property, is included to guarantee the non-negative water heights during the computation. One- and two-dimensional numerical experiments are presented to demonstrate the performance of the proposed methods.

非线性浅水方程（SWE）被广泛用于模拟河流和沿海地区的非恒定水流，在海洋和水利工程中得到了广泛的应用。在这项工作中，我们针对具有非平坦底部地形的SWE，提出了在正交规则的任意选择下，熵稳定，平衡良好且保持阳性的不连续Galerkin（DG）方法。在Chan（J Comput Phys 362：346–374，2018）中，引入了一种类似于SBP的微分算子来构造离散熵保守DG方法。我们将此思想扩展到SWE，并通过添加其他耗散项来建立熵稳定方案。仔细考虑了源项，以确保所得方法的均衡特性。一个简单的保持正性的限制器，包含与熵稳定属性兼容的值，以确保计算过程中的非负水位。一维和二维数值实验被提出来证明所提出的方法的性能。