High order methods are becoming increasingly popular in shallow water flow modeling motivated by their high computational efficiency (i.e. the ratio between accuracy and computational cost). In particular, Discontinuous Galerkin (DG) schemes are very well suited for the resolution of the shallow water equations (SWE) and related models, being a competitive alternative to the traditional finite volume schemes. In this work, a novel framework for the construction of DG schemes using augmented Riemann solvers is proposed. Such solvers incorporate the source term at cell interfaces in the definition of the Riemann problem, allowing definition of two different inner states in the so-called star region. The benefits of this family of solvers lie in the exact preservation of the Rankine–Hugoniot condition at cell interfaces at the discrete level, ensuring the preservation of equilibrium solutions (i.e. the well-balanced property) without requiring extra corrections of the numerical fluxes. The semi-discrete DG operator becomes nil automatically under equilibrium conditions, provided the use of suitable quadrature rules. The proposed scheme is applied to the Burgers' equation with geometric source term and to the SWE. The numerical results evidence that the proposed scheme achieves the prescribed convergence rates and preserves the equilibrium states of relevance with machine precision.

高阶方法由于其高计算效率（即精度与计算成本之比）而在浅水流建模中变得越来越流行。特别是，非连续伽勒金（DG）方案非常适合浅水方程（SWE）和相关模型的解析，是传统有限体积方案的有竞争力的替代方案。在这项工作中，提出了使用增强的黎曼求解器构造DG方案的新颖框架。这样的求解器在黎曼问题的定义中在单元界面处结合了源项，从而允许在所谓的星形区域中定义两个不同的内部状态。这个求解器系列的好处在于，可以精确地在离散级别的单元界面上保留Rankine–Hugoniot条件，确保平衡溶液的保留（即平衡性能），而无需对通量进行额外的校正。如果使用适当的正交规则，则半离散DG算子在平衡条件下将自动变为零。拟议的方案适用于具有几何源项的Burgers方程和SWE。数值结果表明，该方案可以达到规定的收敛速度，并保持与机器精度相关的平衡状态。拟议的方案适用于具有几何源项的Burgers方程和SWE。数值结果表明，该方案可以达到规定的收敛速度，并保持与机器精度相关的平衡状态。拟议的方案适用于具有几何源项的Burgers方程和SWE。数值结果表明，该方案可以达到规定的收敛速度，并保持与机器精度相关的平衡状态。