We present a novel class of locally conservative, entropy stable and well-balanced discontinuous Galerkin (DG) methods for the nonlinear shallow water equation with a non-flat bottom topography. The major novelty of our work is the use of velocity field as an independent solution unknown in the DG scheme, which is closely related to the entropy variable approach to entropy stable schemes for system of conservation laws proposed by Tadmor [22] back in 1986, where recall that velocity is part of the entropy variable for the shallow water equations. Due to the use of velocity as an independent solution unknown, no specific numerical quadrature rules are needed to achieve entropy stability of our scheme on general unstructured meshes in two dimensions. The proposed DG semi-discretization is then carefully combined with the classical explicit strong stability preserving Runge-Kutta (SSP-RK) time integrators [13] to yield a locally conservative, well-balanced, and positivity preserving fully discrete scheme. Here the positivity preservation property is enforced with the help of a simple scaling limiter. In the fully discrete scheme, we re-introduce discharge as an auxiliary unknown variable. In doing so, standard slope limiting procedures can be applied on the conservative variables (water height and discharge) without violating the local conservation property. Here we apply a characteristic-wise TVB limiter [5] on the conservative variables using the Fu-Shu troubled cell indicator [10] in each inner stage of the Runge-Kutta time stepping to suppress numerical oscillations.

翻译：我们为非线性浅水方程式提出了一个新颖的本地保守、稳定且平衡的不连续Galerkin(DG)方法类别,该类方法具有非线性浅水方程式的不负平面地貌。我们工作的主要新颖之处是使用速度字段作为DG办法中未知的一种独立解决方案,这与Tadmor [22] 1986年提出的对保护法律体系体系的加密稳定办法的增缩变量方法密切相关,该方法回顾速度是浅水方程式的增缩变量的一部分。由于使用速度作为不为人知的独立解决方案,因此不需要具体的数字二次等式规则来实现我们在一般非结构型 meshes上的计划的增缩稳定性。拟议的DG半分化方法与古典的强势稳定保存 Runge-Kutta(SSP-RKRK) 时间聚合者[13] 密切结合,以便产生一种本地保守、平衡和真实的、保持完全离析的变异性方案。在这里,通过简单的缩缩缩缩缩缩的系统,在不使用一个不易动的递增缩的递的递进的递进的递进的递进的递进式系统中,可以使用一个不动的递进的递进的递进式的递进的递进的递进的递进式的递进的递进的递进式的递进式的递进式系统。