论文标题
浅水方程的基于高阶速度的不连续的盖尔金方案：局部保护，熵稳定性，均衡性质和阳性保存
A high-order velocity-based discontinuous Galerkin scheme for the shallow water equations: local conservation, entropy stability, well-balanced property, and positivity preservation
论文作者
论文摘要
我们提出了一类新型的本地保守，熵稳定且平衡良好的不连续的Galerkin（DG）方法，用于非线性浅水方程 非平板底部地形。我们工作的主要新颖性是将速度场用作DG方案中未知的独立解决方案，该解决方案与1986年Tadmor [22]提出的熵稳定方案的熵可变方法密切相关，在1986年，回忆起速度是shallow water shallow Water equareation entropy变量的一部分。由于将速度用作独立解决方案未知，因此不需要特定的数值正交规则来实现我们方案的熵稳定性，以二维的一般非结构化网格。 然后将所提出的DG半差异与经典的明确稳定性保留runge-kutta（ssp-rk）时间积分器[13]相结合，以产生本地保守的，良好的平衡且积极性，并保留完全离散的方案。在这里，借助简单的缩放限制器实施了积极保护属性。在完全离散的方案中，我们将放电作为一个辅助变量。这样，可以在保守的变量（水高和排放）上应用标准坡度限制程序，而无需违反当地的保护特性。在这里，我们在runge-kutta时间的每个内阶段中使用Fu-Shu陷入困境的单元格指示器[10]在保守变量上应用特征性的TVB限制器[5]，以抑制数值振荡。
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.
