This paper aims to simulate viscoplastic flow in a shallow-water regime. We specifically use the Bingham model in which the material behaves as a solid if the stress is below a certain threshold, otherwise, it moves as a fluid. The main difficulty of this problem is the coupling of the shallow-water equations with the viscoplastic constitutive laws and the high computational effort needed in its solution. Although there have been many studies of this problem, most of these works use explicit methods with simplified empirical models. In our work, to accommodate non-uniform grids and complicated geometries, we use the discontinuous Galerkin method to solve shallow viscoplastic flows. This method is attractive due to its high parallelization, h- and p-adaptivity, and ability to capture shocks. Additionally, we treat the discontinuities in the interfaces between elements with numerical fluxes that ensure a stable solution of the nonlinear hyperbolic equations. To couple the Bingham model with the shallow-water equations, we regularize the problem with three alternatives. Finally, in order to show the effectiveness of our approach, we perform numerical examples for the usual benchmarks of the shallow-water equations.

翻译：本文旨在模拟浅水 regime 下的粘塑性流动。我们具体采用宾汉模型，该模型中材料在应力低于某一阈值时表现为固体，否则作流体运动。该问题的主要难点在于浅水方程与粘塑性本构律的耦合，以及求解所需的高计算成本。尽管已有大量相关研究，但多数工作采用简化经验模型的显式方法。本工作中，为适应非均匀网格和复杂几何，我们采用间断伽辽金法求解浅水粘塑性流动。该方法因具备高度并行性、h 和 p 自适应能力以及捕捉激波的特性而具有吸引力。此外，我们通过数值通量处理单元间界面的间断性，确保非线性双曲方程的稳定解。为将宾汉模型与浅水方程耦合，我们采用三种替代方案对问题进行正则化处理。最后，通过浅水方程常规基准算例的数值实验，展示了本文方法的有效性。