不规则沟渠浅水流动的高阶平衡有限差分AWENO格式

High-Order Well-Balanced Finite Difference AWENO Schemes for Shallow Water Flows along Channels with Irregular Geometry

摘要: 在本文中，我们提出了具有不规则几何形状和非平坦底部地形的开放沟渠中浅水方程的高阶有限差分加权本质无振荡(Alternative Weighted Essentially Non-Oscillatory) AWENO格式。所提出的格式保持了静水稳态解的良好平衡性质，即在通量梯度和源项之间存在精确平衡时，在离散状态保持稳态。与具有恒定横截面的传统浅水方程相比，由于沟渠不规则几何形状引起的影响，构建良好平衡格式并不是一项简单的工作。为了保持良好平衡性质，我们首先重新构造源项，然后使用具有Lax-Friedrichs通量的静水重构方法，最后借助一种新颖的源项逼近方法离散源项。基准数值示例被应用来验证所得格式的良好性能：平衡性能，高阶精度，以及对不连续解的高分辨率。

Abstract: In this paper, we present high-order finite alternative weighted essentially non-oscillatory (AWENO) schemes for shallow water flows along open channels with irregular geometry and over a non-flat bottom topography. The proposed schemes maintain the well-balanced property for the still water steady-state solutions, namely preserving steady state at the discrete level, when there is an exact balance between the flux gradient and the source term. Compared with the traditional shallow water equations with constant cross-section, the construction of the well-balanced schemes is not trivial work due to the effect induced by the irregular geometry of the channels. To preserve the well-balanced property, we first reformulate the source term, then use the hydrostatic reconstruction (HR) method with Lax-Friedrichs (LF) flux, and finally discrete the source term with the help of a novel source term approximation. Benchmark numerical examples are applied to validate the good performances of the resulting schemes: well-balanced property, high order accuracy, and high resolution for the discontinuous solutions.