多层浅水波方程的间断伽辽金法及其在泥石流等自然灾害中的应用

The multi-layer shallow water equations have been successfully applied to many hydromechanics problems where vertical structure plays an important role in the flow. The primary barrier to the use of these equations more broadly has been the complexity and computational cost of the required solvers. The multi-layer shallow water equations have still water stationary solution and the water depth is always non-negative. However, most of traditional numerical methods cannot maintain the still water stationary solution and the non-negativity of the water depth. Therefore, we will develop a high order positivity-preserving and well-balanced discontinuous Galerkin method for solving the multi-layer shallow water equations in this project. Moreover, we consider the two-phase shallow granular flow equations which are a transformation of the two-layer shallow water equations in the application of natural disasters such as landslides and debris flows. Since they are not always hyperbolic, we will design a high order positivity-preserving and well-balanced central discontinuous Galerkin method for them, which avoids to solve the Riemann problems. In order to predict the impact speed and overflow domain of the landslides and debris flows timely, we will present a high order positivity-preserving and well-balanced reconstructed central discontinuous Galerkin method which can reduce the computational cost by nearly half.

多层浅水波方程已经成功的应用到许多流体力学问题中，在这些问题中，流体竖直方向的特征扮演了比较重要的角色。阻碍这些方程被广泛应用的是方程求解器的复杂性和高昂的计算费用。双层浅水波方程具有静水稳定解，水深总是非负的，但是大多数数值算法并不能保持静水稳定解和水深的非负性，因此本项目将发展一个高阶保正且保持平衡的间断伽辽金法。另外考虑两相颗粒薄层流方程，其是双层浅水波方程在滑坡和泥石流等自然灾害中应用后的一个变化模型，由于该方程不总是双曲型的，所以本项目将设计一个高阶保正且保持平衡的中心间断伽辽金法，其避免了黎曼问题的求解。为了及时的预测出滑坡和泥石流等自然灾害的冲击速度和泛滥区域，本项目将设计一个高阶保正且保持平衡的重构的中心间断伽辽金法，其将减少一半左右的计算时间。

浅水波方程已经成功的应用到许多水力工程问题中。在本项目中，研究了双层浅水波方程和易腐蚀底部上浅水波方程的高阶数值方法。具体地说包括：首先针对双层浅水波方程和易腐蚀底部的浅水波方程提出了一个保正且保持平衡的中心间断伽辽金法。然后在非结构网格上建立了中心间断伽辽金法，该方法用来解决来一些守恒律问题，设计了一个满足极值原理的中心间断伽辽金法来求解守恒律方程和保正的中心间断伽辽金法来求解可压欧拉方程。第三，提出了一个修正的中心间断伽辽金法来求解浅水波方程，该方法能大大减少计算量。通过该项目，发表了9篇SCI论文，培养了三个研究生。

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