论文标题
具有非平台底部地形的浅水方程的高阶均衡阳性移动网格DG方法
A high-order well-balanced positivity-preserving moving mesh DG method for the shallow water equations with non-flat bottom topography
论文作者
论文摘要
提出了一种重新分区型自适应移动网格不连续的Galerkin方法，用于具有非底线底部地形的浅水方程的数值解决方案。井的平衡特性对于模拟在湖泊稳定状态上的扰动波（例如湖泊上的波浪或深海海啸波）至关重要。为了确保井的平衡和阳性性能，在使用坡度限制，阳性保存限制以及网格之间的数据传输时会讨论策略。特别是，建议将DG插值方案用于从旧网格到新的网格的流量变量和底部地形的插值，以及每次施用在水深度上的积极保护限制的每次应用，根据水深的底层形象，对水深进行了高阶校正。此外，与基于常用的熵函数相比，基于平衡变量和水深度的网格适应性可提供更多的结果。提出了一个和两个空间维度中的数值示例，以证明该方法的良好平衡和阳性性能及其捕获对湖泊重度稳态的小扰动的能力。
A rezoning-type adaptive moving mesh discontinuous Galerkin method is proposed for the numerical solution of the shallow water equations with non-flat bottom topography. The well-balance property is crucial to the simulation of perturbation waves over the lake-at-rest steady state such as waves on a lake or tsunami waves in the deep ocean. To ensure the well-balance and positivity-preserving properties, strategies are discussed in the use of slope limiting, positivity-preservation limiting, and data transferring between meshes. Particularly, it is suggested that a DG-interpolation scheme be used for the interpolation of both the flow variables and bottom topography from the old mesh to the new one and after each application of the positivity-preservation limiting on the water depth, a high-order correction be made to the approximation of the bottom topography according to the modifications in the water depth. Moreover, mesh adaptation based on the equilibrium variable and water depth is shown to give more desirable results than that based on the commonly used entropy function. Numerical examples in one and two spatial dimensions are presented to demonstrate the well-balance and positivity-preserving properties of the method and its ability to capture small perturbations of the lake-at-rest steady state.
