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.

中文翻译：

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非平底地形浅水方程的一种高阶均衡保正性移动网格DG方法

针对非平底地形浅水方程的数值求解，提出了一种重新分区型自适应移动网格不连续伽辽金法。良好平衡特性对于模拟静止湖稳态上的扰动波（例如湖上的波浪或深海中的海啸波）至关重要。为了确保良好的平衡性和正性保持特性，讨论了使用斜率限制、正性保持限制和网格之间的数据传输的策略。特别是，建议使用 DG 插值方案对从旧网格到新网格的流变量和底部地形进行插值，并且在每次应用对水深的正性保留限制后，根据水深的变化对底部地形的近似值进行高阶修正。此外，与基于常用熵函数的网格自适应相比，基于平衡变量和水深的网格自适应显示出更理想的结果。提供了一个和两个空间维度的数值示例，以证明该方法的良好平衡和正性保持特性及其捕获静止湖稳态小扰动的能力。