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. Mesh adaptivity is realized using a moving mesh partial differential equation and a metric tensor based on the equilibrium variable and water depth. A motivation for the latter is to adapt the mesh according to both the perturbations of the lake-at-rest steady state and the water depth distribution. 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. They also show that the mesh adaptation based on the equilibrium variable and water depth give more desirable results than that based on the commonly used entropy function.

针对具有非平坦底部地形的浅水方程组的数值解，提出了一种重分区型自适应运动网格间断Galerkin方法。良好的平衡特性对于模拟静止湖上的摄动波（例如湖上的波或深海中的海啸波）至关重要。为了确保良好的平衡性和保正性，讨论了使用斜率限制，保正性限制以及网格之间的数据传输的策略。特别是，建议将DG插值方案用于从旧网格到新网格的流量变量和底部形貌的插值，并且在每次应用对水深的正向保留限制后，根据水深的变化，对底部地形的近似值进行高阶校正。使用移动网格偏微分方程和基于平衡变量和水深的度量张量实现网格自适应性。后者的动机是根据静止湖静止状态的扰动和水深分布来调整网格。给出了在一维和二维空间中的数值示例，以证明该方法的良好平衡性和保正性，以及捕获静态湖微扰动的能力。他们还表明，基于平衡变量和水深的网格自适应比基于常用熵函数的网格自适应得到更理想的结果。