This paper proposes high-order accurate well-balanced (WB) energy stable (ES) adaptive moving mesh finite difference schemes for the shallow water equations (SWEs) with non-flat bottom topography. To enable the construction of the ES schemes on moving meshes, a reformulation of the SWEs is introduced, with the bottom topography as an additional conservative variable that evolves in time. The corresponding energy inequality is derived based on a modified energy function, then the reformulated SWEs and energy inequality are transformed into curvilinear coordinates. A two-point energy conservative (EC) flux is constructed, and high-order EC schemes based on such a flux are proved to be WB that they preserve the lake at rest. Then high-order ES schemes are derived by adding suitable dissipation terms to the EC schemes, which are newly designed to maintain the WB and ES properties simultaneously. The adaptive moving mesh strategy is performed by iteratively solving the Euler-Lagrangian equations of a mesh adaptation functional. The fully-discrete schemes are obtained by using the explicit strong-stability preserving third-order Runge-Kutta method. Several numerical tests are conducted to validate the accuracy, WB and ES properties, shock-capturing ability, and high efficiency of the schemes.

翻译：本文提出了高精度保平衡（WB）、能量稳定（ES）的自适应移动网格有限差分格式，用于具有非平坦底地形的浅水方程（SWEs）。为在移动网格上构建ES格式，引入了一种SWEs的重新表述，将底地形作为随时间演化的附加守恒变量。基于修正的能量函数推导了相应的能量不等式，随后将重新表述的SWEs及能量不等式转换至曲线坐标系。构造了两点能量守恒（EC）通量，并证明了基于该通量的高阶EC格式具有保平衡性，能够保持静止湖泊状态。通过向EC格式添加适当耗散项，推导出高阶ES格式，这些耗散项经过全新设计以同时维持WB和ES特性。自适应移动网格策略通过迭代求解网格适应泛函的欧拉-拉格朗日方程实施。采用显式强稳定性保持三阶龙格-库塔方法获得全离散格式。通过多个数值算例验证了格式的精度、WB与ES特性、激波捕捉能力及高计算效率。