We present an adaptive well-balanced positivity preserving central-upwind scheme on quadtree grids for shallow water equations. The use of quadtree grids results in a robust, efficient and highly accurate numerical method. The quadtree model is developed based on the well-balanced positivity preserving central-upwind scheme proposed in [A. Kurganov and G. Petrova, Commun. Math. Sci., 5 (2007), pp. 133--160]. The designed scheme is well-balanced in the sense that it is capable of exactly preserving "lake-at-rest" steady states. In order to achieve this as well as to preserve positivity of water depth, a continuous piecewise bilinear interpolation of the bottom topography function is utilized. This makes the proposed scheme capable of modelling flows over discontinuous bottom topography. Local gradients are examined to determine new seeding points in grid refinement for the next timestep. Numerical examples demonstrate the promising performance of the central-upwind quadtree scheme.

中文翻译：

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用于浅水方程的四叉树网格上的自适​​应良好平衡正保持中心迎风方案

我们在四叉树网格上为浅水方程提出了一种自适应良好平衡的正性保持中央迎风方案。四叉树网格的使用产生了一种稳健、高效和高度准确的数值方法。四叉树模型是基于 [A. Kurganov 和 G. Petrova，Commun。数学。Sci., 5 (2007), pp. 133--160]。设计的方案在某种意义上是非常平衡的，因为它能够准确地保持“静止湖”的稳定状态。为了实现这一点并保持水深的积极性，使用了底部地形函数的连续分段双线性插值。这使得所提出的方案能够对不连续底部地形上的流动进行建模。检查局部梯度以确定下一个时间步长网格细化中的新种子点。数值例子证明了中央上风四叉树方案的有希望的性能。