An adaptive well-balanced positivity preserving scheme on quadtree grids for shallow water equations
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We present an adaptive well-balanced positivity preserving 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 bi-linear interpolation of the bottom topography function is utilized. 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.
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