  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.
