A Robust Two-Dimensional Shallow Flow Model with Adaptive Quadtree Mesh

A two-dimensional shallow flow model is developed by integrating a positivity-preserving, well-balanced central-upwind scheme with a block-structured quadtree AMR grid implemented in the Afivo open-source framework. A non-negative water depth reconstruction ensures second-order spatial accuracy and the robust treatment of wetting and drying, while coarse-grid fluxes at refinement boundaries are obtained by summing the corresponding fine-grid fluxes, thereby guaranteeing strict mass conservation between refinement levels. Mesh refinement is driven by gradients in water surface elevation, which focus resolution on regions of rapid flow variation, thereby improving both accuracy and computational efficiency. Model validation through benchmark problems and the Malpasset dam-break event show close agreement with analytical solutions, laboratory measurements, and previous numerical simulations, while achieving substantial reductions in computational cost.