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The shallow-water equations model hydrostatic flow below a free surface for cases in which the ratio between the vertical and horizontal length scales is small and are used to describe waves in lakes, rivers, oceans, and the atmosphere. The equations admit discontinuous solutions, and numerical solutions are typically computed using high-resolution schemes. For many practical problems, there is a need to increase the grid resolution locally to capture complicated structures or steep gradients in the solution. An efficient method to this end is adaptive mesh refinement (AMR), which recursively refines the grid in parts of the domain and adaptively updates the refinement as the simulation progresses. Several authors have demonstrated that the explicit stencil computations of high-resolution schemes map particularly well to many-core architectures seen in hardware accelerators such as graphics processing units (GPUs). Herein, we present the first full GPU-implementation of a block-based AMR method for the second-order Kurganov---Petrova central scheme. We discuss implementation details, potential pitfalls, and key insights, and present a series of performance and accuracy tests. Although it is only presented for a particular case herein, we believe our approach to GPU-implementation of AMR is transferable to other hyperbolic conservation laws, numerical schemes, and architectures similar to the GPU.

We consider three high-resolution schemes for computing shallow-water waves as described by the Saint-Venant system and discuss how to develop highly efficient implementations using graphical processing units (GPUs). The schemes are well-balanced for lake-at-rest problems, handle dry states, and support linear friction models. The first two schemes handle dry states by switching variables in the reconstruction step, so that bilinear reconstructions are computed using physical variables for small water depths and conserved variables elsewhere. In the third scheme, reconstructed slopes are modified in cells containing dry zones to ensure non-negative values at integration points. We discuss how single and double-precision arithmetics affect accuracy and efficiency, scalability and resource utilization for our implementations, and demonstrate that all three schemes map very well to current GPU hardware. We have also implemented direct and close-to-photo-realistic visualization of simulation results on the GPU, giving visual simulations with interactive speeds for reasonably-sized grids.