  We consider the depth-integrated non-hydrostatic system derived by Yamazaki
et al. An efficient formally second-order well-balanced hybrid finite volume
finite difference numerical scheme is proposed. The scheme consists of a
two-step algorithm based on a projection-correction type scheme initially
introduced by Chorin-Temam [15]. First, the hyperbolic part of the system is
discretized using a Polynomial Viscosity Matrix path-conservative finite volume
method. Second, the dispersive terms are solved by means of compact finite
differences. A new methodology is also presented to handle wave breaking over
complex bathymetries. This adapts well to GPU-architectures and guidelines
about its GPU implementation are introduced. The method has been applied to
idealized and challenging experimental test cases, which shows the efficiency
and accuracy of the method.
