δ-SW-SPH: A diffusive and well-balanced SPH scheme for the shallow water equations

We present the delta-SW-SPH, a numerical scheme that extends the delta-SPH model (Marrone et al., 2011) to the Shallow Water (SW) equations. The new solver introduces a diffusive correction to the continuity equation in terms of free surface elevation. The use of diffusive terms has proven effective in stabilizing SPH discretizations of the Euler equations, leading to robust and widely adopted numerical schemes. Here, we show that the same strategy ensures stable and accurate solutions in the SW equations framework, without requiring additional stabilization techniques. The governing equations are expressed in an Eulerian reference frame through a formulation that exactly preserves the well-balanced property. The method is validated against reference solutions obtained with a finite volume scheme, demonstrating stability, convergence under mesh refinement, and model's sensitivity to the diffusive parameter. The energy dissipation and noise attenuation properties are also analyzed in comparison with the original -SPH model. The capability of the method to accurately reproduce flows over non-uniform bathymetry is demonstrated, and the model is finally applied to a large-scale scenario involving tsunami propagation in the Ionian Sea, highlighting its potential for geophysical flow simulations