A Diffusive and Well-Balanced SPH Scheme for the Shallow Water Equations

The use of diffusive corrections in the continuity equation has become an effective strategy to stabilize Smoothed Particle Hydrodynamics formulations of the Euler equations, enabling robust and reliable numerical schemes which are nowadays widely adopted. In this work, we show that the same approach can be extended to the Shallow Water Equations, providing stable and satisfactory solutions without the need for additional stabilization strategies. The governing equations are expressed in an Eulerian reference frame through a formulation that exactly preserves the well-balanced property. The proposed method is assessed through validation tests against reference solutions obtained with a finite volume scheme. Four benchmark tests are considered, each targeting a distinct feature of the model. The first set of experiments analyzes convergence behavior under mesh refinement, investigates the sensitivity of the solution to the diffusive tuning parameter, and demonstrates the capability of the method to accurately reproduce problems involving non-uniform bathymetry. Finally, the model is applied to a large-scale scenario involving tsunami propagation in the Ionian Sea, highlighting its potential for practical applications involving geophysical flows. The results suggest a viable pathway towards the development of new SPH solvers in this context.