Numerical corrections based on diffusive terms for SPH schemes

Extending the authors' work on the use of artificial diffusive terms to improve the Smoothed Particles Hydrodynamics (SPH) schemes, a new system of equations has been defined which contains diffusive terms in both the continuity and energy equations. The novel discrete equations, at the leading order, coincide with a standard SPH scheme with artificial viscosity. A proper state equation is used to associate the internal energy variation to the pressure field and to increase the speed of sound when strong deformations/compressions of the fluid occur. The increase of the sound speed is associated to the shortening of the time integration step and, therefore, allows a greater accuracy during both breaking and impact events. Moreover, the diffusive term inside the continuity equation allows reducing the high frequency numerical acoustic noise and smoothing the pressure field. The model has been tested using different free surface flows clearly showing to be robust, efficient and accurate. An in-depth analysis of the CPU time cost and comparisons with the standard SPH scheme is provided.