A novel system of equations has been defined which contains diffusive terms in both the continuity and energy equations and, at the leading order, coincides with a standard weakly-compressible 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 larger accuracy during both breaking and impact events. Moreover, the diffusive terms allows reducing the high frequency numerical acoustic noise and smoothing the pressure field. Finally, an enhanced formulation for the second-order derivatives has been defined which is consistent and convergent all over the fluid domain and, therefore, permits to correctly model the diffusive terms up to the free surface. The model has been tested using different free surface flows clearly showing to be robust, efficient and accurate. An analysis of the CPU time cost and comparisons with the standard SPH scheme is provided.