proposed to remove some inconsistencies close to the free-surface. In this case the adopted diffusive term is equivalent to a high-order differential operator on the density field. The scheme is called δ-SPH and preserves all the conservation properties of the SPH without diffusion (e.g., linear and angular momenta, total energy, see ) along with a smooth and regular representation of the density and pressure fields. In the third group there are those SPH schemes which employ numerical fluxes obtained through Riemann solvers to model the particle interactions. Riemann solver technique For an SPH method based on Riemann solvers, an inter-particle Riemann problem is constructed along a unit vector pointing form particle to particle . In this Riemann problem the initial left and right states are on particles and , respectively. The and states are The solution of the Riemann problem results in three waves emanating from the discontinuity. Two waves, which can be shock or rarefaction wave, traveling with the smallest or largest wave speed. The middle wave is always a contact discontinuity and separates two intermediate states, denoted by and . By assuming that the intermediate state satisfies and , a linearized Riemann solver for smooth flows or with only moderately strong shocks can be written as where and are inter-particle averages. With the solution of the Riemann problem, i.e. and , the discretization of the SPH method is where . This indicates that the inter-particle average velocity and pressure are simply replaced by the solution of the Riemann problem. By comparing both it can be seen that the intermediate velocity and pressure from the inter-particle averages amount to implicit dissipation, i.e. density regularization and numerical viscosity, respectively. Since the above discretization is very dissipative a straightforward modification is to apply a limiter to decrease the implicit numerical dissipations introduced by limiting the intermediate pressure by where the limiter is defined as Note that ensures that there is no dissipation when the fluid is under the action of an expansion wave, i.e. , and that the parameter , is used to modulate dissipation when the fluid is under the action of a compression wave, i.e. . Numerical experiments found the is generally effective. Also note that the dissipation introduced by the intermediate velocity is not limited. Incompressible approach Viscosity modelling In general, the description of hydrodynamic flows require a convenient treatment of diffusive processes to model the viscosity in