This is the so-called weak-compressibility assumption. This corresponds to a Mach number smaller than 0. Since only small density variations occur, a linear equation of state can be adopted . Usually the weakly-compressible schemes are affected by a high-frequency spurious noise on the pressure and density fields. Through the years, several techniques have been proposed to get rid of this problem. They can be classified in three different groups:.
The schemes of the first group apply a filter directly on the density field to remove the spurious numerical noise. The most used filters are the MLS Moving Least Squares and the Shepard filter  which can be applied at each time step or every n time steps. The more frequent is the use of the filtering procedure, the more regular density and pressure fields are obtained. On the other hand, this leads to an increase of the computational costs.
In long time simulations, the use of the filtering procedure may lead to the disruption of the hydrostatic pressure component and to an inconsistency between the global volume of fluid and the density field. Further, it does not ensure the enforcement of the dynamic free-surface boundary condition.
The first schemes that adopted such an approach were described in  and in  where the diffusive term was modelled as a Laplacian of the density field. A similar approach was also used in . In  a correction to the diffusive term of  was 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 . In the third group there are those SPH schemes which employ numerical fluxes obtained through Riemann solvers to model the particle interactions   .
In general, the description of hydrodynamic flows require a convenient treatment of diffusive processes to model the viscosity in the Navier—Stokes equations. It needs special consideration because it involves the laplacian differential operator. Since the direct computation does not provide satisfactory results, several approaches to model the diffusion have been proposed.
Introduced by Monaghan and Gingold  the artificial viscosity was used to deal with high Mach number fluid flows. The artificial viscosity also has shown to improve the overall stability of general flow simulations. Therefore, it is applied to inviscid problems in the following form. It is possible to not only stabilize inviscid simulations but also to model the physical viscosity by this approach.
To do so. For low Reynolds numbers the viscosity model by Morris  was proposed.
Smooth Particle Applied Mechanics - The State of the Art
Often in astrophysics, one wishes to model self-gravity in addition to pure hydrodynamics. The particle-based nature of SPH makes it ideal to combine with a particle-based gravity solver, for instance tree gravity code ,  particle mesh , or particle-particle particle-mesh. The discrete element method , used for simulating granular materials , is related to SPH.
From Wikipedia, the free encyclopedia. Method of hydrodynamics simulation. This section is empty.
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Smooth Particle Applied Mechanics: The State Of The Art
Liquid drops and surface tension with smoothed particle applied mechanics. Correct reproduction of group-induced long waves. In: Proc. Papanastasiou TC. Flows of Materials with Yield. Journal of Rheology present , 31 5 , Applied Ocean Research, 50, 1— Simulation of caisson breakwater movement using SPH.
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