Conservative Smoothing of Velocities

It can be shown that the SPH method is unstable in tension. The instability is shown to result from an effective stress with a negative modulus (imaginary sound speed) being produced by the interaction between the constitutive relation and the kernel function, and is not caused by the numerical time integration algorithm. 1 D.S Balsara 2 states, use special filtering of velocities (so called conservative smoothing, because momentum quantities are not modified):(1)
V i ( s m o o t h e d ) = v i + α c s j 2 m j ρ i + ρ j ( v j v i ) W i ( j ) + W j ( i ) 2
1 Swegle J.W., Hicks D.L., and Attaway S.W., 「Smoothed Particle Hydrodynamics: Stability Analysis」, Journal of Computational Physics, Vol. 116, pp. 123-134, 1995.
2 Balsara D.S., 「Von Neumann Stability Analysis of Smoothed Particle Hydrodynamics Suggestions for Optimal Algorithms」, Journal of Computational Physics, Vol. 121, pp. 357-372, 1995.