# 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):

$${V}_{i}\left(smoothed\right)={v}_{i}+{\alpha}_{cs}{\displaystyle \sum _{j}\frac{2{m}_{j}}{{\rho}_{i}+{\rho}_{j}}\left({v}_{j}-{v}_{i}\right)\frac{{W}_{i}\left(j\right)+{W}_{j}\left(i\right)}{2}}$$

^{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.