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)

$$V_i\left(smoothed\right)=v_i+{\alpha }_{cs}{\displaystyle \sum _j\frac{2m_j}{{\rho }_i+{\rho }_j}\left(v_j-v_i\right)\frac{W_i\left(j\right)+W_j\left(i\right)}{2}}$$