I wish correct some evident errors.
I use a three-dimensional grid for entropy calculus, then:
[math]
0 = \rho_{000} \Delta \rho_{000}+\sum_{ijk} \rho_{ijk} \epsilon_{ijk}
[/math]
the isoentropic discrete equation is:
[math]
0 = \rho_{000} \Delta \rho_{000}+\sum_{ijk} \rho^2_{ijk} \Delta_i \Delta_j \frac{v_{ijk}}{\Delta_i \Delta_j \Delta_k} \Delta t
[/math]
[math]
0 = \rho_{000} \frac{\Delta \rho_{000}}{\Delta t}+\frac{\rho^2_{ijk} v_{ijk}-\rho^2_{-i,-j,-k} v_{-i.-j.-k}}{\Delta_k}+\cdots
[/math]
[math]
0 = \rho \frac{\partial \rho}{\partial t}+\nabla \cdot \left(\rho^2 \vec v\right)
[/math]
then the entropy equation can be write
[math]
0 = \frac{1}{2} \frac{\partial \rho^2}{\partial t}+\nabla \cdot \left(\rho^2 \vec v\right)
[/math]
or
[math]
0 = \rho \frac{\partial \rho}{\partial t}+2 \rho \vec v \cdot \nabla \rho+\rho^2 \nabla \cdot \vec v=\rho \left(\frac{\partial \rho}{\partial t}+2 \vec v \cdot \nabla \rho+\rho \nabla \cdot \vec v\right)
[/math]
my search of this aesthetic beauty (simmetry, semplicity, etc), lead me to error in the differential equation.