Mass Conservation

Im pretty new to the LBM method and i’ve got a (most probable very silly) question about the basic algorithm:

What is being done to assure the conservation of mass ?

Assume u’ve got a squarish grid consisting of 9 cells.
Further assume the distribution-functions of the middle cell to be zero except those ones representing the fraction of paricles that are not moving.
If now there is even just one single distribution-function of an adjacent cell with direction towards the middle cell that is different from zero, after the stream-step the Density of the middle Fluid Cell is >1.
And this cannot be regulated by the following collision-step.

The only chance, i saw was to normalize the distribution-functions at the end of a LBM step by the sum of all distribution functions of one cell so that the density will become 1 again.
But this is either problematic if a cell totally empties and together with common “Free Surface Boundary Conditions”, such a treatment could cause a loss of mass not just in very artificial situations.

*Imagine a flat surface fluid in a pot-shaped vessel and all interface-cells to have distribution-functions equal to zero except the ones heading straight to the bottom. This would cause them to empty. The underneath fluid-cells would absorb the additional mass by the above suggestet normalisation and thereafter they would be changed into interface-cells themselved (which again would have distributionfunctions different from zero only with the bottom-direction). Then the whole precedure would start again at * until the vessel would only contain one layer that would be reflected at the bottom (obstacle boundary) and the vessel would fill up infinite (unless, i dont know, the equilibrium-functions would damp the distribution-functions in such a way, that the surface would be convergent to some height-level

If sb would have a suggestion/answer on this id be very glad.

any reason why you want mass to be conserved locally? fluids don’t work this way. the mass does not remain the same on a site after streaming, which is ok and consistent with the physics of a fluid. mass is conserved locally during collision, and globally during streaming. global mass conservation means that the sum of mass over all sites is constant. global conservation during streaming is an immediate consequence of local conservation during collision.