Minimal number of discrete velocities for a flow description and internal structural evolution of a shock wave

A fluid flow is described by fictitious particles hopping on homogeneously distributed nodes with a given finite set of discrete velocities. We emphasize that the existence of a fictitious particle having a discrete velocity among the set in a node is given by a probability. We describe a compressible thermal flow of the level of accuracy of the Navier-Stokes equation by 25 or 33 discrete velocities for two-dimensional space and perform simulations for investigating internal structural evolution of a shock wave.