  The discretized equilibrium distributions of the lattice Boltzmann method are
presented by using the coefficients of the Lagrange interpolating polynomials
that pass through the points related to discrete velocities and using moments
of the Maxwell-Boltzmann distribution. The ranges of flow velocity and
temperature providing positive valued distributions vary with regulating
discrete velocities as parameters. New isothermal and thermal compressible
models are proposed for flows of the level of the isothermal and thermal
compressible Navier-Stokes equations. Thermal compressible shock tube flows are
simulated by only five on-lattice discrete velocities. Two-dimensional
isothermal and thermal vortices provoked by the Kelvin-Helmholtz instability
are simulated by the parametric models.
