A recursive mathematical formulation of the discrete-velocity Boltzmann equation (DVBE) under the Bhatnagar-Gross-Krook (BGK) approximation is introduced. This formulation allows us to formally express the solution of the DVBE as an infinite sum over successive particle derivatives of the distributions associated with local equilibrium. A Chapman-Enskog multiple-scales expansion shows that this sum can be safely truncated beyond the second order if the Navier-Stokes level of description is requested. Therefore, the distribution functions depend only on the first and second-order derivatives of the related equilibrium distributions. This defines a general framework to design low-memory kinetic schemes for the evolution of the distribution functions based solely on flow variables that are sufficient to define local equilibrium. As an example, a family of mass-conserving numerical schemes is introduced by dicretizing the particle derivatives by backward finite differences. Interestingly, a so-called simplified Lattice Boltzmann method introduced by Chen et al. in 2017 can be recast in this family. Numerical simulations highlight a level of numerical dissipation that is generally higher than the level obtained with a standard Lattice Boltzmann scheme, as expected by approximating derivatives by finite differences. Nevertheless, we show by using a von Neumann analysis that it is possible to parametrize our scheme, according to the relaxation coefficient of the DVBE, to reduce significantly its numerical dissipation and improve its spectral properties. Beyond these specific developments, we believe that this framework can also be of interest to connect macroscopic and kinetic representations, e.g. when dealing with initial and boundary conditions or in hybrid simulations matching Navier-Stokes and Lattice Boltzmann schemes.
&Mathematics with Applications