From discrete Boltzmann equation to compressible linearized Euler equations

This paper concerns the asymptotic analysis of the linearized Euler limit for a general discrete velocity model of the Boltzmann equation. This is done for any dimension of the physical space, for densities which remain in a suitable small neighbourhood of global Maxwellians. Providing that the initial fluctuations are smooth, the scaled solutions of discrete Boltzmann equation are shown to have fluctuations that locally in time converge weakly to a limit governed by a solution of linearized Euler equations. The weak limit becomes strong when the initial fluctuations converge to appropriate initial data. As applications, the two-dimensional 8-velocity model and the one-dimensional Broadwell model are analyzed in detail.