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We compute the continuum thermo-hydrodynamical limit of a new formulation oflattice kinetic equations for thermal compressible flows, recently proposed in[Sbragaglia et al., J. Fluid Mech. 628 299 (2009)]. We show that thehydrodynamical manifold is given by the correct compressible Fourier-Navier-Stokes equations for a perfect fluid. We validate the numericalalgorithm by means of exact results for transition to convection inRayleigh-B\'enard compressible systems and against direct comparison withfinite-difference schemes. The method is stable and reliable up to temperaturejumps between top and bottom walls of the order of 50% the averaged bulktemperature. We use this method to study Rayleigh-Taylor instability forcompressible stratified flows and we determine the growth of the mixing layerat changing Atwood numbers up to At ~ 0.4. We highlight the role played by theadiabatic gradient in stopping the mixing layer growth in presence of highstratification and we quantify the asymmetric growth rate for spikes andbubbles for two dimensional Rayleigh- Taylor systems with resolution up to Lx\times Lz = 1664 \times 4400 and with Rayleigh numbers up to Ra ~ 2 \times10^10.

Lattice Boltzmann Methods for thermal flows: continuum limit and applications to compressible Rayleigh-Taylor systems