The lattice Boltzmann model for the compressible Euler equations is proposed together with its rigorous theoretical background. The proposed model has completely overcome the defects of the previous model that the specific-heat ratio cannot be chosen freely. The macroscopic variables obtained from the solution are shown to satisfy, in the limit of the small Knudsen number, the compressible Euler equations if the variation of the solution is moderate. This is the case where no shock waves or contact discontinuities appear. In contrast, when the solution makes steep variation at several localized regions due to the appearance of shock waves and contact discontinuities, the corresponding macroscopic variables satisfy the weak form of the Euler equations. Their derivation is carried out rigorously by taking into account the scale of variation of the solution correctly. This is the first study that has laid the theoretical foundation of the lattice Boltzmann model for the simulation of flows with shock waves and contact discontinuities. Numerical examples and the error estimates are also given, which are consistent with the above theoretical arguments.

The existing lattice Boltzmann method multispeed thermal models show a limited accuracy. This paper proposes a two-dimensional multispeed thermal model for the finite-difference lattice Boltzmann method (FDLBM). To recover correct fluid equations, up to fourth orders of local flow velocity should be retained in the local equilibrium distribution function and tensors of particle velocities should have up to seventh rank isotropy. In the FDLBM, particle velocities can be selected independently from the lattice configuration. Therefore, particle velocities of octagonal directions, which have up to seventh rank isotropic tensors, are adopted. The proposed model was verified by two simulations. The model showed excellent numerical stability in addition to strict accuracy.

We have developed a lattice Boltzmann model for the compressible Navier-Stokes equations with a flexible specific-heat ratio. Several numerical results are presented, and they agree well with the corresponding solutions of the Navier-Stokes equations. In addition, an explicit finite-difference scheme is proposed for the numerical calculation that can make a stable calculation with a large Courant number.

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Multispeed thermal models of the lattice Boltzmann method (LBM) that have a single relaxation [Bhatnagar-Gross-Krook (BGK)] scheme have been proposed by several authors. While these models are intended to correctly represent heat characteristics and compressibility, most of them do not provide satisfactory accuracy. This paper discusses how to construct a correct model. Thermally correct two-dimensional and three-dimensional multispeed LBM BGK models are proposed. The models are verified by simulations of Couette flow, evolution from circularly distributed temperature, and normal shock wave. The results show exact agreement with the theoretical predictions. The numerical stability of the model is demonstrated by the simulation of recovery from a random fluctuation.