Lattice Boltzmann method for the compressible Euler equations

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.