Alternative method to construct equilibrium distribution functions in lattice-Boltzmann method simulation of inviscid compressible flows at high Mach number

A method is proposed to construct an equilibrium density distribution function in the simulation of compressible flows at high Mach number by the lattice-Boltzmann method. In this method, the conventional Maxwellian distribution function is replaced by a circular function which is very simple and satisfies all needed statistical relations to recover the compressible Navier-Stokes equations. The circular function is then distributed to the lattice velocity directions by Lagrangian interpolation in such a way that all the needed statistical relations are exactly satisfied when the integral in the phase space is replaced by the summation in the context of the lattice-Boltzmann (LB) method. In this framework, the equilibrium distribution functions and the associated lattice velocity model can be derived naturally without assuming specific forms. Two LB models with adjustable specific heat ratio, respectively, for one-dimensional (1D) and two-dimensional (2D) compressible flows are shown in the paper. Some test cases of compressible flows with strong shock waves are simulated to validate the present approach. Excellent results are obtained. Note that in the simulation, the total variation diminishing (TVD) scheme was used to capture the discontinuity with coarse mesh.