We propose using the Maxwell iteration to derive the hydrodynamic equations from the lattice Boltzmann equation (LBE) with an external forcing term. The proposed methodology differs from existing approaches in several aspects. First, it need not explicitly introduce multiple-timescales or the Knudsen number, both of which are required in the Chapman–Enskog analysis. Second, it need not use the Hilbert expansion of the hydrodynamic variables, which is necessary in the asymptotic analysis of the LBE. The proposed methodology assumes the acoustic scaling (or the convective scaling) ${\delta }_t\sim {\delta }_x$, thus ${\delta }_t$ is the only expansion parameter in the analysis of the LBE system, and it leads to the Navier–Stokes equations in compressible form. The forcing density derived in this work can reproduce existing forcing schemes by adjusting appropriate parameters. The proposed methodology also analyzes the numerical accuracy of the LBE. In particular, it shows the Mach number $\text{Ma}$ should scale as $O\left({\delta }_t^{1/3}\right)$ to maintain the truncation errors due to $\text{Ma}$ and ${\delta }_t$ in balance when ${\delta }_t\to 0$, so that the LBE can converge to the expected hydrodynamic equations effectively and efficiently.

In this work, we present an alternative derivation of the Navier-Stokes equations from Bhatnagar-Gross-Krook models of the lattice Boltzmann method with diffusive scaling. This derivation is based on the Maxwell iteration and can expose certain important features of the lattice Boltzmann solutions. Moreover, it will be seen to be much more straightforward and logically clearer than the existing approaches including the Chapman-Enskog expansion.

Based on the theory of asymptotic analysis, we prove that the viscous stress tensor computed with the lattice Boltzmann equation (LBE) in a two-dimensional domain is indeed second-order accurate in space. We only consider simple bounce-back boundary conditions which can be reduced to the periodic boundary conditions by using the method of image. While the LBE with nine velocities on two-dimensional square lattice (i.e., the D2Q9 model) and with the Bhatnagar-Gross-Krook collision model is used as an example in this work, our proof can be extended to the LBE with any linear relaxation collision models in both two and three dimensions.

In this Brief Report, we show that balanced chemical reaction systems governed by the law of mass action have an elegant conservation-dissipation structure. From this structure a number of important conclusions can be easily deduced. In particular, with the help of this structure we can rigorously justify the classical partial equilibrium approximation in chemical kinetics.

In this paper, we give a systematic stability analysis of the quadrature-based moment method (QBMM) for the one-dimensional Boltzmann equation with BGK or Shakhov models. As reported in recent lite...