Author Description

To improve the numerical stability of the Lattice Boltzmann method, Karlin et al. [I. V. Karlin, F. B ̈osch, and S. S. Chikatamarla, Phys. Rev. E 90, 031302(R) (2014)] proposed the entropic multiple relaxation time (EMRT) collision model. The idea behind EMRT is to construct an optimal post-collision state by maximizing its local entropy value. The critical step of the EMRT model is to solve the entropy maximization problem under certain constraints, which is often computationally expensive and even not feasible. Karlin et al. attempted to solve the constraint maximum entropy problem by assuming that all the higher-order moments relax to an equilibrium state at the same rate. However, this solution does not realize the EMRT model in its general form. To solve the entropy maximization problem in its general form where the higher-order moments could relax to equilibrium states at different rates, we propose to employ the perturbation theory and obtain an asymptotic solution. With mathematical analysis on special cases under relaxed constraints, we discover the unperturbed form of the original problem and derive the asymptotic solution. We show that the asymptotic solution gives a good approximation to the optimal states; thus, our approach provides a novel and efficient way to solve the constraint maximum entropy problem in the EMRT model. Further, we expand this method to the 3D case. The EMRT model exhibit good stability performance where the relaxation of higher order moments is optimal with respect to the entropy function. However, this choice might not be optimal to the accuracy. In 3D Taylor Green vortex, the simulation results of EMRT model show that the quantity of enstrophy is more sensitive to the resolution than the commonly used BGK model.