Accuracy of two-dimensional high-order lattice Boltzmann method with regularization for transition flows

We study the enhancement of the accuracy of the high-order lattice Boltzmann method (LBM) in both the order of equilibrium-distribution expansion and the degree of precision of the Gauss–Hermite quadrature. Furthermore, we investigate the accuracy of high-order regularization. We utilize two-dimensional Kolmogorov flow and Taylor–Green vortex flow over a wide range of Knudsen numbers (Kn) as the benchmark. The degree of precision of the Gauss–Hermite quadrature is assessed up to 33, the order of the equilibrium-distribution expansion is up to 14, and the order of regularization is up to 10. We compare the results to the direct simulation Monte Carlo (DSMC) method. The results indicate an improvement from increasing the degree of precision but not from increasing the order of equilibrium-distribution expansion. At Kn ∼ 0.1, the results from LBM with regularization agree with the theoretical solution. At Kn ≳ 1, the results from LBM with regularization contain oscillation, which is reduced as we increase the order of regularization. For a given order of regularization, there exists a numerical stability threshold for the degree of precision. This is in contrast to the order of equilibrium-distribution expansion, which has been known to have no stability restriction on the degree of precision used. When the simulation with regularization is stable, the degree of precision does not contribute to the accuracy of the simulation result. Conversely, the simulation without regularization, which should be equivalent to keeping the nonequilibrium distribution to infinite order, is stable no matter what degree of precision is chosen. Moreover, the accuracy of the simulation without regularization depends on the degree of precision used.