Three ways to lattice Boltzmann: A unified time-marching picture

It is shown that the lattice Boltzmann equation (LBE) corresponds to an explicit Verlet time-marching scheme for a continuum generalized Boltzmann equation with a memory delay equal to a half time step. This proves second-order accuracy of LBE with respect to this generalized equation, with no need of resorting to any implicit time-marching procedure (Crank-Nicholson) and associated nonlinear variable transformations. It is also shown, and numerically demonstrated, that this equivalence is not only formal, but it also translates into a complete equivalence of the corresponding computational schemes with respect to the hydrodynamic equations. Second-order accuracy with respect to the continuum kinetic equation is also numerically demonstrated for the case of the Taylor-Green vortex. It is pointed out that the equivalence is however broken for the case in which mass and/or momentum are not conserved, such as for chemically reactive flows and mixtures. For such flows, the time-centered implicit formulation may indeed offer a better numerical accuracy.

Figure 1

Schematic of the conceptual links between the three equivalent LB formulations for a continuum fluid with viscosity $\nu$. The diagram straddles across three logical layers: continuum hydrodynamics, continuum kinetic theory, and discrete-kinetic theory.

Figure 2

Isocontours of the velocity magnitude for a laminar flow between parallel plates with orthogonal screens periodically interspaced. The computational domain repeats periodically along the $x$ axis and the screen is located at 4/15 of the total length. The mesh size is $N_x=150N_y=30$.

Figure 3

Cuts of the velocity and pressure profiles as computed with the LB scheme with $\theta =0,1/2$, as compared with analytical solution (solid line). No appreciable difference between the two numerical solutions with $\theta =0,1/2$ is visible at the scale of the plot.

Figure 4

Error on the kinematic viscosity for the case of the Taylor-Green vortex as a function of the grid resolution. No appreciable difference between the two numerical solutions with $\theta =0,1/2$ is visible at the scale of the plot.

Figure 5

Error on the discrete distribution function for the case of the Taylor-Green vortex, for $\theta =0$ (open symbols) and $\theta =1/2$ (solid squares) at various grid resolutions, $N_x∊\left\{25,50,100,200,400\right\}$. The numerical Mach number was fixed at $m=1/5$, corresponding to numerical Knudsen numbers in the following range: $2.5\times {10}^{-3}\le k\le 4.0\times {10}^{-2}$.