Macroscopic finite-difference scheme based on the mesoscopic regularized lattice-Boltzmann method

In this paper, we develop a macroscopic finite-difference scheme from the mesoscopic regularized lattice Boltzmann (RLB) method to solve the Navier-Stokes equations (NSEs) and convection-diffusion equation (CDE). Unlike the commonly used RLB method based on the evolution of a set of distribution functions, this macroscopic finite-difference scheme is constructed based on the hydrodynamic variables of NSEs (density, momentum, and strain rate tensor) or macroscopic variables of CDE (concentration and flux), and thus shares low memory requirement and high computational efficiency. Based on an accuracy analysis, it is shown that, the same as the mesoscopic RLB method, the macroscopic finite-difference scheme also has a second-order accuracy in space. In addition, we would like to point out that compared with the RLB method and its equivalent macroscopic numerical scheme, the present macroscopic finite-difference scheme is much simpler and more efficient since it is only a two-level system with macroscopic variables. Finally, we perform some simulations of several benchmark problems, and find that the numerical results are not only in agreement with analytical solutions, but also consistent with the theoretical analysis.

Figure 1

The numerical and analytical solutions of velocity $\mathbf{u}=(u_1,u_2)$ at different positions [(a) $u_1$ and (b) $u_2$. Solid line denotes the analytical solution, and symbol represents the numerical solution].

Figure 2

A comparison of the convergence rate among the present, M-RLB, and RLB schemes for the four-roll mill problem.

Figure 3

The numerical and analytical solutions of $\varphi$ at different times [solid line denotes the analytical solution, and symbol represents the numerical solution].

Figure 4

A comparison of the convergence rate among the present, M-RLB, and RLB schemes for the isotropic two-dimensional CDE.

Figure 5

The numerical and analytical solutions of (a) the velocity component $u_1$ and (b) the temperature field $T$ [solid line denotes the analytical solution, and symbol represents the numerical solution].

Figure 6

A comparison of the convergence rate among the present, M-RLB, and RLB schemes for the thermal Poiseuille flow with $Pr=0.25$ [(a) the velocity component $u_1$ and (b) the temperature field $T$].