A lattice Bhatnagar-Gross-Krook (BGK) model [10] [11] is briefly introduced here for the purpose of describing the semi-implicit scheme in the following section. The BGK model is defined by the following equation:

where is a particle distribution function representing the probability of finding a fluid particle with a velocity at location and time. is the relaxation time. is the number of velocities. is an equilibrium distribution function.

Velocity is determined by a lattice structure. For 2D regular cell models, the most commonly used lattice structure is D2Q9 according to the DdQb notation of Qin et al. [11] . Here, is the space dimensions, while is the number of velocities including the zero velocity,.

The equilibrium distribution function for the D2Q9 lattice model is given by

where is the weight factor in the direction. The weight factors for the D2Q9 model are 4/9 for the rest particles, 1/9 for particles streaming to the face connected neighbors and 1/36 for particles streaming to the edge connected neighbors and while is the speed of sound, and

for D2Q9 model. The relaxation time is related to the viscosity by

The boundary condition most commonly used in lattice Boltzmann method is the bounce-back method. The bounce-back scheme is easy to implement and make the method ideal for simulating fluid flows in complicated geometries [8] . However, the bounce-back scheme is only first-order in numerical accuracy at the boundaries, which is not consistent with the order of the LBM in interior points. The non-equilibrium extrapolation scheme is adopted here. The basic idea is to decompose the distribution function into equilibrium part and non-equili- brium part. The scheme is of second-order accuracy in both time and space region [12] .

The macroscopic variables such as density and velocity, which are required for equilibrium distribution function, can be obtained from the following equations:

From Equation (1), the fluid particle evolution involves two distinct steps:

Collision:

Propagation:

Macroscopic variables, collision and propagation are the three time consuming steps during the lattice Boltzmann simulation process. The main difference between the standard scheme and semi-implicit scheme in code implementation is that, the former scheme needs three loops for three steps, while the latter scheme needs only one loop in all the nodes for all three steps. When the distribution functions for node were computed, the distribution functions of its neighboring node are used by standard scheme for all directions, but the distribution functions of its neighboring nodes are used by semi-implicit scheme for the red directions (shown in Figure 1). The accuracy and efficiency of semi-impli- cit scheme will be discussed through the simulation of cavity flow in the next section.

The simulation results computed by semi-implicit scheme were compared with previous works done by Ghia et al. [8] and Hou et al. [9] . First, the locations of primary and secondary vortices are listed in Table 1 and Table 2. The convergence criterion, Er is choosen 10⁻⁶ for all simulations.

Numerical simulations were carried out for, 1000, 2000, 5000, 7500. Figure 2 shows the changes of stream function for each Reynolds numbers, the locations of vortices with different level can be seen clearly from these figures, for example, the third level vortices is turned out on the upper left corner of the square cavity when. Table 1 shows the locations of the center of primary vortices and the pair of lower vortices for different Reynolds numbers. The locations of the secondary vortex in the upper left corner for and 7500 are listed in Table 2. These result show good agreement with Ghia et al. [8] and Hou et al. [9] .

For all simulations, steady state is reached when the convergence criterion, Er is less than 10⁻⁶ for successive 1000 simulation steps. Table 3 shows the iterations that both schemes are needed. Ef in Table 3 equals, which shows the saving proportion of iterative steps of the semi-implicit

Note: a, Ghia [8] b, Hou [9] , c, semi-implicit scheme.

Note: a, Ghia [8] b, Hou [9] , c, semi-implicit scheme.

scheme. The iterations can be saved about 30%, when Re less than 3000, and the advantage of semi-implicit scheme becomes more obvious as the Re increases. The standard scheme is not converged once Re > 3400. On the contrary, the iterations of semi-implicit scheme is approximately linear to Re (Figure 3).