1. Building an LBM Model
The following is a mixture of MATLAB and C. Care must be taken, because array indexing begins at 1 in MATLAB and at 0 in C.

2. Define velocity vectors1 2 3 4 5 6 7 8
D2Q9 e1 e2 e3 e4 e5 e6 e7 e8
%define es
ex(0)= 0; ey(0)= 0
ex(1)= 1; ey(1)= 0
ex(2)= 0; ey(2)= 1
ex(3)=-1; ey(3)= 0
ex(4)= 0; ey(4)=-1
ex(5)= 1; ey(5)= 1
ex(6)=-1; ey(6)= 1
ex(7)=-1; ey(7)=-1
ex(8)= 1; ey(8)=-1
In MATLAB, can write ex(0+1)=0, ex(1+1)=1, etc.

3. Problem definition LY=10 LX=20 tau = 1 g=0.00001 %set solid nodes
is_solid_node=zeros(LY,LX)
for i=1:LX
is_solid_node(1,i)=1
is_solid_node(LY,i)=1
end LY LX
% if ~is_interior_solid_node(j,i)

4. Initialize density and fs (assuming zero velocity)
%define initial density and fs
rho=ones(LY,LX);
f(:,:,1) = (4./9. ).*rho;
f(:,:,2) = (1./9. ).*rho;
f(:,:,3) = (1./9. ).*rho;
f(:,:,4) = (1./9. ).*rho;
f(:,:,5) = (1./9. ).*rho;
f(:,:,6) = (1./36.).*rho;
f(:,:,7) = (1./36.).*rho;
f(:,:,8) = (1./36.).*rho;
f(:,:,9) = (1./36.).*rho;

5. // Computing macroscopic density, rho, and velocity, u=(ux,uy).
for( j=0; j<LY; j++) {
for( i=0; i<LX; i++)
u_x[j][i] = 0.0;
u_y[j][i] = 0.0;
rho[j][i] = 0.0;
if( !is_solid_node[j][i])
for( a=0; a<9; a++)
rho[j][i] += f[j][i][a];
u_x[j][i] += ex[a]*f[j][i][a];
u_y[j][i] += ey[a]*f[j][i][a]; }
u_x[j][i] /= rho[j][i];
u_y[j][i] /= rho[j][i];

7. Periodic Boundaries
On boundary, ‘neighboring’ point is on opposite boundary
ip = ( i<LX-1)?( i+1):( 0 );
in = ( i>0 )?( i-1):( LX-1);
jp = ( j<LY-1)?( j+1):( 0 );
jn = ( j>0 )?( j-1):( LY-1);
where
LHS = (COND)?(TRUE_RHS):(FALSE_RHS);
means
if( COND) { LHS=TRUE_RHS;} else{ LHS=FALSE_RHS;}

8. // Compute the equilibrium distribution function, feq.
for( j=0; j<LY; j++) {
for( i=0; i<LX; i++)
if( !is_solid_node[j][i])
rt0 = (4./9. )*rhoij;
rt1 = (1./9. )*rhoij;
rt2 = (1./36.)*rhoij;
ueqxij = uxij; //Can add forcing here; see MATLAB code
ueqyij = uyij;
uxsq = ueqxij * ueqxij;
uysq = ueqyij * ueqyij;
uxuy5 = ueqxij + ueqyij;
uxuy6 = -ueqxij + ueqyij;
uxuy7 = -ueqxij + -ueqyij;
uxuy8 = ueqxij + -ueqyij;
usq = uxsq + uysq;

9. feqij[0] = rt0*( 1. - f3*usq);
feqij[1] = rt1*( 1. + f1*ueqxij + f2*uxsq f3*usq);
feqij[2] = rt1*( 1. + f1*ueqyij + f2*uysq f3*usq);
feqij[3] = rt1*( 1. - f1*ueqxij + f2*uxsq f3*usq);
feqij[4] = rt1*( 1. - f1*ueqyij + f2*uysq f3*usq);
feqij[5] = rt2*( 1. + f1*uxuy5 + f2*uxuy5*uxuy5 - f3*usq);
feqij[6] = rt2*( 1. + f1*uxuy6 + f2*uxuy6*uxuy6 - f3*usq);
feqij[7] = rt2*( 1. + f1*uxuy7 + f2*uxuy7*uxuy7 - f3*usq);
feqij[8] = rt2*( 1. + f1*uxuy8 + f2*uxuy8*uxuy8 - f3*usq); }

10. Collide // Collision step. for( j=0; j<LY; j++)
for( i=0; i<LX; i++)
if( !is_solid_node[j][i])
for( a=0; a<9; a++) {
fij[a] = fij[a] + ( fij[a] - feqij[a])/tau; }

11. Bounceback Boundaries
Bounceback boundaries are particularly simple
Played a major role in making LBM popular among modelers interested in simulating fluids in domains characterized by complex geometries such as those found in porous media
Their beauty is that one simply needs to designate a particular node as a solid obstacle and no special programming treatment is required

12. Two type of solids:

13. Bounceback Boundaries

14. Collision (solid nodes): Bounceback
// Standard bounceback.
temp = fij[1]; fij[1] = fij[3]; fij[3] = temp;
temp = fij[2]; fij[2] = fij[4]; fij[4] = temp;
temp = fij[5]; fij[5] = fij[7]; fij[7] = temp;
temp = fij[6]; fij[6] = fij[8]; fij[8] = temp;

15. // Streaming step.
for( j=0; j<LY; j++) {
jn = (j>0 )?(j-1):(LY-1)
jp = (j<LY-1)?(j+1):(0 )
for( i=0; i<LX; i++)
if( !is_interior_solid_node[j][i])
in = (i>0 )?(i-1):(LX-1)
ip = (i<LX-1)?(i+1):(0 )
ftemp[j ][i ][0] = f[j][i][0];
ftemp[j ][ip][1] = f[j][i][1];
ftemp[jp][i ][2] = f[j][i][2];
ftemp[j ][in][3] = f[j][i][3];
ftemp[jn][i ][4] = f[j][i][4];
ftemp[jp][ip][5] = f[j][i][5];
ftemp[jp][in][6] = f[j][i][6];
ftemp[jn][in][7] = f[j][i][7];
ftemp[jn][ip][8] = f[j][i][8]; }

16. Incorporating Forcing (gravity)
Gravitational acceleration is incorporated in a velocity term
ueqxij = u_x(j,i)+tau*g; %add forcing