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Equation of Propagation
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ID:(1152, 0)


Rebound in walls orthogonal to the network
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If the collision does not occur at the point of the network, but at a distance \Delta:

\\n\\nthen the function must consider the offset by weighting the contributions\\n\\n

$f_i(x_f,t+\delta t)=\displaystyle\frac{(1-\Delta)f_{-i}(x_f,t)+\Delta(f_{-i}(x_b,t)+f_{-i}(x_{f2},t)}{1+\Delta}$

ID:(8499, 0)


Rebound on walls with inclination
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If the wall shows an inclination with respect to the network it must be modeled in a more complex form:


More general edge

First, an approximate boundary must be defined to allow the necessary edge equations to be established. Then they must be applied in the process of steraming.

ID:(8500, 0)


Streaming
Equation

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In the streaming process the particles are moved according to their velocity directions to neighboring cells

$f_i(\vec{x},t)\leftarrow f_i(\vec{x}+ce_i\delta t,t+\delta t)$

where \vec{x} is the position, t time, \vec{e} _i the direction of the grid and c the speed.

ID:(9150, 0)


Example of Streaming Equations
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In the case of a D2Q9 system we have the 9 values f_i that we have named as O, N, E, S, W, NE, SE, SW, NW. If the number of particles in position (n,m) is denoted as f_i(j,k) we have that the equations are

```

N[x,y] = N[x,y-1]

NW[x,y] = NW[x+1,y-1]

E[x,y] = E[x-1,y]

NE[x,y] = NE[x-1,y-1]

S[x,y] = S[x,y+1]

SE[x,y] = SE[x-1,y+1]

W[x,y] = W[x+1,y]

SW[x,y] = SW[x+1,y+1]

```

ID:(9151, 0)