3 - Projection to divergence free vector field (Algo)

Input vector field \(w=(w^x,w^y)\)

Note: we assume in the following \(\Delta x = \Delta y = 1\)

1 - Compute \(d = \mathrm{div}(w)\)
- \(d_{x,y} = (w^x_{x+1,y}-w^x_{x-1,y} + w^y_{x,y+1}-w^y_{x,y-1})/2\)

2 - Compute \(q\) in solving \(\triangle q=b\)

- \((q_{x+1,y}+q_{x-1,y}-2\, q_{x,y}) + (q_{x,y+1}+q_{x,y-1}-2\, q_{x,y})=d_{x,y}\)
- \(\Rightarrow 4 q_{x,y} = q_{x+1,y}+q_{x-1,y}+q_{x,y+1}+q_{x,y-1} - d_{x,y}\)

- ex. Numerical iterations using Gauss Seidel

Initialize \(q=0\)
For \(i=[1..N_{\max}]\)
- \(q_{x,y} = 1/4 \left(q_{x+1,y}+q_{x-1,y}+q_{x,y+1}+q_{x,y-1} - d_{x,y}\right)\)

3 - Compute \(u=w-\nabla q\)
- \(u_{x,y} = w_{x,y} - \left(q_{x+1,y}-q_{x-1,y}\,,\;q_{x,y+1}-q_{x,y-1}\right)/2 \)