content_image/414078.jpg content_image/414077.jpg content_image/414076.jpg content_image/414074.jpg content_image/414075.jpg
where we have defined b$_{j}$$_{+}$ 1 2 = ( b$_{j}$ + b$_{j}$$_{+1}$ ) / 2 and $^{b$_{j}$}$− 1 2 = ( $^{b$_{j}$}$− 1 + b$_{j}$ ) / 2. This can be easily generalize to three di- mensions as 

<img src='content_image/414078.jpg'>

Then the discrete version of Eq. (A3) is 

<img src='content_image/414077.jpg'>

in which 

<img src='content_image/414076.jpg'>

Then the Newton-Gauss-Seidel iteration says that we can obtain a new (and often more accurate) solution of u , u new $_{i,j,k}$, using our knowledge about the old (and less accu- rate) solution u old i,j,k as 

<img src='content_image/414074.jpg'>

The old solution will be replaced by the new solution to u$_{i,j,k}$ once the new solution is ready, using the red-black Gauss-Seidel sweeping scheme. Note that 

<img src='content_image/414075.jpg'>

