Updates of the Functions, or Relaxation

To solve the Poisson equation iteratively, we make an initial guess for the function values over the domain and then update Eq. [5] numerically. We first write the spatial part of Eq. [5] in FD form  

we replace the denominator on the left side (dx) by x and call this the time-step size. Eq. [23] can then be written as an update equation  

One simple modification leads to two other update schemes. As the solver scans through the grid, the values at grid points are typically updated in sequence. Thus, when we update the function at the (i) point, the (i — 1) value has already been changed. Eq. [25] can then be altered to 

This formula can be approximated on the grid by utilizing the ED methods discussed above. 

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