In finding a numerical solution to the two-dimensional Poisson equation, we used row ordering to define the finite difference equations. There is a better way, called red-black ordering. Think of the...

In finding a numerical solution to the two-dimensional Poisson equation, we used row ordering to define the finite difference equations. There is a better way, called red-black ordering. Think of the grid as a chessboard. Begin in the lower left corner of the grid of unknowns and color
 red. Now alter red and black until you have colored

 Draw the board corresponding to
 and color the interior grid points.

 In the five-point central difference scheme, if
 is red, what color are its four neighbors What color are the neighbors of a black grid point

 In terms of indices
 when is

_ij
red, and when is it black

 To apply Gauss-Seidel, begin with red grid point and apply the difference formula to all the red points. Now start with the black grid point
 and apply the difference formula to the black points. Sketch an algorithm that implements this version of Gauss-Seidel.

 Explain why this ordering allows the iteration to be parallelized.points