# Finite difference method for heat equation

1. Feb 19, 2013

### ericm1234

This question is more general than the title:
When constructing the finite difference mesh, and assigning all those points to it, then taking into account the initial condition and boundary conditions for say, the heat equation, my book does not seem to be consistent on what happens with the 2 bottom corner points, say discrete points $v^{0}_{0}$ and $v^{0}_{N}$ where superscript 0 is the bottom row (in say, time, the y-axis), and subscript N is all the way to right of the x-values (say, space coordinate).
Sometimes it seems the book give these 2 points to the initial condition, sometimes to the boundary conditions, and sometimes, I dare say, to NEITHER. So is there a consistent method for assigning these 2 points? Obviously all the points in between on this bottom 'row' are given to the initial condition; just not sure about these 2 points in particular.

And oh yeah, this is related to Crank-Nicholson; but I can not find a decent derivation.