Re: Basic question about Neumann boundary conditions


In article <1135011450.558978.229510@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
goodchild.trevor@xxxxxxxxx writes:
>Please ignore the following part of my previous message -- the
>derivative is with respect to x, not t:
>
>"However,
>the boundary condition itself seems to say 'the value at x=0 never
>changes.'"
>
>I'd still like to know the correct way to enforce the boundary
>condition, though.
>

u_x(0,t)=0 says "there is no flow of heat at this end"
hence your idea to set u(0,t)=u(0,0) is false.
the implementation with finite differences is quite easy:
you introduce an artifial gridline x=-delta_x and give it the number -1
then yuo have unknowns at the grid lines
-1,0,1,..,n-1 with delta_x = 1/n
since at the gridline number n you have the dirichlet data.
now you use the differnce star for the differnetial equations on the
grid lines 0,...,n-1. on the right you can use the dirichlet value
but on the left you use for every timestep the condition

u(-delta_x,t)=u(delta_x,t)

(value on grid line -1 = value on grid line 1)
which approximates the condition u_x(0,t)=0 of second order
there remains (in your case) a tridiagonal linear system
(for sure you will use an implicit solver).
hth
peter
.

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