Problem 53975. Compute the effective conductivity of more heterogeneous aquifers
I've added a hint to help with this problem.
Thanks, but I thought that what you describe is what I was doing! ... or at least I was equivalently solving for the 2D pressure field, conserving mass at each note. But the differences from your solution were a a bit larger than the tolerance limits in 2 cases.
William, one difference between our approaches is that I applied the no-flow condition on the north and south boundaries. It doesn't appear that you did. Would you please try that and see if your code passes the remaining tests?
The if-statements "if j ~= 1 ..." and "if j ~= nr ..." were intended to address the north and south boundaries. Don't you think they do that? They set the flow into the north and south boundaries to zero by leaving their terms out of the design matrix C( , ).
By setting the heads in the 1st and nr-th rows equal to the heads in the 2nd and (nr-1)st rows, respectively, I imposed zero-gradient conditions at the N and S boundaries so that v_y would be zero.
You have a small gradient (as computed by a first-order finite difference approximation of the derivative.) The head contours should be perpendicular to the boundaries, but a couple of yours meet the N/S boundaries at angles slightly different from perpendicular. As you might imagine, our head contours are close but not quite the same. I'll try to send an image through your profile page.
Chris, Actually, I think my solution correctly implements the zero-gradient condition. It seems to me that the actual difference in the solutions is that mine effectively mapped the domain with (nc+2)x(nr+2) nodes (where K has dimensions (nr,nc) ). Your solution used a different number, roughly 1% different. That led to a numerical difference on the order of 1% in 2 of the tests.
OK, thanks for the explanation. I increased the tolerance a bit to account for the different numerical approaches and rescored your solution.
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