Discretizing PDE using the 9-point Laplacian with variable coefficients

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Discretizing PDE using the 9-point Laplacian Problem

Here are the notes described: https://pdfs.semanticscholar.org/771c/15fba0308428ae1a2bf8db9c8f8d1b63eb3d.pdf

My Attempt: My Attempt

This is what I tried, but then I realized that this would not end with 9-points. I tried to use the centered difference approximation for the points immediately around the center and then again for the points next to those.

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