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# Nine-point stencil for 2D laplacian

By: Gijs Bellaard

Suppose you have some sufficiently smooth black-box function $$f: \mathbb{R}^2 \rightarrow \mathbb{R}$$ of which you want to know the laplacian at the point $$x$$. We introduce the following shorthand: $f = f(x)$ $f_{i,j} = f(x+ih,y+jh)$ where $$h$$ is the spacing. Then the laplacian at the point of interest is: $\Delta f \approx \frac{1}{3h^2} \left( f_{-1,1} + f_{0,1} + f_{1,1} + f_{-1,0} - 8f + f_{1,0} + f_{-1,-1} + f_{0,-1} + f_{1,-1} \right)$ with an error of $$\mathcal{O}(h^2)$$. It's not actually better than the five point stencil in accuracy but it does help smooth out the grid artifacts.