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  "body": "\n## Summary\n\nA five-point stencil is a set of five grid points used to approximate derivatives by finite differences.\nOn a line, it is a point with two neighbours on each side, and it gives fourth-order formulas for the first and second derivative.\nIn the plane, it is a point with its four nearest neighbours, and it gives the standard second-order approximation of the [[Laplacian|0E0B6RA8K0]].\nThe planar version is the basic discretisation of the Poisson equation in the [[finite difference method|X7HYEBA5SA]].\n\n## Explanations\n\n### L1 human\n\nPicture a square grid and a point on it together with its neighbours to the left, right, above and below: these five points form a plus sign.\nAdding the four neighbour values and subtracting four times the centre value measures how much the centre differs from the average around it.\nFor $u(x, y) = x^2 + y^2$ at the origin with spacing $1$, this gives $1 + 1 + 1 + 1 - 0 = 4$, which equals the Laplacian $u_{xx} + u_{yy} = 4$.\n",
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