Five-point Stencil - Two Dimensions

Two Dimensions

In two dimensions, if for example the size of the squares in the grid is h by h, the five point stencil of a point (x, y) in the grid is

forming a pattern that is also called a quincunx. This stencil is often used to approximate the Laplacian of a function of two variables:

The error in this approximation is O(h 2), which may be explained as follows:

From the 3 point stencils for the second derivative of a function with respect to x and y:

\begin{array} {l} \frac{\partial ^2 f}{\partial x^2}= \frac{f\left(x + \Delta x,y\right) + f\left(x - \Delta x,y\right) - 2f(x,y)}{\Delta x^2} - 2\frac{f^{(4)}(x,y)}{4!}\Delta x^2 + \cdots \end{array}

\begin{array} {l} \frac{\partial ^2 f}{\partial y^2}= \frac{f\left(x, y + \Delta y\right) + f\left(x, y - \Delta y\right) - 2f(x,y)}{\Delta y^2} - 2\frac{f^{(4)}(x,y)}{4!}\Delta y^2 + \cdots \end{array}

If we assume :

\begin{array} {ll} \nabla^2 f &= \frac{\partial ^2 f}{\partial x^2}+\frac{\partial ^2 f}{\partial y^2}\\ \nabla^2 f &= \frac{f\left(x + h,y\right) + f\left(x - h,y\right) + f\left(x, y + h\right) + f\left(x, y - h\right) - 4f(x,y)}{h^2} - 4\frac{f^{(4)}(x,y)}{4!}h^2 + \cdots\\ \nabla^2 f &= \frac{f\left(x + h,y\right) + f\left(x - h,y\right) + f\left(x, y + h\right) + f\left(x, y - h\right) - 4f(x,y)}{h^2} + O\left(h^2\right)\\ \end{array}

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