 Mehrstellen method

A Mehrstellen method increases the accuracy of a compact finite difference method discretization by using the differential equation to express its leading truncation terms in terms of the prescribed source. For $\Delta u=f$, the square-grid nine-point finite-difference stencil for the Laplacian satisfies $D_{9}u=\Delta u+h^2{\Delta }^{2}u/12+O(h^4)$. Replacing ${\Delta }^{2}u$ by $\Delta f$ and approximating this source derivative gives $D_{9}U=(I+h^2{D}_5/12)f$, with fourth-order normalized consistency of a numerical method while retaining a compact nine-point matrix. The uncorrected nine-point finite-difference stencil operator is only second-order accurate on general functions.