{c}
{title2=$D_9U=(I+\tfrac{h^2}{12}D_5)f$}

= Mehrstellenverfahren
{c}
{synonym}

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 $D_9u=\Delta u+h^2\Delta^2u/12+O(h^4)$. Replacing $\Delta^2u$ by $\Delta f$ and approximating this source <derivative> gives $D_9U=(I+h^2D_5/12)f$, 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.


 Mehrstellen method (source code)