Mehrstellen method ($D_{9}U=(I+\frac{h^2}{12}{D}_5)f$)

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.

On a square grid, let $C_h$ sum the four axial neighbors and $K_h$ the four diagonal neighbors. With $M_h=2I/3+C_h/12$ and $L_h=(-10I/3+2C_h/3+K_h/6)/h^2$, the method of lines has fourth-order normalized spatial numerical consistency. Its mass symbol is $(4+\cos \xi +\cos \eta )/6\ge 1/3$, and its diffusion symbol is $[-10+4(\cos \xi +\cos \eta )+2\cos \xi \cos \eta ]/(3h^2)\le 0$. Every continuous-time mode therefore contracts, and Parseval identity gives mesh-independent discrete L2 stability. No time-step restriction is involved until a time integrator is selected.