Compact fourth-order Helmholtz stencil ($h^{-2}{\Gamma }_{9}u+\lambda M_{h}u=0$)

Let ${\Gamma }_9$ have center weight $-10/3$, axial weights $2/3$ and diagonal weights $1/6$, and let $M_h$ have center weight $2/3$ and axial weights $1/12$. Taylor expansion gives $h^{-2}{\Gamma }_{9}u=\Delta u+h^2{\Delta }^{2}u/12+O(h^4)$ and $M_{h}u=u+h^2\Delta u/12+O(h^4)$. On a smooth solution of the constant-coefficient Helmholtz equation, the second-order term becomes $h^2\Delta (\Delta u+\lambda u)/12=0$. Thus the normalized PDE residual is fourth order, while the original unscaled stencil residual is sixth order. This local cancellation alone does not prove uniform global accuracy at or near a resonance.