Compact semidiscrete nine-point diffusion ($M_h\dot{U}=L_{h}U$)

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.