= Compact semidiscrete nine-point diffusion
{title2=$M_h\dot U=L_hU$}
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 $(4+\cos\xi+\cos\eta)/6\geq1/3$, and its diffusion symbol is $[-10+4(\cos\xi+\cos\eta)+2\cos\xi\cos\eta]/(3h^2)\leq0$. 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.
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