Dissipative second-order forward advection stencil ($D_{+,2}=(-3I+4S-S^2)/(2h)$)

For the equation $u_t=u_x$, the second-order forward difference has Fourier symbol $[-(1-\cos \theta {)}^2+i\sin \theta (2-\cos \theta )]/h$. The nonpositive real part gives a contractive semidiscrete evolution in the discrete L2 norm. Equivalently, for the unitary lattice shift $S$, $\mathrm{Re}⟨u,D_{+,2}u⟩=-\Vert (S-I{)}^{2}u{\Vert }^2/(4h)$. A centred difference in another direction contributes only a skew-adjoint term, preserving this energy estimate.