Dissipative second-order forward advection semidiscretization ($u_m^{\prime }=h^{-1}(-3u_m/2+2u_{m+1}-u_{m+2}/2)$)

For periodic indexing let $S$ be the unitary forward shift. The operator is $A=-3I/2+2S-S^2/2$. A discrete Fourier transform diagonalizes it with eigenvalue $\lambda (\theta )=-3/2+2e^{i\theta }-e^{2i\theta }/2$ and real part $-(1-\cos \theta {)}^2$. Consequently $\Vert e^{tA/h}{\Vert }_2\le 1$ for every $t\ge 0$, uniformly in the grid. Equivalently $A+A^{\ast }=-\frac{1}{2}(2I-S-S^{\ast }{)}^2$, proving decay of the discrete energy directly. The result is semidiscrete stability; a further time discretization must also have adequate absolute stability.