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Let $S$ be the periodic forward-shift matrix, $(Su{)}_m=u_{m+1}$ with indices modulo $M$. The semidiscrete operator isFor ${\omega }_{\ell }=e^{2\pi i\ell /M}$, $\ell =0,\dots ,M-1$, the vector $v^{(\ell )}=(1,{\omega }_{\ell },\dots ,{\omega }_{\ell }^{M-1}{)}^T$ satisfies $S{v}^{(\ell )}={\omega }_{\ell }{v}^{(\ell )}$. These vectors divided by $\sqrt{M}$ form the unitary discrete Fourier transform basis. Hence the normal matrix $A$ has eigenvaluesEvery Fourier coefficient is multiplied by $e^{t{\lambda }_{\ell }/h}$ of modulus at most one. Parseval's identity therefore provesThe same estimate holds for differences of solutions and is uniform in the mesh. This is the required semidiscrete stability of a numerical method. The mean mode has eigenvalue zero and is conserved, consistently with periodic advection.

One can also verify the energy estimate without diagonalization. Since $S^{\ast }=S^{-1}$,The dissipative second-order forward advection semidiscretization thus damps nonconstant modes rather than amplifying them. This does not assert stability of an arbitrary further time-stepping method; its absolute stability region must contain the scaled eigenvalues. No nonnegativity condition on the periodic boundary value is needed: the printed trailing $t\ge 0$ specifies the time range.