Eigenvalue stability analysis of a finite difference method

For a normal amplification matrix $Q$, the bound $\rho (Q)\le 1$ with semisimple unit-modulus eigenvalues controls every power $Q^n$. For a nonnormal family, eigenvalues alone do not control transient growth or give a mesh-uniform bound; eigenvector conditioning, pseudospectra, or a direct energy estimate is then needed.

On a finite inflow grid, explicit upwinding has a triangular amplification matrix whose only eigenvalue is $1-\mu$. For $1<\mu <2$ this eigenvalue lies inside the unit disk, although interior high-frequency data are amplified by $|1-2\mu |>1$ before reaching the boundary. This shows why eigenvalue analysis without uniform normality can give the wrong stability range.