von Neumann stability analysis

Remarks

The proof uses the discrete-time Fourier transform $\hat u(\xi) = \Delta x \sum_j u_j e^{-i j \xi}$, which turns the scheme into $\hat u^{n+1}(\xi) = g(\xi)\, \hat u^n(\xi)$, so $\hat u^n = g^n \hat u^0$. By Parseval's identity, $\norm{u^n}$ equals the $L^2$ norm of $g^n \hat u^0$ up to a constant factor, so stability is equivalent to a bound on $\sup_\xi \abs{g(\xi)}^n$ for $n \Delta t \le T$. This bound holds exactly when $\abs{g} \le 1 + K_0 \Delta t$, because $(1 + K_0 \Delta t)^n \le e^{K_0 T}$. Example: the upwind scheme $u_j^{n+1} = u_j^n - \nu (u_j^n - u_{j-1}^n)$ for $u_t + a u_x = 0$, with $\nu = a \Delta t / \Delta x$, has $\abs{g(\xi)}^2 = 1 - 2 \nu (1 - \nu)(1 - \cos \xi)$; this is at most $1$ for all $\xi$ exactly when $0 \le \nu \le 1$, which is the Courant–Friedrichs–Lewy condition for this scheme. For systems of equations, the analogous condition on the spectral radius of the amplification matrix is necessary but not sufficient in general. For variable coefficients or boundaries, the analysis is only a heuristic test.

Going deeper

For a student

The analysis is the practical tool to test a new difference scheme: substitute $u_j^n = g^n e^{i j \xi}$ and solve for $g$. Combined with consistency, stability gives convergence by the Lax equivalence theorem. The term $K_0 \Delta t$ allows schemes for equations with lower-order terms, such as $u_t = u_{xx} + u$, whose exact solutions also grow.