13.2 Stability: the von Neumann method
The scheme is linear with constant coefficients, so it acts on each Fourier mode separately. Substituting \(u_j^n = G^ne^{ij\theta }\), \(0 \leq \theta \leq \pi \), the mode is reproduced with the amplification factor \[ G(\theta ) = 1 + r(e^{i\theta } - 2 + e^{-i\theta }) = 1 - 4r\sin ^2\frac \theta 2 . \] The scheme is stable in the sense of von Neumann if \(|G(\theta )| \leq 1 + C\Delta t\) for all \(\theta \), so that no mode can grow faster than a fixed exponential. For FTCS, \(G\) ranges from \(1\) down to \(1 - 4r\), so the condition is \(1 - 4r \geq -1\): \[ r = \frac {\kappa \,\Delta t}{\Delta x^2} \leq \frac 12 . \] If \(r > \frac 12\), the highest mode \(\theta = \pi \), the sawtooth \((-1)^j\), is multiplied at each step by \(1 - 4r < -1\), and any trace of it, even rounding error, grows exponentially and swamps the solution (Software 13.5.1). The restriction \(\Delta t \leq \Delta x^2/2\kappa \) is severe: halving \(\Delta x\) requires four times as many steps.
Theorem 13.2.1 (Convergence of FTCS). Let \(u\) be a solution of \(u_t = \kappa u_{xx}\) on \([0, 1] \times [0, T]\) with bounded derivatives \(u_{tt}\) and \(u_{xxxx}\), and boundary values imposed exactly. If \(r \leq \frac 12\), the FTCS solution with exact initial values satisfies \[ \max _{j,\,n\Delta t \leq T}|u_j^n - u(x_j, t_n)| \leq T\Bigl (\frac {\Delta t}2\max |u_{tt}| + \frac {\kappa \Delta x^2}{12}\max |u_{xxxx}|\Bigr ) . \]
Proof. Let \(e_j^n = u_j^n - u(x_j, t_n)\). The exact solution satisfies the scheme up to \(\Delta t\,\tau _j^n\), where by Taylor’s theorem \(|\tau _j^n| \leq \frac {\Delta t}2\max |u_{tt}| + \frac {\kappa \Delta x^2}{12}\max |u_{xxxx}| =: \tau \). Subtracting, \[ e_j^{n + 1} = re_{j - 1}^n + (1 - 2r)e_j^n + re_{j + 1}^n - \Delta t\,\tau _j^n . \] When \(r \leq \frac 12\) all three coefficients are non-negative and sum to \(1\), so \(\max _j|e^{n + 1}| \leq \max _j|e^n| + \Delta t\,\tau \). Since \(e^0 = 0\), \(\max _j|e^n| \leq n\Delta t\,\tau \leq T\tau \). □
The proof used only that the new value is a convex combination of old ones: a discrete maximum principle, the counterpart of Theorem 6.2.1. The general statement is the Lax equivalence theorem (Lax and Richtmyer, 1956): for a consistent linear scheme for a well-posed linear problem, stability is necessary and sufficient for convergence.
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