Lax–Wendroff method

Remarks

For $f(u) = a u$, the two-step form reduces to the scheme of clause 1. The first step is a Lax–Friedrichs step over half a time step on the staggered grid. Example: for clause 1, the Fourier mode $e^{i j \xi}$ is multiplied by $g = 1 - i \nu \sin \xi - \nu^2 (1 - \cos \xi)$, and a short computation gives $\abs{g}^2 = 1 - 4 \nu^2 (1 - \nu^2) \sin^4 (\xi / 2)$. So $\abs{g} \le 1$ for all $\xi$ exactly when $\abs{\nu} \le 1$, and the scheme is stable under this condition. Taylor expansion shows that the local truncation error is $O(\Delta x^3)$ for fixed $\nu$, so the method is second-order accurate. Non-example: dropping the last term gives the forward-time centred-space scheme, which is only first order in time and has $\abs{g}^2 = 1 + \nu^2 \sin^2 \xi > 1$ for $\sin \xi \ne 0$, so it is unstable.