Lax-Wendroff method for linear advection - Stability analysis

Recall how the Lax-Wendroff method is obtained in the constant-speed case [1]:

• a Taylor series in time is written: $$ u(x,t_{n+1}) = u(x,t_{n}) + \Delta t\, u_t(x,t_{n}) + \frac{1}{2}\Delta t^2\, u_{tt}(x,t_{n}) + \dots $$
• the time derivatives are eliminated using the PDE: $u_t = -c u_x$ and $u_{tt} = c^2 u_{xx}$.
• the spatial derivatives are replaced by central finite-difference approximations.

Thus, the following scheme is obtained: $$ u_j^{n+1} = u_j^n - c\Delta t \frac{D_x u_j^n}{2 \Delta x} + \frac{c^2\Delta t^2}{2} \frac{\delta_x^2 u_j^n}{\Delta x^2} \, . $$ This method can be adapted to the variable velocity case.

Let us analyze the stability of the Lax-Wendroff scheme above (constant-speed case). Assuming a perturbation of the form $u_j^{n} = \xi^n \text{e}^{\text i k x_j}$, one has \begin{aligned} \xi &= 1 - c \Delta t \frac{\text{e}^{\text i k \Delta x} - \text{e}^{-\text i k \Delta x}}{2 \Delta x} + \frac{c^2 \Delta t^2}{2} \frac{\text{e}^{\text i k \Delta x} - 2 + \text{e}^{-\text i k \Delta x}}{\Delta x^2} \\ &= 1 - \text{i} \kappa \sin (k\Delta x) + \kappa^2 \left(\cos (k\Delta x) - 1\right) . \end{aligned} where $\kappa = c\frac{\Delta t}{\Delta x}$ is the Courant number. Thus, the squared modulus of the amplification factor is \begin{aligned} |\xi|^2 &= (1 - \kappa^2 (1-\cos(k\Delta x)))^2 + \kappa^2 (1-\cos^2(k\Delta x))\\ &= 1 - 2\kappa^2 (1-\cos(k\Delta x)) + \kappa^2 (1-\cos^2(k\Delta x)) + \kappa^4 (1-\cos(k\Delta x))^2 \\ &= 1 - \kappa^2 (1-\kappa^2) (1-\cos(k\Delta x))^2 \\ &= 1 - 4\kappa^2 (1-\kappa^2) \sin^4\left(\tfrac{1}{2}k\Delta x\right) . \end{aligned} Finally, the Lax-Wendroff scheme is Neumann-stable provided that $|\xi|^2 \leq 1$, which implies that the CFL condition $|\kappa| \leq 1$ is satisfied.

[1] R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.