I hope you can agree that the Lax-Wendroff scheme for the $3$-dimensional wave equation (or consider any system of $4$ linear conservation laws ) can be written as: \begin{align*} Q_{i,j,k}^{n+1}&= \Big(I- \frac{\Delta t^2}{\Delta x^2}A^2-\frac{\Delta t^2}{\Delta y^2}B^2-\frac{\Delta t^2}{\Delta z^2}C^2\Big)Q_{i,j,k}^{n} \\ & \ +(\frac{\Delta t^2}{2\Delta x^2}A^2- \frac{\Delta t}{2\Delta x}A) Q_{i+1, j, k}^n- (\frac{\Delta t^2}{2\Delta x^2}A^2- \frac{\Delta t}{2\Delta x}A)Q_{i-1, j, k}^n \\ &\ +(\frac{\Delta t^2}{2\Delta y^2}B^2 - \frac{\Delta t}{2\Delta y}B) Q_{i, j+1, k}^n- ( \frac{\Delta t^2}{2\Delta y^2}B^2- \frac{\Delta t}{2\Delta y}B)Q_{i, j-1, k}^n \\ &\ + (\frac{\Delta t^2}{2\Delta z^2}C^2 - \frac{\Delta t}{2\Delta z}C)Q_{i, j, k+1}^n- (\frac{\Delta t^2}{2\Delta z^2}C^2- \frac{\Delta t}{2\Delta z}C)Q_{i, j, k-1}^n \\ &\quad + \frac{\Delta t^2}{8\Delta x\Delta y}(AB + BA)(Q_{i+1, j+1, k}^n- Q_{i-1, j+1, k}^n)- (Q_{i+1, j-1, k}^n- Q_{i-1, j-1, k}^n) \\ & \quad + \frac{\Delta t^2}{8\Delta x\Delta z}(AC + CA)(Q_{i+1, j, k+1}^n- Q_{i-1, j, k+1}^n)- (Q_{i+1, j, k-1}^n- Q_{i-1, j, k-1}^n) \\ &\quad + \frac{\Delta t^2}{8\Delta z\Delta y}(CB + BC)(Q_{i, j+1, k+1}^n- Q_{i, j+1, k-1}^n)- (Q_{i, j-1, k+1}^n- Q_{i, j-1, k-1}^n) \end{align*} Where $Q_{i,j,k}^{n+1}\in \mathbb{R}^4$ is a vector of $4$ components, at the updated one step time $t_{n+1}= (n+1)\Delta t$ is given as linear combination of values of a $19$-points stencil in the space, And $A, B, C$ are constant $4\times 4$ matrices.
This same method can be viewed as a finite volume method for updating the cell average $Q_{i,j,k}$ , resulting from defining numerical fluxes at the six faces of the cell in a natural way, based on the $19$ nearby cell values. [1]
My question: How would we conduct Von Neumann stability analysis for this scheme ?
We can have $Q^{n+1}e^{I(i\theta+ j \phi + k\psi)}= G Q^ne^{I(i\theta+ j \phi + k\psi)}$ but $G$ here is a $4\times 4$ matrix and then how can we describe the condition $|G|\leq 1$ ?
Is it even plausible to claim this ?
For the case of a single differential equation: $|G|$ is the module of a complex number, and $|G|\leq 1$ is a sufficient condition for stability of the scheme.
Does is make sense to assume that the amplification factor $G$ have to satisfy $|Q^{n+1}|= |G Q^n|\leq |Q^n|$ and $|G|$ should be the induced norm from the $2$-norm on $\mathbb{R}^4$ ?
Is there any reference on how to do stability analysis for systems of more than one equation?
[1] R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.
Pick a propagation direction defined by the unit vector ${\bf n} = (n_x, n_y, n_z)$. Next, inject the Fourier Ansatz $$ {\bf Q}_{i,j,k}^n = \hat {\bf Q}\; \text{e}^{\text i (\omega t_n - \kappa {\bf n}\cdot {\bf x}_{i,j,k})} $$ where $t_n = n \Delta t$ and ${\bf x}_{i,j,k} = (i \Delta x, j \Delta y, k \Delta z)$ is the position vector. Here, $\text i$ denotes the imaginary unit; $\omega$, $\kappa$ represent the angular frequency and wavenumber, respectively. It follows that $$ {\bf Q}_{i,j,k}^{n+1} = G\, {\bf Q}_{i,j,k}^{n}, \quad {\bf Q}_{i+\delta i,j+\delta j,k+\delta k}^{n} = \text{e}^{-\text i \kappa {\bf n}\cdot \delta{\bf x}}\, {\bf Q}_{i,j,k}^{n} $$ where $G = \text{e}^{\text i \omega \Delta t}$ is the (scalar) amplification factor and $\delta {\bf x} = (\delta i \Delta x, \delta j \Delta y, \delta k \Delta z)$. The scheme's updating formula in OP yields the eigenvalue problem $$ \begin{aligned} {\bf 0}= &\left[ {\bf I}- \tfrac{\Delta t^2}{\Delta x^2}{\bf A}^2-\tfrac{\Delta t^2}{\Delta y^2}{\bf B}^2-\tfrac{\Delta t^2}{\Delta z^2}{\bf C}^2 \right.\\ & \ + \left(\tfrac{\Delta t^2}{2\Delta x^2}{\bf A}^2- \tfrac{\Delta t}{2\Delta x}{\bf A}\right) (\text{e}^{-\text i \kappa n_x \Delta x} - \text{e}^{\text i \kappa n_x \Delta x}) \\ &\ + \left(\tfrac{\Delta t^2}{2\Delta y^2}{\bf B}^2 - \tfrac{\Delta t}{2\Delta y}{\bf B}\right) (\text{e}^{-\text i \kappa n_y \Delta y} - \text{e}^{\text i \kappa n_y \Delta y}) \\ &\ + \left(\tfrac{\Delta t^2}{2\Delta z^2}{\bf C}^2 - \tfrac{\Delta t}{2\Delta z}{\bf C}\right) (\text{e}^{-\text i \kappa n_z \Delta z} - \text{e}^{\text i \kappa n_z \Delta z}) \\ &\ + \tfrac{\Delta t^2}{8\Delta x\Delta y}({\bf AB} + {\bf BA}) \left((\text{e}^{-\text i \kappa (n_x\Delta x+n_y\Delta y)}- \text{e}^{-\text i \kappa (-n_x\Delta x+n_y\Delta y)}) + \dots - G\, {\bf I} \right] {\bf Q}_{i,j,k}^{n} \end{aligned} $$ for the amplification factor $G$. Von Neumann stability is ensured if $|G| \leq 1$ for all ${\bf n}$.