I am doing a Von Neumann analysis on a staggered finite difference scheme (for Maxwell's Equations).
The finite difference scheme is:
$$ \mathbf{u}_v|^{n+2}_{i,j} - \mathbf{u}_v|^{n}_{i,j} = - A \frac{\lambda_y}{2} \left[ (\mathbf{u}_v|^{n-1}_{i+1,j+1} - \mathbf{u}_v|^{n-1}_{i+1,j-1}) + (\mathbf{u}_v|^{n-1}_{i-1,j+1} - \mathbf{u}_v|^{n-1}_{i-1,j-1}) \right] + B \frac{\lambda_x}{2} \left[ (\mathbf{u}_v|^{n-1}_{i+1,j+1} - \mathbf{u}_v|^{n-1}_{i-1,j+1}) + (\mathbf{u}_v|^{n-1}_{i+1,j-1} - \mathbf{u}_v|^{n-1}_{i-1,j-1}) \right] \\+ \lambda_y^2 (\mathbf{u}_v|^{n}_{i,j+2} - 2\mathbf{u}_v|^{n}_{i,j} + \mathbf{u}_v|^{n}_{i,j-2}) \\+ \lambda_x^2 (\mathbf{u}_v|^{n}_{i+2,j} - 2\mathbf{u}_v|^{n}_{i,j} + \mathbf{u}_v|^{n}_{i-2,j}) $$
where the matrices $\mathbf{u}_v$, $A$ and $B$ are:
$$ A = \begin{bmatrix} 0 & 1 & 0\\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}, B = \begin{bmatrix} 0 & 0 & 1\\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix}, \mathbf{u}_v = \begin{bmatrix} u_1\\ u_2 \\ u_3 \end{bmatrix} $$
and $\lambda_x = \frac{\Delta t}{\Delta x}, \lambda_y = \frac{\Delta t}{\Delta y}$
The values of $\mathbf{u}_1|^n_{i,j}$ are computed with even values of $i,j,n$, and the values of $\mathbf{u}_2|^n_{i,j}, \mathbf{u}_3|^n_{i,j}$ are computed with the odd values of $i,j,n$. The caveat is that you need to know more values for initial conditions.
By applying the Fourier inversion formula, and assuming for simplicity that $\Delta x = \Delta y$ we get (see Strikwerda, Finite Difference Schemes and Partial Differential Equations 2ed.):
$$ G^3 + 4 \{ \lambda^2[\sin^2(\theta_x) + \sin^2(\theta_y)] - 1 \} G = 4i\lambda[A \sin(\theta_y)\cos(\theta_x) - B \sin(\theta_x)\cos(\theta_y)], $$
where $G$ is the amplification matrix. I think I have understood properly the concept of Von Neumann analysis, and I want to derive sufficient stability conditions for $\lambda$.
The problem is that there are too many possible cases to consider for each branch of the roots of the polynomial and for each value of $\theta_x$ and $\theta_y$, and though the explicit value of $G$ may be very hard to find, some good estimates for $\lambda$ may be possible.
