I am studying the 1D wave equation: $$ \frac{\partial^2u}{\partial t^2} - a^2 \frac{\partial^2u}{\partial x^2} = 0 $$
And solving it numerically with this implicit finite differences discretization:
$$ \frac{u^{n+1}_{j}-2u^{n}_{j}+u^{n-1}_{j}}{{\Delta}t^2} - \frac{a^2}{4h^2}( (u^{n+1}_{j+1}-2u^{n+1}_{j}+u^{n+1}_{j-1})+2(u^{n}_{j+1}-2u^{n}_{j}+u^{n}_{j-1})+(u^{n-1}_{j+1}-2u^{n-1}_{j}+u^{n-1}_{j-1}))=0 $$
I'd like to show that this method is stable. I've tried to use the Von Neumman method as applied here but the algebraic manipulations get way too tricky. After getting to a 2nd degree equation like in the link, I am not able to simplify it and show that $|G| < 1$, and thus the method is uncondinally stable. Is there a simpler way to do this. Any pointer to a nice reference on this are very welcome.