Consider the PDE $$ \partial_t u + a\partial_x u=0, -\infty<x<\infty, t>0$$ $$ u(x,0)=u_0(x), $$
where $a>0$ is a constant.
I want to show that the difference scheme
$$ \frac{U_j^{n+1}-U_j^n}{\Delta t} + a\left( \frac{U_{j+1}^n-U_{j-1}^n}{2\Delta x} \right)=0$$
is not stable. Unfortunately, I did not completely understand what exactly I need to show in order to conclude that this scheme is not stable. I think the stability is related to convergence. From my notes and the textbook, however, it is not clear how to show stability. There is a lot of overloaded notation and I can't find my way into understanding how this works. I'm more of a pure math type person, and computational math sometimes seems quite vague for me. Can someone please help me?
The Lax-Richtmeyer equivalence theorem states that a consistent finite difference method is convergent if and only if the finite difference scheme is stable. Hence it is correct when you say that it is a matter of convergence in the sense that it is enough to exhibit one initial data for which you are able to prove that the numerical solution does not converge to the analytic solution of the PDE.
The standard machinery to prove stability is the so called Von Neumann analysis. Let $\Delta x = h$ and let $$U_j^n = \frac{1}{\sqrt{2\pi}}\int^{\pi/h}_{-\pi/h}e^{-ijh \xi}\widehat{U^n}(\xi)\,d\xi. \tag 1$$ Write the scheme in update form and use $(1)$ to find the relation $$\widehat{U^n}(\xi) = (g(h\xi,\Delta t,\Delta x))^n\widehat{U^0}(\xi).$$Then one can prove the following result.
If you carry on the computation for FTCS you'll get $$g(\theta) = 1 - ia\frac{\Delta t}{\Delta x}\sin\theta.$$ As you can easily verify, $|g(\theta)| > 1$ for $\theta \neq 0$.