I've got this numerical scheme for the 1D linear advection equation $u_t + au_x = 0$: $$\dfrac{U_{j, n+1} - U_{j, n}}{\Delta t} + a \dfrac{U_{j+2, n} - U_{j+1, n}}{\Delta x} = 0.$$ Tried von Neumann analysis but got $a \Delta t / \Delta x \neq 0 \Leftrightarrow \lambda > 1$. If I understand correctly, the CFL condition is $a < 0$, $a\Delta t/\Delta x \in [-1, 0)$. So what are other available analyses to determine the range of $a\Delta t/\Delta x$ so that the scheme will be stable? Thanks in advance.
Edit: tried error analysis that denote the error at $x_j, t_{n+1} = e_{j, n+1} = U_{j, n+1} - u(x_j, t_{n+1})$, and then $|e_{j, n+1}| \leq (1 + k)\max(|e_{j, n}|, |e_{j+1, n}|, |e_{j+2, n}|) + |T_{j, n}| \Delta t = \cdots \leq \sum_{i = 0}^n (1 + k)^i |T_{j, n}|\Delta t$, where $k = -a\Delta t/\Delta x > 0$, and $T_{j, n}$ is the time-related truncation error, but I cannot make sure the $\sum_{i = 0}^n (1 + k)^i$ is bounded...
I think the discretized PDE should read for $a<0$ $$ \frac{U_j^{n+1} - U_j^n}{\Delta t} + a \frac{U^n_{\color{red}{j + 1}} - U^n_{\color{red} j }}{\Delta x} = 0.$$
A detailed proof why the CFL condition $$ | a | \frac{\Delta t}{\Delta x} \leq 1$$ implies stability is given in this Lecture Notes, Lemma 2.3. This proof is for periodic or zero boundary conditions.