Suppose that we are given a numerical scheme.
In order to find the CFL condition , we set $U_j^n= \lambda ^ne^{ik x_j}$ and put it into the numerical scheme.
I have shown that the given method is unstable since $|\lambda|>1$.
In this case, how do we calculate the CFL condition?
I get $| \lambda |= \sqrt{1+\gamma^2 \nu^2 \sin^2{(kh)}}$. This can't be $<1$. What do we do in this case?
The given equation is $u_t+ \gamma u_x=0, \gamma>0$. We know that $k \in \mathbb{R}, \nu=\frac{\tau}{h}$ where $\tau$ is the step of the discretization of time.
EDIT: Also suppose that we have a numerical scheme , put into it $U_j^n= \lambda^n e^{ik x_j}$ and get $a_i^n \nu \leq 1$. Can we then say that the method is stable if and only if $a_i^n \nu \leq 1$ ?