Consider the following Cauchy (periodic) PDE, where $x\in (0,1]$. $$u_t = u_x, \quad u(x,0) = f(x)$$ The characteristics of the PDE would be of the form $x+t =c$ ($c$ a constant). The "wave" propagates from right to left and the speed of propagation is $dx/dt = -1$.
A von Neumann analysis of the three centered, forward, and backward semi-discretized numerical schemes for this PDE, $$ (u_i)_t = \frac{1}{2h}(u_{i+1}-u_{i-1})$$ $$(u_i)_t = \frac{1}{h} (u_{i+1}-{u_i})$$ $$ (u_i)_t = \frac{1}{h} (u_i-u_{i-1}) $$ where $h =\Delta x$, reveals that the first two schemes are stable and the third one is unstable.
Is there any relationship between the wave propagation direction and these results? The direction in which the information propagates seems to have an effect on the stability of a numerical scheme (in particular on these three ones.) If so, I'd appreciate an (intuitive) explanation.