Stability requires that the absolute value of the error decreases in each time step, or $$ \left|\frac{\epsilon^{\,n+1}}{\epsilon^{\,n}}\right| < 1 $$ This condition is a stability criterion, the error must decay (or at least not grow) from one time step to the next

Since $\sin^2$ $\leq 1$ the amplification factor is always less than 1, and we find that it is bigger than $-1$ if $$ \frac{\Delta t D}{\Delta x^2} \leq \frac{1}{2} $$ Therefore, for a given diffusion coefficient $D$ and spatial resolution $\Delta x$ the method is stable if the time step $\Delta t$ is small enough

Since the amplification factor has the form $1+i$ , the absolute value of this complex number is always larger than unity and the method is unconditionally unstable for pure advection

We must have $$ \frac{\Delta t D}{\Delta x^2} \leq \frac{1}{2}, \quad \frac{U^2 \Delta t}{D} \leq 2 $$ for the numerical approximation of the discrete approximation of the advection-diffusion equation \(\boxed{\frac{f_{j}^{\,n+1} - f_{j}^{\,n}}{\Delta t} + U \left( \frac{f_{j+1}^{\,n} - f_{j-1}^{\,n}}{2 \Delta x} \right) = D \left( \frac{f_{j+1}^{\,n} - 2 f_{j}^{\,n} + f_{j-1}^{\,n}}{\Delta x^{2}} \right)}\) to remain stable. Notice that high velocity and low viscosity lead to instability according to the second restriction