Consider this problem
Apply the Fourier stability test to the difference equation $$ u_m^{n+1}=\frac{1}{2}\left(2-5 \mu+6 \mu^2\right) u_m^n+\frac{2}{3} \mu(2-3 \mu)\left(u_{m-1}^n+u_{m+1}^n\right)-\frac{1}{12} \mu(1-6 \mu)\left(u_{m-2}^n+u_{m+2}^n\right), $$ where $m \in \mathbb{Z}$. Deduce that the test is satisfied if and only if $0 \leq \mu \leq \frac{2}{3}$.
This looked relatively innocent until I calculated $$ H(\theta) = \frac{1}{6}(6-16\mu + 24 \mu^2)-\frac{1}{3}\mu (1-6 \mu)\left(\sin \theta -\frac{2(2-3\mu}{1-6\mu}\right)^2+ \frac{4}{3} \mu \frac{(2-3 \mu)^2}{1-6\mu}. $$ The method requires us to show that $| H| \leq 1$ if and only if $\mu$ is in the above range. This seems like it’s going be very hard as the issue is that depending on the value of $\mu$ I have to take $\sin $ accordingly to maximise. Moreover, I lack the absolute value which will surely provide difficulties.
Question: How do I proceed?
Reference: Here is a reference for the method
