Hi I am having problem discussing stability of a finite difference scheme for a pde.
I was trying to solve point (ii) of this exercise: what I tried first is eigenvalue analysis. I wrote the method in the form $$ u'' = P_{\Delta x} u $$ and noticed the matrix $P_{\Delta x}$ is $TST$ with $\frac{-2}{\Delta x} + \alpha$ along the main diagonal and $\frac{1}{\Delta x}$ along the adjacent diagonals. Then I applied the following result
Suppose $P_{\Delta x}$ is normal and $\exists \beta \in \mathbb{R} \; : \; \lambda \leq \beta \;\; \forall \Delta x >0 \;\; \forall \lambda \in \sigma(\frac{1}{2}(P_{\Delta x}+ P^*_{\Delta x})).$ Then the method is stable.
According to this result and knowing the eigenvalues of $TST$ matrices, it looks to me the method is always stable. However the solution says it is stable only for $\alpha \leq 0$. What am I missing?