Multiplying the scheme \begin{equation*} \dfrac{v_m^{n+1} - v_m^n}{k} + \dfrac{a}{2}\left(\dfrac{v_{m+1}^{n+1} - v_m^{n+1}}{h} + \dfrac{v_m^n - v_{m-1}^n}{h}\right) = 0 \end{equation*} by $v_m^{n+1} + v_m^n$ and summing over all values of $m$, obtain the relation \begin{equation*} \sum_{m = -\infty}^\infty \left[\left(1 - \frac{a \lambda}{2}\right) |v_m^{n+1}|^2 + \frac{a \lambda}{2} v_m^{n+1} v_{m+1}^{n+1}\right] = \sum_{m = -\infty}^\infty \left[\left(1 - \frac{a \lambda}{2}\right) |v_m^n|^2 + \frac{a \lambda}{2} v_m^n v_{m+1}^n\right]. \end{equation*} Conclude that the scheme is stable for $a\lambda < 1$.
Note that $\lambda = \frac{k}{h}$. Question is from Strikwerda's Finite Difference Schemes and PDEs, page 33.
I obtained second equality but couldn't go further. Without posting the whole solution, tips are welcome.
Thanks for your time.
