Good day :)
I'm trying to show for the advection equation $q_t+uq_x=0$ with $u\geq 0$, for which the
- downwind scheme $Q_i^{n+1} = Q_i^n-\frac{u\Delta t}{\Delta x}(Q_{i+1}^n-Q_i^n)$
- centred scheme $Q_i^{n+1} = Q_i^n-\frac{1}{2}\frac{u\Delta t}{\Delta x}(Q_{i-1}^n-Q_{i+1}^n)$
are both unstable under the CFL condition $|\frac{u\Delta t}{\Delta x}|\leq 1$ and under the discrete $L_1$ norm given by $||Q^{n+1}||_1=\Delta x\sum_i|Q^{n+1}_i|$.
I'm always ending with the stability.
Edit: Let's say I want to show stability of the upwind method $Q_i^{n+1}=Q_i^n-\frac{u\Delta t}{\Delta x}(Q_i-Q_{i-1}^n)$ with the discrete $L_1$ norm:
$$\begin{align*} ||Q^{n+1}||_1&=\Delta x\sum_i|Q_i^{n+1}|\\ &=\Delta x\sum_i|Q_i^n-\frac{u\Delta t}{\Delta x}(Q_i-Q_{i-1}^n)|\\ &=\Delta x\sum_i|(1-\frac{u\Delta t}{\Delta x})Q_i^n+\frac{u\Delta t}{\Delta x}Q_{i-1}^n|\\ &\leq\Delta x\left((1-\frac{u\Delta t}{\Delta x})\sum_i|Q_i^n|+\frac{u\Delta t}{\Delta x}\sum_i|Q_{i-1}^n|\right)\\ &=\left((1-\frac{u\Delta t}{\Delta x})||Q^n||_1+\frac{u\Delta t}{\Delta x}||Q^n||_1\right)\\ &=||Q^n||_1 \end{align*} $$
Doing the same for the two above also yields $||Q^{n+1}||_1\leq||Q^n||_1$. So something is wrong in my calculation...
In these notes (Lemma 2.3) there is a proof that shows that the central scheme is unconditionally unstable for zero or periodic boundary conditions. There, stability is not measured in terms of the $L^1$ but in the $L^2$-like energy-norm $$\Vert Q \Vert_{L^2}^2 := \frac{\Delta x}{2} \sum_i Q_i^2 \quad .$$ In a very similar way, you can show that the Downwind method is also unconditionally unstable. Multiply the Downwind scheme with $Q_i^n$ and use the identity $d_2(d_1 - d_2) = \frac{1}{2} \Big( d_1^2 - d_2^2 - (d_1 - d_2)^2\Big)$ also for the $(Q_{i+1}^n - Q_i^n )Q_i^n$ term: \begin{align} \frac{(Q_i^{n+1})^2}{2} = \frac{(Q_i^{n})^2}{2} + \frac{(Q_i^{n+1} - Q_i^n)^2}{2} + \frac{u \Delta t}{\Delta x} \Bigg[ \frac{(Q_{i+1}^n - Q_i^n)^2}{2} + \frac{(Q_i^n)^2}{2} - \frac{(Q_{i+1}^n)^2}{2}\Bigg] \end{align} now sum over $i= 1 \dots N$ and multiply by $\Delta x$ to obtain \begin{align} \Vert Q_i^{n+1} \Vert_2^2 &= \Vert Q_i^{n}\Vert_2^2 + \Delta x \sum_{i=1}^N \Bigg( \frac{(Q_i^{n+1} - Q_i^n)^2}{2} \Bigg) \\ &+ u \Delta t \Bigg[ \sum_{i=1}^N \Bigg( \frac{(Q_{i+1}^n - Q_i^n)^2}{2} \Bigg) + \sum_{i=1}^N \Bigg( \frac{(Q_i^n)^2}{2} - \frac{(Q_{i+1}^n)^2}{2} \Bigg) \Bigg] \\ &= \Vert Q_i^{n}\Vert_2^2 + \Delta x \sum_{i=1}^N \Bigg( \frac{(Q_i^{n+1} - Q_i^n)^2}{2} \Bigg) \\ &+ u \Delta t \Bigg[ \sum_{i=1}^N \Bigg( \frac{(Q_{i+1}^n - Q_i^n)^2}{2} \Bigg) + \frac{(Q_1^n)^2 - (Q_{N+1}^n)^2}{2} \Bigg] \end{align} Now for periodic $Q_1 = Q_{N+1}$ or identical $Q_1 = Q_{N+1}$ boundary conditions, you have the estimate $$\Vert Q_i^{n+1} \Vert_2^2 \geq \Vert Q_i^{n} \Vert_2^2$$ for all choices $\Delta t, \Delta x >0$ and $u\geq 0$.
In your case, if you want to try to work with the $L^1$-norm I think you simply lack the tools to proceed further. Consider e.g. the downwind: \begin{align} ||Q^{n+1}||_1&=\Delta x\sum_i|Q_i^{n+1}|\\ &=\Delta x\sum_i \bigg|Q_i^n-\frac{u\Delta t}{\Delta x}(Q_{i+1} -Q_{i}^n)\bigg|\\ &=\Delta x\sum_i\bigg|\bigg(1+\frac{u\Delta t}{\Delta x}\bigg)Q_i^n- \frac{u\Delta t}{\Delta x}Q_{i+1}^n\bigg| \end{align} Here it becomes difficult to proceed further, at least I do not know how. The energy / $L^2$ norm circumvents the issues of the absolute value.