Lax-Friedrichs scheme stability necessary sufficient We have the advection equation $q_t+aq_x = 0$ with constant velocity $a$ and the Lax-Friedrichs scheme
$Q_j^{n+1}=\frac{1}{2}(Q_{j-1}^n+Q_{j+1}^n)+\frac{ak}{2h}(Q_{j-1}^n-Q_{j+1}^n)$.
How to show that $|\frac{ak}{h}|\leq 1$ is necessary for stability and sufficient for stability in the discrete $L^1$ norm?
Edit:
The discrete $L^1$ norm is defined as
$\|Q^{n+1}\|_1 = h\sum\limits_j|Q_j^{n+1}|$.
So far I've noted the above as:
$\|Q^{n+1}\|_1=h\sum\limits_j\left|\frac{1}{2}(1-\frac{ak}{h})Q_{j-1}^n + \frac{1}{2}(1+\frac{ak}{h})Q_{j+1}^n\right|$
How should I take it from here?
 A: Necessity:
The stated condition is precisely the CFL condition.
Sufficiency:
It appears that you have a couple of sign errors in your final expression. It should read
\begin{align}
\| Q^{n+1} \|_1 &= h \sum_j \left| \frac{1}{2} \left( 1 + \frac{ak}{h} \right) Q^n_{j-1} + \frac{1}{2} \left( 1 - \frac{ak}{h} \right) Q^n_{j+1} \right| \\
%
&\leq \frac{h}{2} \left[ \sum_j \left| \left( 1 + \frac{ak}{h} \right) Q_{j-1}^n \right| + \sum_j \left| \left( 1 - \frac{ak}{h} \right) Q_{j+1}^n \right| \right]
\end{align}
Now, suppose that $\left| \frac{ak}{h} \right| \leq 1$. Then it holds that
$$
\left( 1 - \frac{ak}{h} \right) \geq 0 \quad \text{and} \quad \left( 1 + \frac{ak}{h} \right) \geq 0.
$$
Thus, these terms can be extracted from the absolute values to get
\begin{align}
\| Q^{n+1} \|_1 &\leq \frac{h}{2} \left[ \left( 1 + \frac{ak}{h} \right) \sum_j \left| Q_{j-1}^n \right| + \left( 1 - \frac{ak}{h} \right) \sum_j \left| Q_{j+1}^n \right| \right] \\
%
&= \frac{1}{2} \left[ \left( 1 + \frac{ak}{h} \right) \| Q^n \|_1 + \left( 1 - \frac{ak}{h} \right) \| Q^n \|_1 \right] \\
%
&= \| Q^n \|_1.
\end{align}
The method is therefore stable.
