
LaxFriedrichs Stability
Define the LaxFriedrichs scheme for $\displaystyle u_t + a u_x = 0,\; a>0$ as
$\displaystyle v^{n+1}_m = \frac{1}{2}(v_{m+1}^n + v_{m1}^n)  \dfrac{a\lambda}{1 + a^2\lambda^2}(v_{m+1}^n  v_{m1}^n)$ where $\displaystyle \lambda = \frac{k}{h}$, $\displaystyle h$ is the grid spacing in the $\displaystyle x$direction, and $\displaystyle k$ is the grid spacing in the $\displaystyle t$direction.
To determine the stability I first transformed the LaxFriedrichs scheme from spacial domain into the frequency domain, giving me
$\displaystyle g(\theta) = \frac{1}{2}\left(e^{i\theta} + e^{i\theta}\right)  \dfrac{a\lambda}{1 + a^2\lambda^2}\left(e^{i\theta}  e^{i\theta}\right), \text{ where } \theta = \xi h$
$\displaystyle g(\theta) = \cos \theta  2\dfrac{a\lambda}{1 + a^2\lambda^2} i \sin \theta$
For this scheme to be stable we must require $\displaystyle g \le 1 \Rightarrow g^2 \le 1$
$\displaystyle \Rightarrow \cos^2 \theta + 4\dfrac{a^2\lambda^2}{(1 + a^2\lambda^2)^2} \sin^2 \theta \le 1$
And this is where I am stuck. The instructor claims that the LaxFriedrichs scheme is unconditionally stable, but I don't see it. In the literature that I have looked at they claim that the above equation is equivalent to
$\displaystyle 4\dfrac{a^2\lambda^2}{(1 + a^2\lambda^2)^2} \le 1$
Which doesn't seem to imply unconditional stability. I also don't necessarily see where that came from. Any help would be appreciated. Thank you in advance.
EDIT: Thanks to Ackbeet, I see how the inequalities are equivalent. (And I feel stupid for not seeing it)
EDIT2: $\displaystyle 4a^2\lambda^2 \le 1 + 2a^2\lambda^2 +a^4\lambda^4$
$\displaystyle 0 \le 1  2a^2\lambda^2 +a^4\lambda^4 \Rightarrow (a^2\lambda^2  1)^2 \ge 0$ Thus, LaxFriedrichs is unconditionally stable.