Define the Lax-Friedrichs scheme for as

where , is the grid spacing in the -direction, and is the grid spacing in the -direction.

To determine the stability I first transformed the Lax-Friedrichs scheme from spacial domain into the frequency domain, giving me

For this scheme to be stable we must require

And this is where I am stuck. The instructor claims that the Lax-Friedrichs 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

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:

Thus, Lax-Friedrichs is unconditionally stable.