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.