ChapterZero

One of my assignments for this week is to show the stability of the Lax-Friedrichs scheme when applied to the nonlinear 1-d advection equation: up to a time T, where we can assume smoothness of the solution up to T, and that , so CFL holds (here is the time increment over the space increment).

I’ve been attempting to follow the analysis in Strang’s paper; he assumes that the first variation of the difference scheme is stable in his proof, but I must show this is true for this particular scheme. While attempting to show this, I ran into an interesting linear algebra problem– one that I doubt I’ll have solved by Thursday, so I have to find another approach– that leads me to make the following conjecture:

Let be tridiagonal hollow matrices such that and all the entries of lie in . Then there is a constant depending only on , such that for any , .

I haven’t a clue how to prove this– the obvious approach with Gerschgorin’s theorem only bounds the norms of each matrix by 2–, but running a couple of lines of Matlab code:

n = 200;
mp = eye(200);
kstop = 2000;

for k=1:kstop
        b = rand(n-1,1);
        m = diag(b, 1) + diag(b,-1);
        mp = mp * m;
        norms(k) = norm(mp);
end

plot(norms);

and varying n, kstop gives pretty convincing results. I also found that if you let the entries get even slightly above 1, say up to 1.1, the norms blowup.

Any ideas on how to prove this? My next idea is to look at the minimax theorem.

Update This conjecture can’t be true– just take to have a spectral radius > 1, then the norms of the products will diverge. So, maybe it’s just true with high probability. Ugh.