Talk:Courant–Friedrichs–Lewy condition

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[edit] Dead link

The mathworld link is dead, no? --anon

Yup. Used to work a while ago. I removed it now. Thanks. Oleg Alexandrov (talk) 01:11, 25 May 2006 (UTC)

[edit] CFL equations

The 'classic' CFL criteria

\frac {u \cdot \Delta\,t} {\Delta\,x} < Constant

has a value for Constant of 1. However, a reviewer pointed out that this may be tied in with first-order, one-dimensional, finite differences, and that the value of Constant will differ with the finite difference stencil. I haven't managed to locate such a reference. I added the explicit criterion of < 1 to the Wiki page, as when I first used it it had no formula, and I then went off to Google to locate a suitable formula.Bendel boy 09:39, 16 February 2007 (UTC)

That C also depends on the exact equation you are trying to solve. So, do the Courant papers references you posted use C=1? Then we can trust that Courant was not wrong in that reference. :) Oleg Alexandrov (talk) 16:11, 16 February 2007 (UTC)
Yes. (When I first added this material, I read your comment to mean 'please post the conclusions from the original CFL paper.' Now I see that you only meant 'the value of the Constant is 1 for the references you looked at, but this is not universal.') But the example limits posted are for the advection equation. The most common occurrences of the usage of CFL are with advection-dispersion equations, where having some idea of the limits on the page is useful. And you have extended further with your fourth-order limit, to indicate that the limit equation as well as the constant is not universal. Thanks.
Courant.pdf does. But this is derivative, not the original reference, so you may wish to argue that it is looking at some form of special case.
The original reference is less clear, as the famous CFL criterion is not explicit in that paper. They introduce the criterion in terms of a second-order equation,
\frac {\partial^ 2 u} {\partial t^ 2} = \frac {\partial^2 u} {\partial x^ 2}
The text then states that if the time is discretised in h and distance discretised as kh then for k<1 the scheme will not converge as h tends to 0 (p. 228, Section 2, in the English translation). I.e., the distance increment must always be greater than the time increment. So, here we have an explicit statement that
\Delta\,x > \Delta\,t
or
\frac {\Delta\,t} {\Delta\,x} < 1
The paper did not explicitly include a velocity.


For the heat equation,
2 \cdot \frac {\partial u} {\partial t} = \frac {\partial^ 2 u} {\partial x^ 2}
the stability criterion is that \Delta\,t and \Delta\,x^2 should be reduced in proportion. The text is not explicit on a limiting ratio. (See pp. 231-232, Equations 18, 19 and 16. No, it is not a typo on my part: Equation 16 occurs both after Equation 15, and after Equation 19.) So, with a pure diffusion equation not just the value of Constant is in question, because now the CFL criterion becomes
\frac {\Delta\,t} {\Delta\,x^2} < Constant
Bendel boy

Yeah, for the heat equation ut = uxx, it is

\frac {\Delta\,t} {\Delta\,x^2} < C.

I remember that either C=1 or C=1/2. I think C=1 should be enough for your scheme to not blow up (stability) but smaller C (=1/2 or 1/4) is better if you want to make sure your discretized solution is positive, as the analytical solution. Bu those are vague memories.

The moral is that C does depend on the exact equation. For example, if the heat equation is

ut = Kuxx,

then K shows up in the CFL constant.

For that reason, I will now modify the article to replace 1 with C. It is safer that way, and besides, the actual value of C is not as relevant as the fact that C exists. Oleg Alexandrov (talk) 03:31, 17 February 2007 (UTC)

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