Talk:Courant–Friedrichs–Lewy condition

From Wikipedia, the free encyclopedia

[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)

[edit] Examples

First, stiff equation has a nice image (Image:StiffEquationNumericalSolvers.svg) that it sounds like would be relevant here. (Or is that showing numerical stability rather than convergence?) —Ben FrantzDale 17:28, 14 April 2007 (UTC)

Second, is the following an application of the Courant–Friedrichs–Lewy condition?: In molecular dynamics, the computer tracks the position of atoms as they move over time. If two atoms are moving towards each other on a collision course, if the time step were too large, they would pass through each other. —Ben FrantzDale 17:28, 14 April 2007 (UTC)

Third, a similar issue comes up when applying an energy-minimization algorithm (such as conjugate gradient) to a block of atoms, particularly when you want to find the local energy minimum, not the global one. One of the steps is a line search (in 3n-dimensional configuration space). Even though this isn't dynamics, you have a similar problem in that if you move too fast in your line search, you could move away from the local energy minimum. In the case of two atoms in a 1-dimensional box with repulsive ends:

[   *       * ]
    x       y

The configuration space is two-dimensional but the x = y line has infinite energy. Starting from this configuration:

[         *  *]
          x  y

the residual force is greater on atom y, so a line search with an overly-large step size could lead to this:

[   *    *    ]
    y    x

even though that involved passing through x = y. Is this step-size restriction the same as the Courant–Friedrichs–Lewy condition or is this something else? —Ben FrantzDale 17:28, 14 April 2007 (UTC)

Why do you think that any of the topics you mention is relevant here? As far as I know, the term "CFL condition" is only used in the context of numerical PDE methods, mostly finite difference but also finite volume. In fact, I'm considering removing all the links pointing to this page which you added recently. -- Jitse Niesen (talk) 14:11, 16 April 2007 (UTC)
As I understand it, this is a constraint on time stepping algorithms to make sure they follow physically plausible paths. As a result, I think it is related to time integration and to solving stiff equations. If I am mistaken, feel free to remove those links I added. —Ben FrantzDale 16:20, 16 April 2007 (UTC)