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I've been studying this FEM theory and for the parabolic problems, there's the analysis of stability of the $\theta$-method.

I followed the analysis and they get this CFL (Courant-Friedrich-Lewy) condition

$$\lambda_{max} \simeq Ch^{-2}$$

What does this CFL condition means? How can I understand this?

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It isn't at all clear what kind of answer you're asking for. Do you understand what it means for a numerical method to be stable? Do you understand why stability is necessary for a method to be useful in practice? Is there some aspect of the statement or derivation of this stability condition that you don't understand? –  Brian Borchers Jan 22 '13 at 13:15
    
A CFL condition is typically stated as a dependence between the temporal step size $\Delta t$ and a spatial step size $\Delta x$. I'm curious where you found this quantity stated as a CFL condition. Could you provide a reference? –  Paul Jan 22 '13 at 14:42
    
So your question is "why is this being called a CFL condition when we're not in the finite difference setting of the original CFL condition?" –  Brian Borchers Jan 22 '13 at 15:28
    
What I was trying to ask is, if someone ask you "What is the CFL condition for an hyperbolic problem" what would you answer. I mean, I know it is related with the time-step $\Delta t$ and the space-step $\Delta x$. But what's the point? If I make the space-step smaller the approximation on times becomes worse? better? nothing happens? @BrianBorchers It's been a while I took Numerical Analysis and this is the first time reading something about Hyperbolic problems in 1D. Maybe an explanation of stability would help. –  BRabbit27 Jan 22 '13 at 17:09
    
@BRabbit27: Your question says "parabolic", but your comment says "hyperbolic" –  Paul Jan 22 '13 at 17:58

1 Answer 1

The CFL condition states that the "mathematical domain of dependence" must be (asymptotically) contained in the numerical domain of dependence. For hyperbolic problems, this provides a bound $\Delta t < C \Delta x$ that is useful at all resolutions. For a parabolic problem, it merely requires that $\Delta t \in o(\Delta x)$ in the limit $\Delta x \to 0$. That is, $\Delta t$ must go to zero strictly faster than $\Delta x$, which causes information to propagate infinitely fast, thus matching the mathematical (continuum) behavior. You cannot conclude based purely on CFL theory that the time step must indeed go to zero at least as fast as $\Delta x^2$. This result is readily established using von Neumann stability analysis.

I recommend Chapter 4 of Trefethen's Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations for further details on this subject.

Improper usage: The term "CFL" is sometimes misused to refer to whatever is the appropriate sharp stability requirement for an explicit method applied to the problem being considered. Indeed, CFL analysis is too weak to provide $\Delta t \sim\Delta x^2$ for parabolic problems or exotic dispersive wave problems (such as VLF Whistler waves).

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+1 for pointing out the common abuse of "CFL". –  David Ketcheson Jan 23 '13 at 11:05

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