Courant–Friedrichs–Lewy condition

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In mathematics, the Courant–Friedrichs–Lewy condition (CFL condition) is a condition for convergence while solving certain partial differential equations (usually hyperbolic PDEs) numerically. It arises when explicit time-marching schemes are used for the numerical solution. As a consequence, the timestep must be less than a certain time in many explicit time-marching computer simulations, otherwise the simulation will produce wildly incorrect results. The condition is named after Richard Courant, Kurt Friedrichs, and Hans Lewy who described it in their 1928 paper^[1]^[2].

For example, if a wave is crossing a discrete grid, then the timestep must be less than the time for the wave to travel adjacent grid points. As a corollary, when the grid point separation is reduced, the upper limit for the time step also decreases.

The CFL condition is commonly prescribed for those terms in PDEs which represent advection (hyperbolic part of the PDE). For one-dimensional case, the CFL condition is given by

$\frac {u \cdot \Delta\,t} {\Delta\,x} < C$ ^[3]^[4]

where

u

is the velocity (L/T)

$\Delta\,t$ is the time step (T)

$\Delta\,x$ is the length interval (L),

and the constant $C$ depends on the particular equation to be solved and not on $Δ t$ and $Δ x .$ The number $\nu = \frac {u \cdot \Delta\,t} {\Delta\,x}$ is called the Courant number.

In the two-dimensional case this becomes ^[4]

$\frac {u_ x \cdot \Delta\,t} {\Delta\,x} + \frac {u_ y \cdot \Delta\,t} {\Delta\,y} < C.$

The CFL condition can be a very limiting constraint on the time step $Δ t,$ to the extent that for certain fourth-order nonlinear partial differential equations it can be of the form

$\frac{\Delta t}{\Delta x^4} < C,$

and efforts are often made to avoid it by using implicit methods.

[edit] References

^ R. Courant, K. Friedrichs and H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik, Mathematische Annalen, vol. 100, no. 1, pages 32–74, 1928.
^ R. Courant, K. Friedrichs and H. Lewy, "On the partial difference equations of mathematical physics", IBM Journal, March 1967, pp. 215-234, English translation of the 1928 German original, download available here
^ TriGrid, oceanic modelling
^ ^a ^b http://lcd-www.colorado.edu/~zuev/R/STUFF_2002/JULIA/courant.pdf Live in Feb 2007; dead in March 2007, but may be restored