# Courant–Friedrichs–Lewy condition

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:$:[itex] - C=c\frac{\Delta t}{\Delta x} \leq 1 + C=c\frac{\Delta t}{\Delta x} \leq C_{max}$ [/itex] (2)
(2)

# Common

It is an important stability criterion for hyperbolic equations.

## The one-dimensional case

For one-dimensional case, the CFL has the following form:

 $C=c\frac{\Delta t}{\Delta x} \leq C_{max}$ (2)

where C is called the Courant number

where the dimensionless number is called the Courant number,

• $u$ is the velocity (whose dimension is Length/Time)
• $\Delta t$ is the time step (whose dimension is Time)
• $\Delta x$ is the length interval (whose dimension is Length).

The value of $C_{max}$ changes with the method used to solve the discretised equation. If an explicit (time-marching) solver is used then typically $C_{max} = 1$. Implicit (matrix) solvers are usually less sensitive to numerical instability and so larger values of $C_{max}$ may be tolerated.

Courant, R., K. O. Fredrichs, and H. Lewy (1928), "Uber die Differenzengleichungen der Mathematischen Physik", Math. Ann, vol.100, p.32, 1928.

Anderson, Lohn David (1995), "Computational fluid dynamics: the basics with applications", McGraw-Hill, Inc.