# Courant–Friedrichs–Lewy condition

(Difference between revisions)
 Revision as of 11:04, 26 August 2012 (view source)Michail (Talk | contribs)← Older edit Revision as of 11:49, 26 August 2012 (view source)Michail (Talk | contribs) Newer edit → Line 11: Line 11: where C is called the ''Courant number'' where C is called the ''Courant number'' + where the [[dimensionless number]] is called the '''Courant number''', + + *$u$ is the velocity (whose [[Dimensional analysis#Definition|dimension]] is Length/Time) + *$\Delta t$ is the time step (whose [[Dimensional analysis#Definition|dimension]] is Time) + *$\Delta x$ is the length interval (whose [[Dimensional analysis#Definition|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. {{reference-paper  | author=Courant, R., K. O. Fredrichs, and H. Lewy | year=1928 | title=Uber die Differenzengleichungen der Mathematischen Physik | rest=Math. Ann, vol.100, p.32, 1928}} {{reference-paper  | author=Courant, R., K. O. Fredrichs, and H. Lewy | year=1928 | title=Uber die Differenzengleichungen der Mathematischen Physik | rest=Math. Ann, vol.100, p.32, 1928}} {{reference-paper  | author=Anderson, Lohn David | year=1995 | title=Computational fluid dynamics: the basics with applications | rest=McGraw-Hill, Inc}} {{reference-paper  | author=Anderson, Lohn David | year=1995 | title=Computational fluid dynamics: the basics with applications | rest=McGraw-Hill, Inc}}

## Revision as of 11:49, 26 August 2012

It is an important stability criterion for hyperbolic equations.

In common case it's written as

 $C=c\frac{\Delta t}{\Delta x} \leq 1$ (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.