1

Schr, ETH Zrich

The Courant-Friedrichs-Levy (CFL) Stability Criterion

Christoph Schr Institut fr Atmosphre und Klima, ETH Zrich

http://www.iac.ethz.ch/people/schaer

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Schr, ETH Zrich

Interpretation of Courant number

!=u"t "x :Courant number:

The Courant number is the dimensionless transport per time step (in units of x)

x j j+1 j+2 j-2 j-1

= 1.5 = 1 = 0.5

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Schr, ETH Zrich

t

n+1

n

n-1

n-2

x i-3

i-2

i-1

i

i+1

i+2

i+3

CFL criterion for Leapfrog scheme

numerical domain of dependence

!"!t + u

!"!x = 0Equation:

!in+1 = !i

n1 " !i+1n #!i1n( ) with "= u$t$xScheme:

physical domain of dependence

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Schr, ETH Zrich

t

n+1

n

n-1

n-2

x i-3

i-2

i-1

i

i+1

i+2

i+3

CFL criterion for Leapfrog scheme

numerical domain of dependence

physical domain of dependence

!"!t + u

!"!x = 0Equation:

!in+1 = !i

n1 " !i+1n #!i1n( ) with "= u$t$xScheme:

stable |ut/x| 1

unstable |ut/x| > 1

5

Schr, ETH Zrich

t

n+1

n

n-1

n-2

x i-3

i-2

i-1

i

i+1

i+2

i+3

CFL criterion for Upstream scheme

numerical domain of dependence

!"!t + u

!"!x = 0Equation:

!in+1 = !i

n "# !in "!i"1

n( ) with #= u$t$xScheme:

physical domain of dependence

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Schr, ETH Zrich

t

n+1

n

n-1

n-2

x i-3

i-2

i-1

i

i+1

i+2

i+3

CFL criterion for Upstream scheme

numerical domain of dependence

physical domain of dependence

!"!t + u

!"!x = 0Equation:

stable 0 ut/x 1

unstable ut/x > 1

unstable ut/x < 0

!in+1 = !i

n "# !in "!i"1

n( ) with #= u$t$xScheme:

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Schr, ETH Zrich

Summary Courant-Friedrichs-Levy (CFL) principle:

The physical domain of dependence must be contained in the numerical domain of dependence!

Is a necessary condition for stability, but not a sufficient condition!

CFL criteria:

Generalized form: Velocity represents the largest velocity in the system, at which information can propagate. May include advective transport and wave propagation:

Example 1: shallow water system: gravity waves

Example 2: non-hydrostatic, compressible atmosphere: sound waves

u !t!x "1 valid for a large category of schemes

c !t!x "1 with

c =max uadv + cgroup( )

cgroup = 331 ms-1 T 273.15 K

cgroup = g H