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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
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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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6
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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7
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