The Courant-Friedrichs-Levy (CFL) Stability ... - ETH - ETH Zürich

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The Courant-Friedrichs-Levy (CFL)
Stability Criterion
Christoph Schär
Institut für Atmosphäre und Klima, ETH Zürich
http://www.iac.ethz.ch/people/schaer
Schär, ETH Zürich

Interpretation of Courant number
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Courant number:
! =u"t "x :
j-2 j-1
α = 1.5
α = 1
α = 0.5
j j+1 j+2
x
The Courant number is the dimensionless transport per time step (in units of Δx)
Schär, ETH Zürich

CFL criterion for Leapfrog scheme
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!"
Equation:
!t + u !"
!x = 0
n
Scheme: ! +1 i = ! n–1 i –" ! i +1
( n n
#! ) i–1 with " = u$t
$x
n+1
t
physical domain
of dependence
n
n-1
numerical domain
of dependence
n-2
Schär, ETH Zürich
i-3 i-2 i-1 i i+1 i+2 i+3
x

CFL criterion for Leapfrog scheme
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!"
Equation:
!t + u !"
!x = 0
n
Scheme: ! +1 i = ! n–1 i –" ! i +1
( n n
#! ) i–1 with " = u$t
$x
t
n+1
physical domain
of dependence
n
n-1
n-2
unstable
|uΔt/Δx| > 1
stable
|uΔt/Δx| ≤ 1
numerical domain
of dependence
Schär, ETH Zürich
i-3 i-2 i-1 i i+1 i+2 i+3
x

CFL criterion for Upstream scheme
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!"
Equation:
!t + u !"
!x = 0
! n+1 i = ! n i "# ! n n
Scheme:
i "! i"1
( ) with # = u $t
$x
t
n+1
physical domain
of dependence
n
n-1
numerical domain
of dependence
n-2
Schär, ETH Zürich
i-3 i-2 i-1 i i+1 i+2 i+3
x

CFL criterion for Upstream scheme
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Equation:
t
n+1
!"
!t + u !"
!x = 0
physical domain
of dependence
Scheme:
! n+1 i = ! n i "# (! n n
i "! i"1 ) with # = u $t
$x
n
n-1
n-2
unstable
uΔt/Δx > 1
stable
0 ≤ uΔt/Δx ≤ 1
unstable
uΔt/Δx < 0
numerical domain
of dependence
Schär, ETH Zürich
i-3 i-2 i-1 i i+1 i+2 i+3
x

Summary
Courant-Friedrichs-Levy (CFL) principle:
The physical domain of dependence must be contained in
the numerical domain of dependence!
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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
u !t
!x "1 valid for a large category of schemes
c !t
!x "1 with c = max u adv + c group
( )
c group =
g H
Schär, ETH Zürich
Example 2: non-hydrostatic, compressible
atmosphere: sound waves
c group = 331 ms -1
T 273.15 K