1.
SWAN Advanced Course
4. Numerics in SWAN
Delft Software Days
28 October 2014, Delft

2.
Contents
• Discretization
• Convergence criteria
• Source term stability
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3.
Discretization
• Numerical schemes for propagation (fully implicit):
• x,y-space : upwind: BSBT (1st order), SORDUP (2nd),
Stelling-Leendertse (3rd)
• time : backward
• V-space : hybrid central / upwind (first order upwind too
diffusive, central scheme prone to wiggles)
• T-space : hybrid central / upwind
• Implicit propagation scheme is unconditionally stable: robust
• 1st order scheme is rather diffusive (take care on large distances)
• accuracy = f ('t, 'x, 'y, 'V, 'T)
• iterative (4 sweep) solution technique
• x,y-space: regular, curvi-linear or unstructured grids
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4.
Propagation (x,y-space)
• To allow for energy crossing the quadrants (refraction,
quads, diffraction):
• Iterative procedure
• Computation is stopped when accuracy criteria are met
(specified by user)
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5.
Convergence criteria
Lake George:
2% criteria vs.
fully-converged
Convergence if:
a.
'H ( i ) 0.02 H ( i ) or 'H ( i ) 0.02 H
( average
)
m 0 m 0 m 0 m 0 b.
'T () i 0.02 T () i or 'T () i 0.02 T
( average
)
m 01 m 01 m 01 m 01 c. Conditions a. AND b. are satisfied in 98% of all wet grid points
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6.
Convergence criteria
Hs
DHSIGN
90%-conv. crit.
default 98%-conv. crit.
Hs
Example: 2003 experiment NCEX
(Levi Gorrell)
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7.
Convergence criteria
• Check iteration behaviour of output quantities
• TEST output
• DHS, DRTM01
• Default not always effective o significant inaccuracies
• Either stronger accuracy than default (2%) or use different
convergence criterium: based on curvature
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8.
Convergence criteria
Curvature-based convergence criteria (Zijlema vd Westhuysen 2005)
i i i i
m m m m
1 1
H H T
T drel
H T
' ( ' H ) / H i [ cur v .ma x ]
and 0 0 ,
0 0
@
i i m 0 m 0 i i
m m
1 1
0 0
in more than [npnts] % of wet points
Haringvliet Estuary Lake George
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9.
Convergence enhancing measures
N c N S E
t
( ) g
ª º ª º ª º
« » « » « » « » « » « »
«¬ »¼ «¬ »¼ «¬ »¼
HF waves have much shorter time scales than LF waves Æ AN=b stiff
Mismatch Î additional measures required
Economically, large computational time steps
w
’
w
N b
A
%
%
Many time scales are involved in evolution of wind waves
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10.
Convergence enhancing measures
This may lead to numerical instabilities. Two solutions:
1. Action density limiter:
restriction of the total change of action density per iteration at
each wave component
D
PM
2 3
2. Under relaxation
g
N
k c
J
V
'
J 0.1
Phillips equilibrium spectrum
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11.
Convergence enhancing measures
2. Under-relaxation: enhancing main diagonal Æ stabilizing effect
G G G G
N i N i
1
ANi b , W
1
DV
W
- pseudo timestep IJ - smaller updates N
- costs computational time
- frequency dependent
- alfa to be set in swan input file (0.002-0.01)
G G G
ADV I

12.
Ni b DV Ni1
• Under-relaxation improves iteration behaviour
• Under-relaxation slows convergence
• Not meaningful for nonstationary computations
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13.
Convergence enhancing measures
Hm0 deep water, fetch = 12.5 km
J 0.1
U10 = 10 m/s
U10 = 30 m/s
• Effect limiter is clear without
under-relaxation
• Under-relaxation improves
iterative behaviour:
• Smoothed
• Reduction of overshoot
• Alteration of limiter activity
• Under-relaxation slows
convergence
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14.
Interpolation
• Boundary conditions where waves enter computational domain
• Measured / computed 2D spectra
• Nesting of SWAN runs
• Nesting with course-grid WAM or WAVEWATCH run
Procedure:
1. Available spectra are normalized first by mean frequency and
direction
2. Linear interpolation of spectra in intermediate locations
3. Resulting spectra are transformed back
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15.
Interpolation
(Bi-linear) interpolation of input grids on computational grid :
• Bathymetry
• Wind field
• Current field
• Water level field
• Bottom friction
WARNINGS:
• Resolve relevant spatial and temporal details
• Input grid should cover computational grid entirely
• Bottom: input grid ~ computational grid
14

16.
And also:
MXITST=0 is useful for checking the input!
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