This document discusses numerical techniques in SWAN (Simulating WAves Nearshore), a wave model. It covers discretization schemes, convergence criteria, source term stability, and interpolation methods. The propagation schemes are fully implicit and use upwind differencing. Convergence is checked using criteria on changes in wave height and period from iteration to iteration. Under-relaxation and action density limiting can improve stability for multi-scale wave simulations. Input data like spectra and environmental fields are interpolated to the computational grid.

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