SWAN Advanced Course4. Numerics in SWAN

Delft Software Days28 October 2014, Delft

Contents

• Discretization

• Convergence criteria

• Source term stability

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Discretization

• Numerical schemes for propagation (fully implicit):

• x,y-space : upwind: BSBT (1st order), SORDUP (2nd),Stelling-Leendertse (3rd)

• time : backward• -space : hybrid central / upwind (first order upwind too

diffusive, central scheme prone to wiggles)• -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, , )• iterative (4 sweep) solution technique• x,y-space: regular, curvi-linear or unstructured grids

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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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Convergence if:

a.

b.

c. Conditions a. AND b. are satisfied in 98% of all wet grid points

Lake George:

2% criteria vs.fully-converged

Convergence criteria

( ) ( ) ( ) ( )0 0 0 0or0.02 0.02i i i average

m m m mH H H H( ) ( ) ( ) ( )01 01 01 01or0.02 0.02i i i average

m m m mT T T T

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Convergence criteria

Hs

DHSIGN

90%-conv. crit.

default 98%-conv. crit.

Hs

Example: 2003 experiment NCEX(Levi Gorrell)

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• Check iteration behaviour of output quantities

• TEST output

• DHS, DRTM01

• Default not always effective significant inaccuracies

• Either stronger accuracy than default (2%) or use differentconvergence criterium: based on curvature

Convergence criteria

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Convergence criteria

Curvature-based convergence criteria (Zijlema & vd Westhuysen 2005)

0 0( ) / [ . m a x ]ii i m mH H curv

1 10 0 0 0

1 10 0

,i i i im m m m

i im m

H H T T drelH T

and

Haringvliet Estuary Lake George

in more than [npnts] % of wet points

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Convergence enhancing measures

HF waves have much shorter time scales than LF waves AN=b stiff

Mismatch additional measures required

Economically, large computational time steps

( )gN c N S Et

N bA

Many time scales are involved in evolution of wind waves

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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 ateach wave component

2. Under relaxation

32PM

g

Nk c Phillips equilibrium spectrum

0.1

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2. Under-relaxation: enhancing main diagonal stabilizing effect

11,

i iiN N AN b

1i iA I N b N

• Under-relaxation improves iteration behaviour

• Under-relaxation slows convergence

• Not meaningful for nonstationary computations

Convergence enhancing measures

- pseudo timestep -> smaller updates N- costs computational time- frequency dependent- alfa to be set in swan input file (0.002-0.01)

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Hm0 deep water, fetch = 12.5 km

0.1

U10 = 10 m/s

U10 = 30 m/s

Convergence enhancing measures

• Effect limiter is clear withoutunder-relaxation

• Under-relaxation improvesiterative behaviour:

• Smoothed

• Reduction of overshoot

• Alteration of limiter activity

• Under-relaxation slowsconvergence

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• 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

direction2. Linear interpolation of spectra in intermediate locations3. Resulting spectra are transformed back

Interpolation

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(Bi-linear) interpolation of input grids on computational grid :• Bathymetry• Wind field• Current field• Water level field• Bottom friction

Interpolation

WARNINGS:• Resolve relevant spatial and temporal details• Input grid should cover computational grid entirely• Bottom: input grid ~ computational grid

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And also:

MXITST=0 is useful for checking the input!

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