Absorbing layers
for shallow water models
Axel Modave
⋆
Universit ´e de Li `ege
Eric Deleersnijder
Universit ´e catholique de Louvain
Eric Delhez
Universit ´e de Li `ege
⋆
A.Modave@ulg.ac.be
Absorbing layer
as open boundary treatment
d
Ingoing waves
Outgoing waves
Absorbing layer
as open boundary treatment
d
Ingoing waves
Outgoing waves
✔
Extension of the computational domain
The fields are subject to a particular treatment
✔
Prescribe progressively the external forcing
The best layer for linear gravity wave
Different kinds of layers
The best layer for linear gravity wave
Different kinds of layers
Comparison of layers
Absorbing layer
for linear gravity wave
Layer
Domain
Basic equations
∂η
∂t
+ h
∂u
∂x
= 0
∂u
∂t
+ g
∂η
∂x
= 0
Absorbing layer
for linear gravity wave
Layer
Domain
s
(x)
Equations
with absorption terms
∂η
∂t
+ h
∂u
∂x
=
−σ η
∂u
∂t
+ g
∂η
∂x
=
−σ u
Absorbing layer
for linear gravity wave
Layer
Domain
s
(x)
Equations
with absorption terms
∂η
∂t
+ h
∂u
∂x
=
−
σ
η
∂u
∂t
+ g
∂η
∂x
=
−
σ
u
The best absorption coefficient
A discrete problem
s
uniform value
Reflection coefficient
1
The best absorption coefficient
A discrete problem
s
uniform value
Reflection coefficient
1
0
In the
continuous context
, the reflection coefficient is
0
if
Z
Layer
The best absorption coefficient
A discrete problem
s
uniform value
Reflection coefficient
Discrete context
Continuous context
1
0
In the
continuous context
, the reflection coefficient is
0
if
Z
Layer
σ
(x) dx = +∞
In the
discrete context
, the fields variations must also be represented by
the discrete scheme
The best absorption coefficient
Optimization procedure
Layer
Domain
Reflection
Incident pulse
The best absorption coefficient
Optimization procedure
Layer
Domain
Reflection
Incident pulse
Minimize
Energy norm
of the reflected signal
Z
Domain
1
2
g
(η − η
reference
)
2
+
1
2
h
(u − u
reference
)
2
dx
The best absorption coefficient
Discrete optimum
Modave and al. Ocean Dynamics (2010)
Absorption coefficient value
x
Schifted hyperbola
Discrete optimum
Shifted hyperbola :
√
gh
δ
x
x − δ
The best layer for linear gravity wave
Different kinds of layers
Flow relaxation scheme (FRS)
An easy-used layer
Martinsen and al. Coastal Eng. (1987)
∂H
∂t
+
∂(Hu)
∂x
+
∂(Hv)
∂y
=
−σ(H − H
ext
)
∂(Hu)
∂t
+
∂(gH
2
/2)
∂x
+
∂(Hu
2
)
∂x
+
∂(Huv)
∂y
− f H v =
−σ(H u − H u
ext
)
∂(Hv)
∂t
+
∂(gH
2
/2)
∂y
+
∂(Huv)
∂x
+
∂(Hv
2
)
∂y
+ f Hu =
−σ(H v − H v
ext
)
Relaxation terms
Adapted FRS
An other easy-used layer
Lavelle and al. Ocean Modell. (2008)
∂H
∂t
+
∂(Hu)
∂x
+
∂(Hv)
∂y
=
−(σ
x
+ σ
y
)(H − H
ext
)
∂(Hu)
∂t
+
∂(gH
2
/2)
∂x
+
∂(Hu
2
)
∂x
+
∂(Huv)
∂y
− f H v =
−σ
x
(Hu − Hu
ext
)
∂(Hv)
∂t
+
∂(gH
2
/2)
∂y
+
∂(Huv)
∂x
+
∂(Hv
2
)
∂y
+ f Hu =
−σ
y
(Hv − Hv
ext
)
Other
relaxation terms
Perfectly matched layer (PML)
A layer with theoretical justification
Hu Comput. Fluids (2008)
∂H ∂t + ∂(Hu) ∂x + ∂(Hv) ∂y = −(σx + σy )(H − H ext ) − qH ∂(Hu) ∂t + ∂(gH2 /2) ∂x + ∂(Hu2 ) ∂x + ∂(Huv)
∂y − f H v =−(σx + σy )(Hu − Hu
ext ) − qHu ∂(Hv) ∂t + ∂(gH2 /2) ∂y + ∂(Huv) ∂x + ∂(Hv2 ) ∂y + f Hu = −(σx + σy )(Hv − Hvext) − qHv
With the additional equations:
∂qH
∂t = σxσy (H − H ext
) + σy ∂(Hu − Huext )
∂x + σx ∂(Hv − Hvext) ∂y ∂qHu ∂t = (σx + σy )(−f Hv + (f Hv) ext
) + σxσy (Hu − (Hu)ext + ˆqHu)
+ σy ∂h(gH2 /2 + Hu2 ) − (gH2 /2 + Hu2)exti ∂x + σx ∂h(Huv) − (Huv)exti ∂y ∂qHv ∂t = (σx + σy )(f Hu − (f Hu) ext ) + σxσy (Hv − (Hv)ext + ˆqHv)
+ σy∂ [(Huv) − (Huv)ext]
∂x + σx ∂h(gH2 /2 + Hv2 ) − (gH2 /2 + Hv2)exti ∂y ∂ ˆqHu ∂t = −f Hv + (f Hv) ext ∂ ˆqHv ∂t = f Hu − (f Hu) ext
The best layer for linear gravity wave
Different kinds of layers
Test case
Collapse of the Gaussian-shaped mound of water
Domain
Layer
x
y
Domain
Layer
x
y
Test case
Collapse of the Gaussian-shaped mound of water
x
y
Domain
Layer
x
y
Domain
Layer
Elevation and error
using the shifted hyperbola
Elevation after
9h
Error with the FRS
−0.1 −0.05 0 0.05 0.1 −0.01 −0.005 0 0.005 0.01
Error with the adapted FRS
Error with the PML
−0.005 0 0.005 0.01 −0.005 0 0.005 0.01
