Wavelet methods For the numerical simulation of incompressible fluids Erwan Deriaz
[email protected]
Laboratoire de Modélisation et Calcul (Grenoble) PhD Director : Valérie Perrier Universität Ulm, Abteilung Numerik, Seminar February 16th 2006
0-0
Wavelets for the Navier-Stokes equations
in a Riesz basis of
Decomposition of
div
[Urban96]:
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$
$
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The Navier Stokes equations can be projected on
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the Leray projector.
)
(
)
where we noted
:
Perspectives: Numerical resolution of incompressible Navier-Stokes equations in dimension 2 and 3 by an adaptive method:
= 2 or 3
div
limit conditions (periodic, Dirichlet homogeneous or non homogeneous)
With a wavelet discretization:
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and
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with
Plan :
I - Divergence-free wavelets
II - Helmholtz decomposition with wavelets
III - Numerical simulations, results
Conclusion - Perspectives
Wavelets [Meyer90,Daubechies92] *
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that forms a basis (sometime orthogonal) for example).
0
'(
Family of functions of a functionnal space (
0.6
0.9
0.5
0.8
0.4
0.7
0.3 0.6 0.2 0.5 0.1 0.4 0 0.3 −0.1 0.2
−0.2
0.1
−0.3 −0.4 −8
−6
−4
−2
0
2
4
6
8
0 −20
−16
−12
−8
−4
Example of a wavelet with its frenquency localisation.
0
4
8
12
16
20
Multiresolution Analysis
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$ $
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.
0
(
+
.
defined by :
$
$
$
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Wavelet space
Riesz basis of
)
$
(3)
(dilation)
$
(2)
)
(1)
is defined as
*
Definition [Mallat87]: A Multiresolution Analysis of verifying : a sequence of closed sub-spaces
Time-frequency partition with wavelets
'
frequency
*
wavelet packets $
.
'
$ $
O
$
time '
Filtering Schema: decomposition – recomposition
Divergence-free wavelets
Proposition (Malgouyres): [Lemarié92] Let be an MRA. If for a certain , then there is an MRA such that: '
'
'
vectorial functions , and that, when translated and dilated, form an inconditionnelle basis of .
Theorem: There exist
'
wavelets :
&
and
'
'
,
,
constructed by tensor products of
,
(
)
0
(
Example of divergence-free wavelets in 2D
3
2.2
1.8 2 1.4
1.0
1
0.6 0
0.2
−0.2 −1 −0.6
−2
−1.0 −2
−1
0
1
2
3
2.2
2.2
1.8
1.8
1.4
1.4
1.0
1.0
0.6
0.6
0.2
0.2
−0.2
−0.2
−0.6
−0.6
−1.0
−1.0
−0.6
−0.2
0.2
0.6
1.0
1.4
1.8
2.2
−1.0
−0.6
−0.2
0.2
0.6
1.0
1.4
1.8
2.2
−1.0 −1.0
−0.6
−0.2
0.2
0.6
1.0
1.4
1.8
2.2
Example of divergence-free wavelets in 3D
Vorticity isosurfaces of the 3D isotropic divergence-free wavelets
Divergence-free wavelet transform
*
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'
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linear combinations Standard wavelet transform
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$ #
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Anisotropic divergence-free wavelets
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is an orthogonal basis change (orthogonal matrix).
the position. the scale and with
The operation on the coefficients:
Anisotropic divergence-free wavelets in -D:
$
$
$
'
*
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$
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.. .
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.. .
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.. ..
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. ..
. .
.. . .. .
, orthogonal. Matrix of size
.. . .. . .. . .. . .. . .. . ..
..
..
n
. .. ..
. .. . .. .
, and Let
II - Helmholtz decomposition Principal
Vector field
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, decomposition with
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'
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and we have
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Hcurl
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H
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are orthogonal in
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and
the functions uniqueness.
In N-S, importance of this decomposition to project the term . H
onto
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Leray projector (in Fourier)
' &%
/
.,
*
*
&%
'
.. .
In Fourier,
et
Wavelet Helmholtz decomposition We want to write:
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with
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,
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Problem : the projectors on the divergence-free wavelet basis and on the gradient wavelet basis are biorthogonal projectors.
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et
Iterative method to find
.
and
Construction of the sequences
'
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Hdiv,0 HN
v (=v 0 )
v0div
Hn v0N
(=v 1) v0n
v1div v2 1 vrot
0 vrot
vect
and
Convergence processus for the sequences with vect .
Hrot,0
Theorem: Convergence in dimension 2 for Shannon wavelets.
, we find a
such that:
If there are
to $
$
Proof (Kai Bittner): Looking at the proximity of convergence criteria.
)
$
(1)
and
)
$
(2)
then
(3)
Numericaly the convergence have been tested successfully on variate 2D and 3D fields.
Erreur L2 en échelle logarithmique
10e4 10e3
en 256 x 256 2 moments nuls
10e2 10e1
en 512 x 512 2 moments nuls
1 10e−1
en 1024 x 1024 2 moments nuls
10e−2 10e−3
en 256 x 256 3 moments nuls
10e−4 10e−5 10e−6 0
4
8
12
16
20
Nombre d’itérations
24
28
32
Problem of the frequency localisation of the wavelets Convergence rate linked (
proportional) to : *
*
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*
&
*
'
'
*
* *
*
*
Problem on the frequency localisation of the wavelets. - in Fourier - ponderation function:
- Size of the compact support:
'
*
'
'
Conclusion : we must get a better localisation in frequence for
.
The wavelet packets
,
Definition :
, packets associated to the scale function
:
$
0
(
$
$
$
$
$
In general, fail to control the frequency localisation.
Fequency target - With the Shannon wavelets
- with the Walsh packets
for
By iteration,
$
$
$ %
Packets modulation "A theoretical study show that we have to target" Ideal Packet :
'
Examples : 1.2
0.05
+
0.8
+
0.04
0.4 0.03 0.0 0.02 -0.4
0.01 -0.8
-1.2
++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++
0.00 6
7
8
9
10
11
12
-3
-2
Quadratic spline wavelet packets with 2 wavelets
-1
0
1
2
3
Packets with 4 quadratic spline wavelets
0.6
0.05
+
0.4
+
0.04
0.2 0.03 0.0 0.02 -0.2
0.01 -0.4
-0.6
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
0.00 5
7
9
11
13
15
17
-3
-2
-1
0
1
2
3
Numerical schema for Navier-Stokes )
Leray projector with wavelets ( give the pressure directly) :
)
operator is linear.
Euler explicite in time :
)
on the wavelet coefficients :
div
div
div
div
)
&
&
&
&
Test with the simulation “fusion of 3 vortices” t=0
t=10
t=20
t=30
- wavelet code splines of degree 1 and 2 the simplest ( Helmholtz) - Runge-Kutta schema of order 2 for the time evolution *
-
grid *
Results are visually identical to a spectral code in
.
t=40
iterations for
Conclusion Assets
- Calculation in
- Scale separation - Non linear approximation
Perspectives - Get adaptativity - Limit conditions - Complex geometries
