Wavelet methods For the numerical simulation of ... - Erwan DERIAZ

Feb 16, 2006 - that forms a basis (sometime orthogonal) of a functionnal .... 8. B. #йи is an orthogonal basis change (orthogonal matrix). ..... Scale separation.
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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]: 

 #

#

  &

 



"

'



!

  $ 

  $

  



  



 $ %   

The Navier Stokes equations can be projected on

  

)



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: 



 













 























 

.

* 

)

/



'

&%



0

(

 

 



$ 



., +



#



and





&

!  "



with

Plan :

I - Divergence-free wavelets

II - Helmholtz decomposition with wavelets

III - Numerical simulations, results

Conclusion - Perspectives

Wavelets [Meyer90,Daubechies92]  *



)

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

$

$

*





 0

(

$ $



 $

$







$







$





 







.





0

(

+

.



defined by :







$

$

$

$



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



*

 

*





'

'







'

 



















*









 #



*







*











'

'

'







'







     



  '











*











#

     

 



 

linear combinations Standard wavelet transform

 & 



 & 







 

 

$ #

$







 

 &



 &





*

 

 *

$ #

$

Anisotropic divergence-free wavelets



 





$

$

$











*

*









'

'















*













 







$

$

$











*

*







 '



'





     

' &%

*

*

















*



* 







 









' &%







$

$





&

&





 $

 $





 $



&











$









*







*



 







*



&







is an orthogonal basis change (orthogonal matrix).

the position. the scale and with

The operation on the coefficients:



Anisotropic divergence-free wavelets in -D:  







$

$

$













'





 





*







$

$













'





 

















.. .

&%

'



  









 



















$

$

$















'



 







              

.. .







$

$













'





 









  









 

















$

$









'















        

.. .



*



$

* 



$











 

 

 





  

  

&







&









 



 

























*





























*



  







&



&%





'

  





























































 











&





























































 



 



























  



 































&









 

























&%















&

  









'



.

   





  

   







'  &

&%

.

.. ..

.

. ..

. .

.. . .. .



















































, orthogonal. Matrix of size

.. . .. . .. . .. . .. . .. . ..

..

..



n

. .. ..

. .. . .. .

, and Let









II - Helmholtz decomposition Principal 

Vector field

* 



)





, decomposition with







 





















 







 



'

&%





'

&%

'

 * 

and we have







 













 '

&%





 

 

Hcurl







 *

)

H

  



'

are orthogonal in

)



and





the functions uniqueness.

 



In N-S, importance of this decomposition to project the term . H



onto





'

&%







Leray projector (in Fourier) 





 

' &%



  







/

.,









 







 



*

 



 













 

















*







 













&%

'









.. .

 

 



In Fourier,

et

















 





Wavelet Helmholtz decomposition We want to write:   



'

&%

with 







 &

,



& 

















'

&%



'

&%

'

&%





Problem : the projectors on the divergence-free wavelet basis and on the gradient wavelet basis are biorthogonal projectors.





'

&%

et



Iterative method to find

.



and



Construction of the sequences





'

&%

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 : *

*

*

*





&

*





 '

'











*

* *



  

*



  





 



*



 



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