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

Feb 16, 2006 - that forms a basis (sometime orthogonal) of a functionnal .... 8. B. #Ð¹Ð¸ is an orthogonal basis change (orthogonal matrix). ..... Scale separation.
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: '

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













'









 



 











'

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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

*



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



'

'

'





'

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

     



  '











*









#

     

 



 

linear combinations Standard wavelet transform

 & 



 & 





 

 

\$ #

\$



 

 &



 &



*

 

 *

\$ #

\$

Anisotropic divergence-free wavelets



 

\$

\$

\$





*

*









'

'









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\$

\$

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*

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 '

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'

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' &%

*

*



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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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







\$

\$

\$



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 





              

.. .





\$

\$



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

  

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

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\$

\$

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.. .



*



\$

* 



\$







 

 

 

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  

  

&



&







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

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

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   

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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

* 



)



, 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) 





 

' &%



  







/

.,









 





 



*

 



 













 















*

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



 











&%

'









.. .

 

 

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