Divergence-free and curl-free wavelets in 2D and 3D, application to turbulent flows Erwan Deriaz
and Val´erie Perrier
Laboratoire de Mod´elisation et Calcul de l’IMAG, BP 53 - 38 041 Grenoble Cedex 9, France Abstract. We investigate the use of compactly supported divergence-free wavelets for the representation of solutions of the Navier-Stokes equations. After reviewing the theoretical construction of divergence-free wavelet vectors, we present in detail the bases and corresponding fast algorithms for two and three-dimensional incompressible flows. We also propose a new method to practically compute the wavelet-Helmholtz decomposition of any (even compressible) flow: this decomposition, which allows the incompressible part of the flow to be separated from its orthogonal complement (the gradient component of the flow) is the key-point for developing divergence-free wavelet schemes for Navier-Stokes equations. Finally, numerical tests validating our approach are presented.
Submitted to: Journal of Turbulence
1. Introduction The prediction of fully-developed turbulent flows represents an extremely challenging field of research in scientific computing. The Direct Numerical Simulation (DNS) of turbulence requires the integration in time of the full nonlinear Navier-Stokes equations, i.e. the computation of all scales of motion. However, at large Reynolds number, turbulent flows generate increasingly small scales: to be realistic, the discretization in space (and also in time) ought to handle a huge number of degrees of freedom. This is impossible with currently available computers in three dimensions. Many attempts have been made, or are underway, to overcome this problem: among these are Vortex Methods which are able to generate very thin scales, or Large Eddy Simulations (LES) and subgrid-scale techniques which separate the flow into large scales, that are explicitly computed, from small scales, that are parametrized or computed statistically. In this context, wavelet bases offer a different approach. They provide an alternate decomposition allowing the intermittent spatial structure of turbulent flows to be represented with only few degrees of freedom. This property comes from the good localization, both in physical and frequency domains, of the basis functions. The wavelet decomposition was introduced in the beginning of the 90s for the analysis of turbulent flows [12, 31, 29]. Wavelet
To whom correspondence should be addressed (
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Div-free and curl-free wavelets for Navier-Stokes
2
based methods for the resolution of the two-dimensional (2D) Navier-Stokes equations appeared later [3, 15, 13, 33, 22, 18], and very recently for three-dimensional (3D) domains [24, 25]. They have also been used to define LES-type methods such as the CVS method [14], adaptive LES methods [17], or to derive 3D models of turbulence [19]. Most of these cited works use a Galerkin, a Petrov-Galerkin or a collocation approach for the vorticity formulation in dimension two, with periodic boundary conditions. However, these approaches are not appropriate for the 3D case with non periodic boundary conditions. An alternative approach was, at the same period, considered first by K. Urban and then investigated by several authors. They proposed to use the divergence-free wavelet bases originally designed by P.G. Lemari´e-Rieusset [27]. Divergence-free wavelet vectors have been implemented and used to analyze 2D turbulent flows [1, 21, 38], as well as to compute the 2D/3D Stokes solution for the driven cavity problem [35, 36]. Since divergence-free wavelets are constructed from standard compactly supported biorthogonal wavelet bases, they can incorporate boundary conditions [9, 37]. The great interest of divergence-free wavelets is that they provide bases suitable to represent the incompressible Navier-Stokes solution, in two and three dimensions. Our present objective is thus to investigate their practical feasibility and amenability. In order to eliminate the pressure, we project the equations onto the space of divergence-free vectors. This (orthogonal) projection is the well-known Leray projector, and can be computed explicitly in Fourier space for periodic boundary conditions. Unfortunately, as already noted by K. Urban [37], the Leray operator cannot be represented simply in terms of divergence-free wavelets, since they form biorthogonal bases (and not orthogonal). The goal of the present paper is to investigate the use of divergence-free wavelets for the simulation of turbulent flows. First, in section 2 we review the basic ingredients of the theory of compactly supported divergence-free wavelet vectors, developed by P.G. Lemari´e-Rieusset [27]. In section 3, we present in detail the bases we implement in two and three dimensions: isotropic bases, as presented in the previous works ([27], [35],[1]), but also anisotropic bases, that are easier to implement. We will see that the choice of the complement wavelet basis is not unique, and this choice induces the values of divergence-free coefficients for compressible flows. We discuss the algorithmic implementation of divergence-free wavelet coefficients in two and three dimensions, leading to fast algorithms (in O( ) operations where is the number of grid points). Section 4 is devoted to the Helmholtz decomposition of compressible fields in a wavelet formulation: the method we present uses both the biorthogonal projectors on divergence-free, and on curl-free wavelets. Our method is an iterative procedure, and we will experimentally prove that it converges. Section 5 addresses the main ingredients of a Galerkin method for the Navier-Stokes equations, based on divergence-free wavelets. Finally, the last section presents numerical tests that validate our approach: nonlinear compressions of 2D/3D incompressible turbulent flows, and the wavelet-Helmholtz decomposition of several examples, such as the computation of the divergence-free part and the pressure arising from the nonlinear term of
Div-free and curl-free wavelets for Navier-Stokes
3
the Navier-Stokes equations. 2. Theory of divergence-free wavelet bases In this section, we review briefly the construction of divergence-free wavelets. Compactly supported divergence-free vector wavelets were originally designed by P.G. Lemari´e-Rieusset, in the context of biorthogonal Multiresolution Analyses (also noted MRA). For the definition and properties of biorthogonal MRA and associated wavelets, we refer to appendix A, and to textbooks [10, 23, 30, 7]. For the theory of divergence-free wavelets, we refer to the articles [27, 36] and to the book of K. Urban [38]. We illustrate the construction with the explicit example of splines of degree 1 and 2. 2.1. Theoretical ground of the divergence-free wavelet vectors Let introduce
Hdiv
div
div
the space of divergence-free vector functions in . The divergence-free wavelets in defined by P. G. Lemari´e-Rieusset [27], provide with Riesz bases of Hdiv . Their construction is based on the existence of two different related by differentiation and integration. one-dimensional multiresolution analyses of
Let be a one-dimensional MRA, with a differentiable scaling function , (meaning that span ), and a wavelet ; one can build a second MRA with a scaling function ( span ) and a wavelet satisfying:
(1)
Example: spline scaling functions and wavelets of degree 1 and 2 Biorthogonal splines provide wavelet bases which are regular, compactly-supported and easy to implement. The scaling functions of the associated MRA are standard B-spline bases, and the wavelets are constructed easily, by linear combinations of translated B-splines. An example of MRA satisfying equation (1) is given by splines of degree 1 ( MRA spaces) and splines of degree 2 ( MRA spaces). In both cases we draw the scaling functions , and their associated wavelets , with shortest support (figure 1).
The isotropic divergence-free wavelets in are then obtained by suitable combinations of tensor products of functions , , and , , fulfilling (1). Following [27], there exist vector functions div Hdiv ( of cardinality ,
!
"
#
$
%
'
&
!
Div-free and curl-free wavelets for Navier-Stokes
4
1.5
1.5
1.3 1.1 1.1 0.7
0.9 0.7
0.3 0.5 −0.1 0.3 0.1
−0.5
−0.1 −0.9 −0.3 −0.5
−1.3 −2
−1
0
1
2
3
4
5
−2
−1
0
1
2
3
4
5
6
Figure 1. Scaling functions and associated even and odd wavelets with shortest support, for splines of degree 1 (left) and 2 (right).
) compactly supported, such that every vector function of cardinality can be uniquely expanded:
div !
"
with
div !
div
div !
"
"
.
!
"
!
Hdiv
"
The generating wavelets div take the general following form in :
Let be given. We have to fix an integer
. Then for every , the vector function div !
$
%
"
such that
&
#
#
&
"
div
!
&
!
"
"
writes:
div !
"
"
where:
!
"
&
"
!
!
%
$
!#
!
&
&
' '(
*
)
)
&
!
!
+
!
with the notation
(2)
&
!
)
"
&
&
!"
!
and
!
!
if if if
!
if if
#
#
being equal to or .
Example: The 2D divergence-free vector scaling function takes the form:
,
div
-
-
.
.
/
/
Div-free and curl-free wavelets for Navier-Stokes
5
and the corresponding isotropic vector wavelets are given by the system:
div
-
&
div
div
.
/
/
&
-
-
.
We display in figure 2 the three generating vector wavelets in the case of spline generators of degree 1 and 2 of figure 1.
div
div 2.2
2.2
1.8
1.8
1.8
1.4
1.4
1.4
1.0
1.0
1.0
0.6
0.6
0.6
0.2
0.2
0.2
−0.2
−0.2
−0.2
−0.6
−0.6
−0.6
−1.0
−1.0 −0.6
−0.2
0.2
0.6
1.0
1.4
1.8
2.2
div
2.2
−1.0
−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
Figure 2. Example of isotropic 2D generating divergence-free spline wavelets
div
(center) and
div
div
(left),
(right).
These wavelets have already been studied by several authors for the analysis of 2D turbulent flows [1, 21], and also to solve the Stokes problem in two and three dimensions [35, 36, 37]. From now on, we will focus on the 2D and the 3D case, and we will present the associated fast algorithms. We will then point out that the expansion of compressible flows following the divergence-free wavelet bases is not uniquely given. Moreover, we will introduce 2D and 3D anisotropic divergence-free wavelets that are easier to implement. 3. Isotropic and anisotropic divergence-free wavelets: practical implementation Isotropic 2D/3D divergence-free wavelet transforms have already been implemented by K. Urban [36, 38], from divergence-free scaling coefficients. Since the computation of
Div-free and curl-free wavelets for Navier-Stokes
6
divergence-free scaling coefficients requires the solution of a linear system, we will propose a different method. We present in detail our 2D/3D divergence-free wavelet decomposition algorithm: it is based on the construction of a non-unique complement vector space. We also introduce anisotropic divergence-free wavelet bases, and their corresponding decomposition algorithms, which differ from previous studies. In the following, we suppose we are given two one-dimensional multiresolution analyses and and , and , their associated (one dimensional) scaling functions and wavelets, satisfying condition (1).
3.1. Isotropic divergence-free wavelet transforms 3.1.1. The 2D case :
The starting point of the construction is a 2D multiresolution analysis of
,
whose 2D vector scaling functions
,
-
are given by:
,
,
,
-
-
-
-
!
with Then a velocity field
(3)
-
"
"
$
!
Following wavelet theory, the family
of this MRA are:
! "
-
-
In the isotropic case, the six canonical generating 2D vector wavelets
#
in
. (4)
, forms a basis of has the following wavelet decomposition: &
Div-free and curl-free wavelets for Navier-Stokes
7
Notice that the first line of the decomposition represents the wavelet decomposition in the MRA of the first component , whereas the second line concerns the wavelet decomposition of in the MRA .
Isotropic generating divergence-free wavelets are then constructed by linear combination of the . More precisely, for each , the divergence-free wavelet div is given uniquely (and follows the general form (2)), whereas one has to build a such that: complement function
!
#
"
!
!
span
!
span
!
span
div !
!
The choice of the functions is not unique. Moreover they cannot be constructed such $ is orthogonal to Hdiv . We prothat the space span pose a choice for , described in appendix B, for which the computation of divergence-free wavelet coefficients div is reduced to a very simple linear combination of the standard . wavelet coefficients !
!
!
#
!
!
!
"
Now, the expansion (4) of the vector function
div
div
can be rewritten:
div
div
div
div
(5)
where the new coefficients are directly expressed from the original ones by equation (33) in appendix B. Appendix C summarizes the algorithm in pseudo code. Remark that the first line of the above decomposition represents a divergence-free part of , whereas the second line is a complement vector function, not orthogonal to the first one.
Remark: div , div . Moreover, we have
and div
are generating functions of the scalar space
div
div
div
Then, the incompressibility condition div
div
is equivalent to
!
, for all
$
#
.
For incompressible flows, since the biorthogonal projectors onto the spaces commute with partial derivatives [27], the divergence-free wavelet coefficients ' are uniquely determined, by the formula in equation (33), appendix B. div Difficulties arise when we want to compute the divergence-free part of a compressible flow. Because of the non-orthogonality between the divergence-free basis and div , the values of the divergence-free wavelet coefficients depend on the its complement choice of the complement basis. We address this problem in section 4, in order to provide a wavelet-Helmholtz decomposition of any flow.
!
"
!
!
Div-free and curl-free wavelets for Navier-Stokes
8
3.1.2. The 3D case The construction and fast algorithms corresponding to 3D divergencefree wavelet bases are obtained in a similar fashion as for the 2D case, except that one has to start with the following vector multiresolution analysis of :
From the 21 canonical generating 3D vector wavelets , one construct 14 generating divergence-free wavelets, and 7 complement functions n . Appendix B indicates their exact forms (which, for symmetry reasons, differ from the general form 2). As for the 2D case, the computation of divergence-free wavelet coefficients of any 3D vector field is given by a short linear combination of standard biorthogonal wavelet coefficients, arising from fast wavelet transforms. ! "
&
#
!
3.2. Anisotropic divergence-free wavelet transforms In this section we construct anisotropic wavelets that are divergence-free. Since we start from one-dimensional wavelets and verifying , we derive easily divergence-free wavelet bases by tensor products of one-dimensional wavelets. We detail in the following the construction of such bases in the 2D and 3D cases.
3.2.1. The anisotropic 2D case Unlike the isotropic case, anisotropic divergence-free wavelets are generated from a single vector function
an div
-
by anisotropic dilations, and translations. The 2D anisotropic divergence-free wavelets are given by:
an div
$
$
-
an div
where is the scale parameter, and parameter. When the indices and vary in , the family
is the position forms a basis of
Hdiv . We introduced as complement functions:
an n
-
since they verify an is orthogonal to an ( being fixed). n div The anisotropic divergence-free wavelet transform of a given vector function similarly as the isotropic one. Starting from anisotropic wavelet decomposition of MRA (see appendix A):
an
an
an
an
works in the
Div-free and curl-free wavelets for Navier-Stokes
9
where:
an
-
an
-
for , are the anisotropic canonical wavelets. Remark that for more simplicity, the dilated functions are not normalized, in -norm. can be expanded onto the new basis: an an an an (6) div div n n
an
(7)
an
3.2.2. The anisotropic 3D case take the form:
an
with the corresponding coefficients:
an an
div an n
In the same way, the anisotropic 3D divergence-free wavelets
an div
-
an div
$
$
$
-
-
an div
with . Unlike the 2D case, we have to choose two functions among the three above, to generate the divergence-free basis. As complement basis we introduce the most of possible orthogonal function to the previous ones:
an n
-
The operations to compute divergence-free coefficients and complement coefficients are similar to the 2D case.
Div-free and curl-free wavelets for Navier-Stokes
10
4. An iterative algorithm to compute the wavelet-Helmholtz decomposition 4.1. Principle of the Helmholtz decomposition
The Helmholtz decomposition [16, 5] consists in splitting a vector function into its divergence-free component div and a gradient vector. More precisely, there exist a potential-function and a stream-function such that:
div
and
div
(8)
Moreover, the functions and are orthogonal in . The stream-function and the potential-function are unique, up to an additive constant. In , the stream-function is a scalar valued function, whereas in it is a 3D vector function. This decomposition may be viewed as the following orthogonal space splitting:
Hdiv where Hdiv
Hcurl
is the space of divergence-free vector functions, and
Hcurl
is the space of curl-free vector functions (if we have to replace by curl in the definition). For the whole space , the proofs of the above decompositions can be derived easily, by mean of the Fourier transform. In more general is the space of domains, we refer to [16, 5]. Notice that one can also prove that Hdiv is the space of gradient functions. functions, whereas H curl The objective now is to generate a wavelet-Helmholtz decomposition. Since in the , we have to work previous sections we have constructed wavelet bases of Hdiv analogously to carry out wavelet bases of Hcurl .
4.2. Construction of a gradient wavelet basis
A definition of wavelet bases for the space Hcurl has already been provided by K. Urban in the isotropic case [37]. We will focus here on the construction of anisotropic curl-free vector wavelets in the 2D case (it goes similarly in the -dimensional case). This construction is very similar to the divergence-free wavelet construction, despite some crucial differences. The starting point here is to search wavelets in the MRA instead of , where the one-dimensional spaces and are related by differentiation and integration (section 2.1). Since Hcurl is the space of gradient functions in , we construct gradient wavelets by taking the gradient of a 2D wavelet basis of the MRA . If we neglect the normalization, the anisotropic gradient wavelets are defined by:
an curl
-
Div-free and curl-free wavelets for Navier-Stokes Thus, when
$
$
,
11
vary in
an curl
, the family
forms a wavelet basis
. of Hcurl an works similarly to The decomposition algorithm on curl-free wavelets curl the one on anisotropic divergence-free wavelets. Starting from the anisotropic wavelet decomposition of a vector function in the MRA :
an,#
an
an,#
an
where the canonical anisotropic vector wavelets are:
an,#
-
an,#
-
By applying the change of basis: an,#
an,#
an curl an
an,#
an,#
an,#
an,#
we obtain:
an curl
an curl
an
an
(9)
where the curl-free wavelet coefficients are obtained from the standard ones by
an curl
an
an
(10)
and have associated complement coefficients
an
an
an
(11)
4.3. Implementation of the Helmholtz decomposition in the wavelet context From now on, our objective is to compute the wavelet decomposition of a given vector function : this means to find a divergence-free component div and an orthogonal curl-free component curl such that:
div where:
div
div
div
curl
curl
curl
curl
are the wavelet expansions onto div-free and curl-free wavelet bases constructed previously (section 3.2.1 and 4.2). We will focus here on 2D anisotropic wavelet bases (and we will omit the superscript ”an” in the notation of the basis functions).
Div-free and curl-free wavelets for Navier-Stokes
12
To provide with such decomposition, we have to overcome two problems: - First, the div-free wavelets and curl-free wavelets form biorthogonal bases in their respective spaces, and as already noticed by K. Urban [37], they do not give rise, in a simple way, to the orthogonal projections div and curl of . As a solution, we propose to construct, in wavelet spaces, two sequences and that converge to div and curl . div curl - The second difficulty is that div-free wavelets are in spaces of the form , whereas curl-free wavelets are in , where and are couples of spaces related by differentiation and integration. These spaces are different, and in order to construct our approximations and , we have to define a precise div curl interpolation procedure between them. In particular, the spaces can be suitably chosen from .
be 4.3.1. Iterative construction of the div-free and curl-free parts of a flow Let a vector function, and suppose that is periodic in both directions, and known on grid points that are not necessarily the same for and . In the following, we will note: - an approximation of in the space , given by some interpolation process. - an approximation of in the space , also given by some interpolation process.
satisfying div
, and div div satisfying curl curl , as follows: curl - We begin with , and we compute div, the divergence-free wavelet decomposition of , and its complement n , by formula (6, 7): We now define the sequences
div
n
Then we compute the difference
div
div
n
n
div at collocation points. Secondly we consider div , and we apply the curl-free wavelet decomposition (9, 10, 11), leading to a curl-free part and its complement:
div
curl
Finally we define pointwise:
div
curl
curl
N
N
curl.
- At step , by knowing at grid points, we are able to construct a divergence free-part of by (6), and , the curl-free component of by (9) ( div curl div div being computed at grid points). The next term of the sequence is again defined pointwise:
div
curl
(12)
Div-free and curl-free wavelets for Navier-Stokes
We iterate this process until
div
, and we obtain:
13
curl
div
div
curl
curl
where the right hand side is an approximation of , which interpolates the data up to an error ( being given). Hdiv,0 HN
v (=v 0 )
v0div
Hn v0N
(=v 1) v0n
v1div v2 v1curl
v0curl
Hcurl,0
Figure 3. Idealistic schematization of the convergence process of the algorithm with HN span N and Hn span n .
Ideally, the iteration converges as indicated on figure 3 . However we are not able to prove the convergence of the sequence . We will demonstrate it experimentally in section 6.4, on arbitrary fields. Nevertheless, we outline some remarks: - The convergence rate depends on the choice of complement functions n , N .
The smaller the -scalar products and , the div curl n N faster the sequences converge. - Ideally, we would like the convergence rate to be independent of the interpolating operators and . We propose below a choice for these operators, based on spline-quasi interpolation, which is satisfactory at relatively-slow convergence rate.
4.3.2. Helmholtz-adapted spline interpolation In this part, we will detail our choice of operators and , in the context of the spline spaces of degree 1 ( ) and 2 ( ) that we introduced earlier.
Div-free and curl-free wavelets for Navier-Stokes
14
Let us suppose the components and of a velocity field are known respectively at and , for . This choice of grid is knot points induced by the symmetry centers of scaling functions of and of (see figure 4).
1.0
0.9
0.8 0.7
0.6 0.5 0.4 0.3 0.2 0.1 0 −1.1
−1
−0.7
0
−0.3
0.1
0.5
1
0.9
Figure 4. The two scaling functions of
1.3
and
2
1.7
2.1
, and their symmetry centers
For given, is chosen as a quasi-interpolation operator (similarly to section 6.1.2) in the spline space
,
1
,
,
2
,
where 1 and 2 are the vector scaling functions introduced in section 3.1.1. The second operator provides a quasi-interpolation of vector functions onto a new spline space . Under interpolation considerations, we define:
span
Hence we can write:
,
span
where 1 .
,
and 2
,
1
,
2
are the 2D vector scaling functions of the MRA
5. A divergence-free wavelet method for the Navier-Stokes equations We present in this section the basics of a divergence-free wavelet numerical method, for the resolution of the incompressible Navier-Stokes equations, written in velocity-pressure formulation (without forcing term): *
(13)
Div-free and curl-free wavelets for Navier-Stokes
15
with periodic or Dirichlet boundary conditions, in a square (or cubic) domain. Our objective is to derive a Galerkin method based on finite dimensional spaces of divergencefree wavelets. Galerkin (or Petrov-Galerkin) methods are variational methods for DNS of turbulence. In the context of Wavelet Galerkin or Petrov-Galerkin methods (including collocation methods), the works [3, 15, 13, 33] propose numerical methods for the resolution of the 2D Navier-Stokes equations in vorticity-stream function formulation, with periodic boundary conditions. However, these methods cannot extend in a simple way to the 3D case nor to Dirichlet boundary conditions. In the -formulation with classical discretizations, like spectral methods (in the non periodic case) [2], finite element methods [16], or (non divfree) wavelets [18, 25, ?], one has to adapt the discretization bases for velocity and pressure, in order to satisfy some inf-sup condition (also called LBB condition), or one has to introduce some stabilization term to avoid spurious modes in the computation of the pressure. Then system (13) leads to a saddle point problem. Depending on the chosen formulation (Galerkin, collocation..) this problem is usually solved by the Uzawa algorithm, or by a splitting method. In any case, numerical difficulties arise in the computation of the pressure, which asks to solve the ill-conditioned linear system of Schur complement, or a Poisson equation. When using divergence-free bases, this difficulty is totally avoided: indeed, the pressure disappears by projecting the first equation of (13) onto the divergence-free vector functions:
.
/
(14)
where denotes the Leray projector. The solution
div
then has the form:
div
After projecting equation (14) onto a finite dimensional wavelet subspace
span
div
$
$
(14) is simply reduced to a system of ordinary differential equations, which can be solved by a classical finite-differences or Runge-Kutta scheme. The main difficulty in this approach is the . / computation of . However, the Helmholtz decomposition of the nonlinear term yields:
.
/
where is the flow pressure. The wavelet Helmoltz decomposition presented in section 4 allows us to write:
.
/
div
.
/
curl
div
curl
.
/
curl
Then we get the divergence-free wavelet decomposition of The second term gives the pressure from the curl-free coefficients of
div as follows.
div
.
Div-free and curl-free wavelets for Navier-Stokes
16
Computation of the pressure: Remember that curl-free wavelets are constructed by
an curl
.
$
$
From the equalities
/
curl
curl
curl
curl
one directly obtains by integration:
curl
Thus the computation of the pressure is no more than a standard anisotropic wavelet reconstruction in , from the curl-free coefficients of the nonlinear term obtained through the wavelet Helmoltz decomposition.
6. Numerical experiments In this section we present numerical results of the application of the divergence-free wavelet decomposition. We begin with the analysis of periodic, numerical, incompressible velocity fields in two and three dimensions, generated by pseudo-spectral codes. First, we have to take care of the initial interpolation of such fields, in order not to violate the incompressibility condition satisfied in Fourier space. Then, after the vizualisation of the divergence-free wavelet coefficients, we will study the compression factor obtained through the wavelet decomposition. In the last part, we investigate, and numerically demonstrate, the convergence of the algorithm presented in section 4.3, which provides the wavelet Helmholtz decomposition of any flow. In order to validate the approach we proposed in section 5, we compute, in wavelet space, the div-free component of a nonlinear term of the Navier-Stokes equations, and we extract the associated pressure. In all the experiments, we use divergencefree wavelets constructed with splines of degrees 1 and 2. 6.1. Approximation of the velocity in spline spaces Usually, the velocity fields are given by grid point values. The first step of the wavelet decomposition consists in interpolating these velocity data on a suitable B-spline space. The problem is that this approximation may not conserve the divergence-free condition, a condition that was satisfied in Fourier space when velocity data come from a spectral code. The spline approximation of data, obtained through spectral methods, introduces a slight error for the divergence-free condition. This difference may be not negligible. For the turbulent of the -norm of the velocity. fields we studied (2D and 3D) the error is about
Div-free and curl-free wavelets for Navier-Stokes
17
Thus we propose two ways to overcome this problem. The first way is to interpolate the velocity in the Fourier domain and to compute exactly its biorthogonal projection on wavelet spaces. The second way is to interpolate on the divergence-free B-spline spaces with the wavelet Helmholtz decomposition detailed in section 4.3.1. 6.1.1. By using the Discrete Fourier Transform Since they are highly accurate, spectral methods are often considered as a reference technique for simulating incompressible turbulent . / flows. For periodic boundary conditions on the cube , the Discrete Fourier Transform is used to decompose the velocity . regular grid, If means the (vector) Discrete Fourier coefficients of on a
"
the velocity expansion in the Fourier exponential basis is:
"
(15)
is:
In this context, the divergence-free condition div
(16)
Assume now that the velocity field to be analyzed (assumed to be 1-periodic in both directions), comes from a spectral method and satisfies the incompressibility condition in Fourier domain (16). To compute its decomposition in a divergence-free wavelet basis of , we have first to approximate in the suitable space introduced in section 3.1.1, where corresponds to . Then we search for an approximate function such that:
For the choice of functions and verifying equation (1), the incompressibility takes the discrete form on the coefficients condition div :
"
(17)
To conserve the incompressibility condition satisfied by , a solution consists in considering as the biorthogonal projection onto the space , since we know that this projector commutes with partial derivatives [27]. This is equivalent to considering that
%
%
%
%
Div-free and curl-free wavelets for Navier-Stokes Replacing
18
by its Fourier expansion (15), it follows that
%
%
%
"
%
%
"
%
%
%
where , denote the (continuous) Fourier transforms of the dual scaling functions , . Finally, we obtain an explicit form for the Discrete Fourier Transform DFT of the coefficients (and in the same way for ):
DFT
%
%
(18)
DFT
%
%
This means that the discrete Fourier transform of coefficients is given by the discrete . / Fourier transform of , multiplied by tabulated values on of the Fourier transform of the duals , . In practice, we don’t know the explicit forms of these functions, except by the infinite product: "
%
%
%
%
"
%
"
%
&
%
Nevertheless, the infinite product converges rapidly, which allows to obtain point values of and , with sufficient accuracy.
%
In three dimensions, one proceeds similarly, by considering the biorthogonal projection of a 3D vector field onto the space .
6.1.2. By quasi-interpolation The spline quasi-interpolation is a good compromise when we have to deal simultaneously with spline approximations of even and odd degrees. In this context, the order of approximation is , by using B-splines of degree [11]. An advantage of the procedure is that it may be applied for any boundary conditions. Let be a B-spline scaling function ( or ). Given the sampling ( ), we want to compute scaling coefficients , of a spline function , that will nearly interpolate the values :
(19)
is an interpolating function if:
Div-free and curl-free wavelets for Navier-Stokes
19
For example, if we consider
(spline of degree 2), the previous condition implies:
In order to avoid the inversion of a linear system, the quasi-interpolation introduces, instead of :
By replacing by in (19), we obtain the following error at each grid point:
&
&
with
/
Therefore, the pointwise error of quasi-interpolation is of order 4, for a sufficiently regular function. 6.2. Analysis of 2D incompressible fields We focus in this part on the analysis of 2D decaying turbulent flows. The first numerical experiment we present analyses the merging of two same sign vortices. It concerns free decaying turbulence (no forcing term). The experiment was originally designed by M. Farge and N. Kevlahan [33], and often used to test new numerical methods [3, 18]. The experiment of [3] was reproduced here by using a pseudo-spectral method, solving the Navier-Stokes equations in velocity-pressure formulation. The initial state is displayed on figure 5 left. In a periodic box, three vortices with a gaussian vorticity profile are present; two are positive with the same intensity, one is negative with half the intensity of the others. The negative vortex is here to force the merging of the
two positive ones. The time step was and the viscosity . The solution is computed on a grid. The vorticity fields at times , , and are displayed on figure 5. The second row of figure 5 displays the absolute values of the isotropic divergence-free wavelet coefficients of the velocity field at corresponding times, renormalized by at scale $ index . As one can see, divergence-free wavelet coefficients concentrate on strong energy zones, which correspond to region of strong variations of the velocity, that is around or in between vortices, or along vorticity filaments.
The second experiment deals with a decaying 2D turbulent field, obtained with an initial state of random phase spectrum. This vorticity field was computed with a spectral code at a resolution (see of [20] for more details). As noticed in [20], there is a Newtonian
.
Div-free and curl-free wavelets for Navier-Stokes
-1.6
-0.8
0
0
20
0.8
&
1.6
Figure 5. Vorticity fields at times , , divergence-free wavelet coefficients of the velocity.
2.4
&
and
, and corresponding
viscosity such that the Reynolds number is of . This field has been kindly provided to us by G. Lapeyre [26] and was published in [20]. After turnover time-scales of the predominant eddies, for a time-scale based upon the total enstrophy of the flow, the vorticity field exhibits the emergence of coherent structures together with strong filamentation of the flow field outside the vortices (Figure 6 left). We show in figure 7, the isotropic and anisotropic divergence-free wavelet coefficients (in -norm) issued from the decomposition of the velocity field displayed on figure 6 (right). We must emphasize that before computing the divergence-free wavelet coefficients, we first had to compute the velocity field from the vorticity field (this is done in Fourier domain). As expected, the wavelet coefficients give insight into the energy distribution over the scales of the flow. As one can see on figure 7 (left), the energy at smallest scale (or highest wavenumbers) is localized along the strong deformation lines, and fits the filamentation between vortices, or with strong changes in vortices. The top-right square corresponding ) exhibits vertical structures, whereas the bottom-left to vertical isotropic wavelets ( div ) exhibits horizontal deformation lines. square corresponding to horizontal wavelets ( div
Div-free and curl-free wavelets for Navier-Stokes
21
Turbulent vorticity field
-20
-10
0
10
Corresponding velocity field
20
Figure 6. Vorticity field for a corresponding velocity field (right).
Isotropic div-free wavelet decomposition
0
&
simulation of decaying turbulence (left), and the
Anisotropic div-free wavelet decomposition
Figure 7. Isotropic (left) and anisotropic (right) divergence-free wavelet coefficients of the velocity field of figure 6 (right).
Now we investigate the compression properties of the divergence-free wavelet analysis: as predicted by the nonlinear approximation theory (see [7]), the compression ratio in the energy-norm is governed by the underlying regularity of the velocity field in some Besov space.
Div-free and curl-free wavelets for Navier-Stokes Let
22
be an incompressible field, its divergence-free wavelet expansion writes:
div
div
div
div
div
div
The nonlinear approximation of relies on computing the best N-term wavelet approximation by reordering the wavelet coefficients: !
div
and introducing
!
div
div
!
div
!
"
!
!
!
!
div
!
!
(20) !
Then we have
(21)
if the quantity is finite, with (this means that - div belongs to the Besov space ). As stated in [7], the evaluated regularity can’t be larger than the order of polynomial reproduction in scaling spaces plus one (that equals the number and of zero moments of the dual wavelet). In our experiment, the dual spline wavelets (see appendix A) have respectively two and three zero moments, which allows us to evaluate regularities only smaller than two. Figure 8 shows the nonlinear compression of divergence-free wavelets, provided on the turbulent field. The curve represents the -error , versus , in log-log plot.
The convergence rate measured on the curve is with the slope of the curve , which shows that the velocity flow belongs to the corresponding Besov space with
. When looking at the compression curve on figure 8, we observe three different zones: - First, large scale wavelets capture large scale structures of the flows. Consequently, the compression progresses slowly and irregularly. - Then we observe a linear slope that represents the nonlinear structure of the turbulent flows. In this region, we are able to evaluate the regularity of the field. - The last region corresponds to an abrupt decrease, due to the fact that the data are discrete.
One can also remark on figure 8 that only % of the coefficients recover about % of the -norm. The same experiment was carried out on the three interacting vortices, but not reported here, since the slope of the curves saturates at , due to the small number of vanishing moments of the wavelets we use (equal to 2 in our experiment): this means that these fields are more regular, and suggests using wavelets with more vanishing moments to optimize the compression factor.
!
!
%
%
Div-free and curl-free wavelets for Navier-Stokes
23
3
10
2
10
slope: p = 0.67
1
L2 error
10
0
10
−1
10
−2
10
−3
10
0
10
1
2
10
10
3
10
4
10
5
10
6
10
7
10
number of coefficients
Figure 8. -error provided by the nonlinear N-best terms wavelet approximation (20): in log-log plot, -error (21) versus for a 2D turbulent flow.
6.3. Analysis of a 3D incompressible field In this part we consider a 3D periodic field, generated from a DNS (spectral method) of freely decaying isotropic turbulence, kindly provided to us by G.-H. Cottet and B. Michaux. The experiment is detailed in [8], and was a Gaussian initial condition for the velocity and collocation points. Figure 9 displays the vorticity isosurfaces corresponding to about of the maximum vorticity at five turnover times (time being expressed in terms of large-eddy turnover time units where is the initial integral scale and ). The Reynold number based on the Taylor microscale is initially and decreases to at (see [8] for more details) .
Figure 9. Isosurface 12.45 of vorticity magnitude after 5 large-eddy turnovers provided by a spectral method [8].
The isotropic divergence-free wavelet decomposition of the corresponding velocity field is computed, and displayed on figures 10 and 11. As explained in section 3.1.2, the isotropic 3D divergence-free wavelet decomposition provides generating wavelets , . div div
!
!
Figure 10 left shows the corresponding renormalized coefficients
div
, whereas figure
Div-free and curl-free wavelets for Navier-Stokes
10 right shows the
$
$
) wavelet coefficients div , are displayed on figure 11 below, for two different generating wavelets: we choose div
, until
24
. The smallest scale (
which corresponds to horizontal structures, and
div
which exhibits vertical ones.
(left) Figure 10. Isosurface of divergence-free wavelet coefficients associated to div and to (right), in absolute value. div
Figure 12 displays the nonlinear compression error: unlike the 2D case, due to the low resolution ( instead of ) of the field, it is difficult to detect a linear part in the curve.
Nethertheless, in a intermediate region, the curve nearly presents a slope of .
6.4. Wavelet-Helmholtz decomposition of velocity fields 6.4.1. Convergence of the method The wavelet Helmholtz decomposition presented in section 4 is tested in order to demonstrate numerically that our algorithm converges. In this part, we apply the method of section 4.3 to random 2D vector fields (defined as mathematical functions, whithout physical meaning), and we study the behaviour of the residual (see section 4.3). These vector fields are constructed with a uniform random , in terms law at each collocation point. Figure 13 displays the -norm of the residual of the number of iterations , in four different situations (different sizes for , different zero moments for the wavelets). We make the following conclusions:
Div-free and curl-free wavelets for Navier-Stokes
Figure 11. Isosurface
(left) and to
div
25
of divergence-free wavelet coefficients associated to
div
(right), in absolute value.
4
10
slope: p=0.48
3
10
2
10
1
10
0
L2 error
10
−1
10
−2
10
−3
10
−4
10
−5
10
−6
10
0
10
1
10
2
3
10
4
10
5
10
6
10
7
10
10
number of coefficients
Figure 12. log-log plot,
-error provided by the nonlinear N-best terms wavelet approximation (20): in -error (21) versus for a 3D turbulent flow.
- For all functions we have tested, the method converges, and the curves show that, except at the early beginning, the convergence is exponential. - The slope of the curve hardly depends on the number of grid-points (it is steeper with fewer grid points). - The convergence rate increases with the number of vanishing moments of the dual wavelets. Now if we want to estimate the numerical cost of the wavelet-Leray projection (divfree part) of a given (compressible) vector field, we have to compare it with the cost of the Leray-projection in Fourier domain, namely , if is the number or grid points. Since the price to pay for a Fast Wavelet Transform is of order operations, and since in numerical experiments the wavelet-Helmholtz algorithm converges in about iterations to , the whole cost is reach an error of which is asymptotically better. In the future, we will investigate the influence of the wavelet bases and of the interpolation projectors, on the convergence rate.
Div-free and curl-free wavelets for Navier-Stokes
26
4
L2 error in a logarithmic scale
3
256 x 256 points 2 zero moments
2 1
512 x 512 points 2 zero moments
0 −1
1024 x 1024 points 2 zero moments
−2 −3
256 x 256 points 3 zero moments
−4 −5 −6 0
4
8
12
16
20
24
28
32
Number of iterations
Figure 13. Convergence curves of the iterative wavelet Helmholtz algorithm.
6.4.2. Wavelet-Helmholtz decomposition of the nonlinear term of NS Since our main objective for further research is to use divergence-free wavelets to solve numerically the Navier-Stokes equations (see section 5) we have to find the wavelet-Helmholtz decomposition
of the nonlinear term . To illustrate the feasibility of our approach, we consider the 2D turbulent field displayed on Figure 6, and we compute the div-free and curl-free
wavelet components of its associated nonlinear term using the wavelet-Helmholtz decomposition: Figure 14 shows the anisotropic wavelet coefficients ( -normalized) of the divergence-free part (left) and of the curl-free part (right) arising from this decomposition. The wavelet coefficients are displayed as indicated in figure 18 of Appendix A. One can notice the appearance of smalles scale wavelet coefficients (especially at the bottom right) in the decomposition of the divergence-free part, by comparison with the decomposition 7: it is obvious that the nonlinear term contributes to the creation of small scales. From the divergence-free wavelet coefficients of figure 14, we reconstruct the
divergence-free part of , and display on figure 15 (left) the vorticity field associated to this velocity. This visualization confirms the creation of small scale structures in the nonlinear term. Figure 15 (right) displays the reconstruction of the pressure, from curl-free wavelet coefficients, as explained in section 5. As expected, low pressures correspond to coherent vortices. To control our results, we have compared the pressure obtained by the wavelet-Helmoltz decomposition, to the one computed in Fourier domain, and we have found a relative error of &
in the -norm, an error which probably arised from the interpolation procedure. On the other hand the difference between the Leray projection (in Fourier space) and the wavelet projection onto the divergence-free space represents % of the -norm. Figure 16 displays the localisation of this error. As one can see, the error is localized around strong gradients of the field, which are zones where the Fourier interpolation and the spline interpolation (preliminary step before projecting in both cases) do not give the same result.
Div-free and curl-free wavelets for Navier-Stokes
0
&
27
Figure 14. Anisotropic wavelet coefficients corresponding to the wavelet-Helmholtz decomposition of : divergence-free coefficients (left), and curl-free coefficients (right).
Figure 15. Vorticity (on the left) and differential pressure (on the right) derived from the wavelet-Helmholtz decomposition of the nonlinear term , with displayed on figure 6.
Conclusion and perspectives We have presented in detail the construction of 2D and 3D divergence-free wavelet bases, and a practical way to compute their associated coefficients. We have introduced anisotropic div-
Div-free and curl-free wavelets for Navier-Stokes
28
Figure 16. Relative error between the divergence-free part of (obtained through Fourier transform) and the one provided by the wavelet-Helmholtz decomposition.
free and curl-free wavelet bases, which are easier to handle. We have shown that these bases make possible an iterative algorithm to compute the wavelet-Helmholtz decomposition of any flow. Thus, numerical tests prove the feasibility of divergence-free wavelets for simulating turbulent flows in two dimensions and three dimensions. A divergence-free wavelet based solver for 2D Navier-Stokes equations in -formulation is underway and will be reported in a forthcoming paper. An important issue that must be addressed is the flexibility of the method: although all numerical tests have been presented in the periodic case, the method extends readily to nonperiodic problems in square or cubic domains by using wavelets incorporating homogeneous boundary conditions like those in [4, 32]. Indeed, the construction of div-free wavelets with non periodic boundary conditions was already done by K. Urban [37]. Another point is that, since we consider the -formulation for the Navier-Stokes equation, and since we are able to compute the Leray projector in the wavelet domain, the method extends easily to the 3D case. At last, we conjecture that this method should be competitive with finite element methods and spectral methods in the non-periodic case, since in this case classical methods involve a saddle point problem, and since wavelet methods take advantage both of the compression properties of the wavelet bases for functions and for operators (in the periodic case, the comparison would be unfair, since the spectral methods are clearly optimal).
Div-free and curl-free wavelets for Navier-Stokes
29
Acknowledgments The authors would like to thank G.H. Cottet and G. Lapeyre for helpfully providing to them numerical turbulent flows for analyses. This work has been supported in part by the European Community’s Improving Human Potential Programme under contract HPRN-CT2002-00286, ”Breaking Complexity”. References [1] C.-M. Albukrek, K. Urban, W. Dahmen, D. Rempfer, and J.-L. Lumley, Divergence-Free Wavelet Analysis of Turbulent Flows, J. of Scientific Computing 17(1): 49-66, 2002. [2] C. Bernardi, Y. Maday, Approximations spectrales de probl`emes aux limites elliptiques (in french), Math´ematiques & Applications 10, Springer-Verlag France, Paris, 1992. [3] P. Charton, V. Perrier, A pseudo-wavelet scheme for the two-dimensional Navier-Stokes equations, Comp. Appl. Math. 15(2): 139-160, 1996. [4] G. Chiavassa and J. Liandrat, On the effective construction of compactly supported wavelets satisfying boundary conditions on the interval, App. Comput. Harmonic Anal., 4(1), 62-73, 1997. [5] Chorin, A.J., and J.E. Marsden, A Mathematical Introduction to Fluid Mechanics, book, 3rd ed., Springer, 1993. [6] A. Cohen, I. Daubechies and J.C. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math., 45, 485-560, 1992 [7] A. Cohen, Wavelet methods in numerical analysis, Handbook of Numerical Analysis, vol. VII, P.G.Ciarlet and J.L.Lions eds., Elsevier, Amsterdam, 2000. [8] G.-H. Cottet, B. Michaux, S. Ossia and G. Vanderlinden, A comparison of spectral and vortex methods in three-dimensional incompressible flows, J. Comp. Phys 175, 1-11, 2002. [9] W. Dahmen, A. Kunoth and K. Urban, A wavelet-Galerkin method for the Stokes problem, Computing 56, 259-302, 1996. [10] I. Daubechies, Ten lectures on Wavelets, SIAM book, Philadelphia, Pennsylvania, 1992. [11] C. De Boor, A Practical Guide to Splines, book, Springer-Verlag New York Inc., 2001. [12] M. Farge, Wavelet transforms and their applications to turbulence, Ann. Rev. Flu. Mech. :395457, 1992. [13] M. Farge, N. Kevlahan, V. Perrier and E. Goirand, Wavelets and turbulence, Proc. IEEE 84(4), 639-669, 1996. [14] M. Farge and K. Schneider, Coherent Vortex Simulation (CVS), A Semi-Deterministic Turbulence Model Using Wavelets, Flow, Turbulence and Combution, 66: 393-426, 2001. [15] J. Fr¨ohlich and K. Schneider, Numerical simulation of decaying turbulence in an adaptive wavelet basis, Appl. Comput. Harmon. Anal., 3: 393-397, 1996. [16] V. Girault, P.A. Raviart, Finite element methods for Navier-Stokes equations, Springer-Verlag Berlin, 1986. [17] D. Goldstein, 0. Vasilyev, and N.K.-R. Kevlahan, Adaptive LES of 3D decaying isotropic turbulence. In Studying turbulence using numerical simulation databases - X, Proceedings of the 2004 summer program. (ed. P. Moin, N. Mansour & P. Bradshaw ), 14 pp. Stanford: CTR., 2004. [18] M. Griebel and F. Koster, Adaptive wavelet solvers for the unsteady incompressible Navier-Stokes equations, Advances in Mathematical Fluid Mechanics, J. Malek and J. Necas and M. Rokyta eds, Springer-Verlag, 2000.
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30
[19] S. Grossmann and M. L¨ohden, Wavelet analysis of Navier-Stokes flow, Z.Phys.B, 100:137-147, 1996. [20] B.L. Hua and P. Klein, An exact criterion for the stirring properties of nearly two-dimensional turbulence, Physica D 113:98-110, 1998. [21] J. Ko, A.J. Kurdila and O.K. Rediniotis, Divergence-free Bases and Multiresolution Methods for Reduced-Order Flow Modeling, AIAA Journal, 38(2): 2219-2232, 2000. [22] F. Koster, M. Griebel, N. Kevlahan, M. Farge and K. Schneider, Towards an adaptive waveletbased 3D Navier-Stokes solver, Numerical flow simulation I, Notes on Numerical Fluid Mechanics, 66, 339-364, E.H. Hirschel eds, Vieweg-Verlag, Braunschweig, 1998. [23] J.-P. Kahane and P.G. Lemari´e-Rieusset, Fourier series and wavelets, book, Gordon & Breach, London, 1995. [24] N. Kevlahan, O.V. Vasilyev, D. Goldstein and A. Jay, A three-dimensional adaptive wavelet method for fluid–structure interaction. In Direct and Large-Eddy Simulation V, (ed. B. J. Geurts, R. Friedrich & O. M´etais). 8 pp. Kluwer, 2004. [25] N.K.-R. Kevlahan and O.V. Vasilyev, An adaptive wavelet collocation method for fluid–structure interaction at high Reynolds numbers. To appear in SIAM J. Sci. Comput., 2005. [26] G. Lapeyre, Topologie de m´elange dans un fluide turbulent g´eophysique (in french), Th`ese de doctorat de l’Universit´e Paris VI, 2000. [27] P.G. Lemari´e-Rieusset, Analyses multi-r´esolutions non orthogonales, commutation entre projecteurs et d´erivation et ondelettes vecteurs a` divergence nulle (in french), Revista Matem´atica Iberoamericana, 8(2): 221-236, 1992. [28] P.G. Lemari´e-Rieusset, Un th´eor`eme d’inexistance pour les ondelettes vecteurs a` divergence nulle (in french), C. R. Acad. Sci. Paris, t. 319, S´erie I, p. 811-813, 1994. [29] J. Lewalle, Wavelet transform of the Navier-Stokes equations and the generalized dimensions of turbulence, Appl. Sci. Res. 51(1-2):109-113, 1993. [30] S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, 1998. [31] C. Meneveau, Analysis of turbulence in the orthonormal wavelet representation, Journal of Fluid Mechanics 232: 469-520, 1991. [32] P. Monasse and V. Perrier, Orthonormal wavelet bases adapted for partial differential equations with boundary conditions, SIAM J. on Math. Analysis 29(4):1040-1065, 1998. [33] K. Schneider, N. Kevlahan and M. Farge, Comparison of an adaptive wavelet method and nonlinearly filtered pseudo-spectral methods for two-dimensional turbulence, Theor. Comput. Fluid Dyn. 9: 191-206, 1997. [34] K. Schneider and M. Farge, Numerical simulation of a mixing layer in an adaptive wavelet basis, C. R. Acad. Sci. Paris, s´erie II, 263-269, 2000. [35] K. Urban, A Wavelet-Galerkin Algorithm for the Driven–Cavity–Stokes–Problem in Two Space Dimensions, RWTH Aachen, Preprint 1994. [36] K. Urban, Using divergence-free wavelets for the numerical solution of the Stokes problem, AMLI’96: Proceedings of the Conference on Algebraic Multilevel Iteration Methods with Applications, 2: 261–277, University of Nijmegen, The Netherlands, 1996. [37] K. Urban, Wavelet Bases in H(div) and H(curl), Mathematics of Computation 70(234): 739-766, 2000. [38] K. Urban, Wavelets in Numerical Simulation, Springer, 2002. [39] O.V. Vasilyev and N.K.-R Kevlahan, An adaptive multilevel wavelet collocation method for elliptic problems, To appear in J. Comput. Phys., 2005
Div-free and curl-free wavelets for Navier-Stokes
31
Appendix A: Multiresolution Analysis (MRA) Multiresolution Analyses (MRA) are approximation spaces allowing the construction of wavelet bases and were introduced by S. Mallat [30]. We recall here some definitions.
Definition (MRA): A Multiresolution Analysis of verifying:
is a sequence of closed subspaces
(1)
$
(2) (Dilation invariance) (3) (Shift-invariance)
There exists a function
is dense in
such that
span
$
is called a scaling function of the MRA. denotes the level of refinement. By virtue of (2),
span . one has
Wavelets appear as bases of complementary spaces such that , where the sum is direct, but not necessarily orthogonal. In this context (called the biorthogonal case introduced by [6]), the choice of spaces is not unique. In each space one can construct
a function , called wavelet such that span . Then the wavelet space decomposition of writes:
has the following wavelet expansion:
and any function
(22)
%
being fixed, we can associate a unique dual pair Biorthogonal basis:
that the following biorthogonality (in space ) relations are fulfilled: and
%
%
%
where wavelet coefficients
%
%
and can be obtained by:
%
%
%
%
$
, such ,
. In (22), the scaling coefficients and the
%
%
Scaling equations and filter design: Since the function
sequence (also called the low pass filter) verifying:
lives in
, there exists a
(23)
Div-free and curl-free wavelets for Navier-Stokes
32
By applying the Fourier transform , (23) rewrites:
where
"
is the transfer function of the filter
Again, because of
.
, the wavelet satisfies a two-scale equation:
(24)
where the coefficients are called the high pass filter. Again the Fourier transform of expressed with the transfer function of filter as:
is
In the same way, the dual functions satisfy scaling equations:
%
(
%
(
%
%
%
%
%
%
%
%
%
%
(25)
And the relations between the transfer functions are:
"
%
%
"
which corresponds to:
%
%
In practice, the filter coefficients and are all what is needed to compute the wavelet decomposition (22) of a given function. Notice that these filters are finite if and only if the functions and are compactly supported. Fast wavelet algorithms provide with the computation of wavelet coefficients in operation when the filters are finite (see Mallat [30]).
Example: symmetric biorthogonal splines of degree 1 A simple example for spaces are the spaces of continuous functions, which are piecewise . / linear on the intervals , for . In this case we can choose as scaling function the hat function . Its transfer function is given by
"
"
(26)
The shortest even dual scaling function associated with %
"
is associated with the filter:
"
(27)
%
The corresponding values of filters and are given in table 1, appendix C. Figure 17 displays the scaling functions and their associated wavelets in this case.
The Fourier transform of a function is defined by
Div-free and curl-free wavelets for Navier-Stokes
33
Multivariate wavelets The above considerations can be extended to multi-dimensions. The simplest way to obtain multivariate wavelets is to use anisotropic or isotropic tensor products of onedimensional functions. The anisotropic two-dimensional wavelets are constructed with tensor products of wavelets at $ $ different scales ,
where and are one-dimensional wavelet bases. Notice that these bases can come from different MRAs ( and ). The anisotropic decomposition is the easiest way to compute a multi-dimensional wavelet transform, as it corresponds to apply one-dimensional wavelet decompositions in each direction. In the two-dimensional case, the algorithm is schematized in figure 18.
Let and , (resp. , ) be the low-pass and the high pass filters corresponding to the scaling function the wavelet , and their biorthogonal functions , (resp. and and their biorthogonal functions , ). We will note in ”pseudo-code”:
%
%
%
%
%
%
%
FWT2 an
%
%
%
%
%
(28)
the 2D anisotropic fast wavelet transform in the periodic case (see [30] for the details of the algorithm) corresponding to the formula:
%
%
being the dual function of
IFWT2 an
. Similarly,
(29)
will denote the inverse transform, which corresponds to the formula:
evaluated at collocation points. Starting from cost of the is of order operations.
collocation points, the computational
1.0
1.5
6
10
0.9
1.3
5
8
0.8
1.1
0.7
0.9
0.6
0.7
4
6
3 4 2 0.5
0.5
2 1
0.4
0.3
0.3
0.1
0.2
−0.1
0 0
0.1 0 −1.0
−0.5 −0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1.0
−4
−2
−0.3
−0.8
−2
−1
−1.0
−0.6
−0.2
0.2
0.6
1.0
1.4
1.8
2.2
−3 −2.0
−1.6
−1.2
−0.8
−0.4
0
0.4
0.8
1.2
1.6
2.0
−6 −1.0
−0.6
−0.2
0.2
0.6
1.0
Figure 17. From left to right: the scaling function with its associated symmetric wavelet with shortest support, and their duals: the dual scaling function and the dual wavelet .
1.4
1.8
2.2
Div-free and curl-free wavelets for Navier-Stokes
,
34
Figure 18. Anisotropic two-dimensional wavelet transform.
In the isotropic case, the two-dimensional wavelets are obtained through tensor products of wavelets and scaling functions or wavelets at the same scale. This produces the following $ : , where: basis of
!
!
#
The interest of this basis remains in the fact that the size of their support is proportional to in each direction, i.e. the basis functions are ”isotropic”. The principle of the associated decomposition algorithm is illustrated by figure 19.
,
,
Figure 19. Isotropic two-dimensional wavelet transform.
As for the anisotropic case, we will note in ”pseudo-code”: !
FWT2
%
%
%
%
(30)
the 2D isotropic fast wavelet transform in the periodic case (see [30] for the details) corresponding to the formula: !
%
!
%
!
being the dual function of
IFWT2
!
!
. Similarly,
(31)
Div-free and curl-free wavelets for Navier-Stokes
35
will denote the inverse transform, which corresponds to the formula:
!
!
!
evaluated at collocation points. Starting from cost of the is of order operations.
collocation points, the computational
!
Appendix B: Divergence-free wavelet formulas In this apprendix, we explicit the formulas of the 2D and 3D isotropic divergence-free wavelets, and the practical way to compute associated coefficients. In two dimensions The isotropic divergence-free wavelets are elementary combinations of the six generating 2D vector wavelets written in formula 3, namely:
*
*
div !
!
/
.
&
/
div
the expansion of a vector :
&
.
div
*
In the new basis
div
*
div
*
*
div
div
div
div
div
(32)
is related to the standard one (4) by the relationship between the new coefficients and the by: original ones !
"
div
'
&
&
&
&
div
div
"
(33)
Div-free and curl-free wavelets for Navier-Stokes
36
In three dimensions
The MRA generating 3D-vector scaling functions:
,
-
provides naturally with 3
-
,
,
-
and 21 generating 3D-vector wavelets:
&
#
#
with
#
"
#
-
"
, and
"
:
-
-
-
$
%
#
!
!
!
!
!
!
It goes similarly for the expression of the wavelets associated to parameters , , and . Let introduce . The isotropic wavelet expansion of a given 3D vector function writes in this basis:
-
,
-
%
-
"
-
-
and
For example, we give below the expressions of wavelets
"
(34)
!
Following section 2.1, there exist 14 kinds of isotropic divergence-free wavelets, with arbitrary possible choices in equation (2). In the following we do not detail all the expressions
Div-free and curl-free wavelets for Navier-Stokes
37
we choose, but only some typical ones:
div
div
div
div
div
div
/
.
/
.
/
/
.
-
-
&
-
-
.
-
&
-
$
%
It goes similarly for all basis function: for each given, two divergence-free wavelets are carried out by linear combination of , , , in order to satisfy the div , divergence-free condition. The complement wavelet n is constructed in order to take care of the symmetry. For example, we consider: #
!
!
&
!
!
"
!
div
div
n
&
&
Div-free and curl-free wavelets for Navier-Stokes
and similarly for
div
div
,
.
&
"
n
and similarly for
div
and
div
"
,
.
&
div
"
div
div
div
"
and
38
n
Now we can rewrite (34):
div !
div !
div !
div !
n !
n !
!
where the divergence-free wavelet coefficients are simply obtained from the standard ones, for example:
div
div
div
div
div
div
The complement coefficients are in this case:
n
n
n
n
&
&
Div-free and curl-free wavelets for Navier-Stokes
39
Appendix C: Pseudo-code In this appendix we summarize in pseudo-code the algorithms for 2D anisotropic div-free and curl-free wavelets. It goes similarly for the isotropic case, and for the 3D case. We start from two MRA and with scaling functions , and wavelets , verifying equation (1):
%
Equation (1) rewrites for the dual functions %
%
%
%
%
%
,
%
,
%
, and
:
(35)
For the transfert functions, it can be traduced by:
%
"
"
%
(36)
The transfert functions corresponding to the spline of degree 1 and 2 of figure 1 are given by [23, 10]:
"
"
"
"
(37)
leading to the following values for the decomposition and reconstruction filters (Table 1 ):
%
%
%
%
Table 1. Decomposition filter ( ) and reconstruction filter ( ) coefficients, associated to piecewise linear splines (left) and the piecewise quadratic splines (right) verifying derivation conditions (1) with shortest supports.
Direct and inverse 2D anisotropic div-free wavelet transforms The notations and formulas are given in section 3.2.1. We suppose we are given known at collocation points a 2D periodized field . / .
Direct transform
FWT2 an
%
%
FWT2 an
%
%
%
%
%
%
Div-free and curl-free wavelets for Navier-Stokes for
div
#
of the coefficients)
#
(Renormalization
40
end In the periodic case, one has to take care of the role played by the 1D scaling function in the 2D anisotropic wavelet basis, and to adapt the above formulas for the corresponding coefficients. The div are the div-free wavelet coefficients of . Remark that if is divergence
free, one obtains (and these coefficients don’t need to be computed).
Inverse transform
Starting from the coefficients div for
div
and
:
div
#
#
end
IFWT2 an
IFWT2 an
Div-free and curl-free wavelets for Navier-Stokes
41
Direct and inverse 2D anisotropic curl-free wavelet transforms The curl-free wavelet transform provide with formulas identical to the div-free transform (see section 4.2), except that one has to permutate the role of and , and replacing the by curl , and div by .
Direct transform
Starting from
at collocation points:
FWT2 an
%
%
%
%
for
(Renormalization
curl
#
%
%
%
%
of the coefficients)
#
FWT2 an
end
The curl one obtains
are the curl-free wavelet coefficients of . Remark that if .
Inverse transform
Starting form the coefficients curl for
and
:
curl
curl
#
is curl-free,
Div-free and curl-free wavelets for Navier-Stokes
#
42
end
IFWT2 an
IFWT2 an
Div-free and curl-free wavelets for Navier-Stokes Figure captions
43
Div-free and curl-free wavelets for Navier-Stokes
44
Figure 1. Scaling functions and associated even and odd wavelets with shortest support, for splines of degree 1 (left) and 2 (right).
Figure generating divergence-free spline wavelets 2. Example of isotropic two-dimensional (left), (center) and (right). div div div
Figure 3. Idealistic schematization of the convergence process of the algorithm with HN span N and Hn span n .
Figure 4. The two scaling functions of
and
, and their symmetry centers
Figure 5. Vorticity fields at times , , divergence-free wavelet coefficients of the velocity.
and
, and corresponding
Figure 6. Vorticity field for a simulation of decaying turbulence (left), and the corresponding divergence-free wavelet coefficients of the velocity field (right).
Figure 7. Isotropic (left) and anisotropic (right) divergence-free wavelet coefficients of the velocity field of figure 6(right).
Figure 8. -error provided by the nonlinear N-best terms wavelet approximation (20): in log-log plot, -error (21) versus for a two-dimensional turbulent flow.
Div-free and curl-free wavelets for Navier-Stokes
45
Figure 9. Isosurface of vorticity magnitude after 5 large-eddy turnovers provided by a spectral method [8].
Figure 10. Isosurface of divergence-free wavelet coefficients associated to (left) div (right) in absolute value. and to div
Figure 11. Isosurface
(left) and to
div
of divergence-free wavelet coefficients associated to
div
(right), in absolute value.
Figure 12. -error provided by the nonlinear N-best terms of wavelet approximation (20): in log-log plot, -error (21) versus for a three-dimensional turbulent flow.
Figure 13. Convergence curves of the iterative wavelet Helmholtz algorithm.
Figure 14. Anisotropic wavelet coefficients corresponding to the wavelet-Helmoltz : divergence-free coefficients (left), and curl-free coefficients (right). decomposition of
Figure 15. Vorticity (on the left) and pressure (on the right) derived from the wavelet-Helmoltz decomposition of the nonlinear term , with displayed on figure 6.
Figure 16. Relative error between the divergence-free part of (obtained through Fourier transform) and the one provided by the wavelet-Helmholtz decomposition.
Figure 17. From left to right: the scaling function with its associated symmetric wavelet and the dual wavelet . with shortest support, and their duals: the dual scaling function
Figure 18. Anisotropic two-dimensional wavelet transform.
