Eric Van Reeth

Pascal Bertolino

Marina Nicolas

Jean-Marc Chassery

Adaptive Edge Orientation Analysis

Keywords: Wavelets, Bandlets, Curvelets, Edge Direction, Image Analysis, Multi-resolution

This paper presents a method that detects edge orientations in still images. Edge orientation is a crucial information when one wants to optimize the quality of edges after different processings. The detection is carried out in the wavelet domain to take advantage of the multi-resolution features of the wavelet spaces, and locally adapts the resolution to the characteristics of edges. Our orientation detection method consists of finding the local direction along which the wavelet coefficients are the most regular. To do so, the image is divided in square blocks of varying size, in which Bresenham lines are drawn to represent different directions. The direction of the Bresenham line that contains the most regular wavelet coefficients, according to a criteria defined in the paper, is considered to be the direction of the edge inside the block. The choice of the Bresenham line drawing algorithm is justified in this paper, and we show that it considerably increases the angle precision compared to other methods as for instance, the method used for the construction of bandlet bases. An optimal segmentation is then computed in order to adapt the size of the blocks to the edge localization and to isolate in each block at most one contour orientation. Examples and applications on image interpolation are shown on real images.

INTRODUCTION

Image processing has become in the last decade a very attractive field of research. Most of the processings concern image denoising, image rescaling, compression or deblurring. More and more advanced concepts have been introduced in order to make these processings converge towards the best image quality. Numerous studies have shown that image quality is strongly correlated to the quality of the image edges. This is the reason why a large number of processings adapt themselves to the edges, and try to improve or preserve the quality of edges as much as possible. Later on, some techniques have focused on taking into account the edge orientation to take advantage of the regularity along the edges. Thanks to these approaches, edges can be seen as long and regular structures instead of discontinuities, which is simpler to process. Among these techniques, we can name the work of Mallat and Le Pennec who created the bandlets, [START_REF] Lepennec | Bandelettes et representation geometrique des images[END_REF] Starck Cands and Donoho who introduced the ridgelets/curvelets, 2 or Do and Vetterli who invented the contourlets. [START_REF] Do | The contourlet transform: an efficient directional multiresolution image representation[END_REF] All these techniques focus on using the regularity of wavelet coefficients in some directions, in order to improve different processings (denoising, inpainting, compression). Other approaches have been developed, like the work of Perona [START_REF] Perona | Orientation diffusions[END_REF] which builds a geometrical flow (non-wavelet based) by iteratively diffusing gradient directions. This last approach has been used by Moloney and Jiang [START_REF] Jiang | A new direction adaptive scheme for image interpolation[END_REF] for interpolation applications. The approach that we present here is wavelet based, and combines multi-resolution features of wavelets with an adaptive segmentation method of the image, to isolate a unique edge direction in each block. The first section of this paper deals with the resolution adaptation that is performed on the image. The second part explains the segmentation method that is performed in order to describe at best the geometrical features of the objects inside the image. An innovative method is then introduced to detect the directions of the edges in blocks, by finding the direction that maximizes the homogeneity of wavelets coefficients. Examples of direction descriptions are shown at the end of the paper. Moreover, applications on image interpolations are also shown. However, the interpolation method will not be discussed here, since it is not the point of this paper. It is however useful to look at such applications since it proves the directions we have detected are coherent with the image content.

Further author information: send correspondence to Eric Van Reeth. E-mail: eric.vanreeth@st.com

OVERVIEW OF THE METHOD

The general method is shown in Figure 1. The resolution adaptation consists of dividing the image into macroblocks and choosing in each macro-block the resolution of the "A Trous" wavelet transform at which the edges are best represented. The size of the macro-blocks must be chosen with regards to the size of the objects inside the image, so that the resolution choice will reasonably be based on one edge of one object. Then, a direction analysis is performed to detect the dominant direction in each macro-block. The regularity of the wavelet coefficients of the macro-block is tested to check if there are other directions inside the macro-block. In the case where there are other directions, a segmentation of the macro-block into four sub-blocks of equal sizes is performed. In each sub-block, the direction detection process is iterated until a direction is isolated in one block or until the minimum block size is reached.

RESOLUTION ADAPTATION

"A trous" Wavelet Algorithm

Wavelets are a great tool to study image contours because they separate high and low frequencies and above this, they offer flexibility regarding resolution adaptation. [START_REF] Lindenberg | Principles for Automatic Scale Selection[END_REF] A complete description of wavelet theory and multiresolution spaces can be found in. [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF] Briefly, when performing a discrete wavelet transform of an image I, the image is separated in two spaces : the approximation space A, and the detail space D. The approximation space contains the low frequencies of I whereas the detail space contains the high frequencies, including the edges of I. By iterating this transform on approximation images, we obtain a frequency description of the image for different resolutions. The transform we use is a dyadic transform, meaning that the resolution is divided by two at each scale. If j is the scale of the wavelet transform, the resolution of the transform spaces decreases with a factor of 2 -j for j = 1, ..., J.

In our study, we use Mallat's algorithm called the "A trous" transform, which has the advantage of holding the same image size across the different scales. Thus, it introduces redundancy at low resolutions which we use as additional information to detect the orientations of edges. An example of a discrete dyadic wavelet decomposition is shown in Figure 2. We observe on this example that at the finest resolution, the detail image contains fine details (fine pant stripes) whereas at lower resolution it contains coarser details (pant border).

Local Resolution Choice

In our study, we use this property in our study to locally adapt the scale of the wavelet transform to the contour characteristics, and thus make the processing at the optimal local resolution. Here is the description of the process. Let D 2 -j be the detail image at resolution 2 -j (or scale j). Note that our study will only be performed on detail images since our study concerns edges, which are high frequency structures that only appear in the detail space. We build a partition of the images D 2 -j by dividing them into macro-blocks. Let Ω n 2 -j be the n-th macro-block of the partition of the image D 2 -j . For each macro-block Ω n 2 -j , we choose the optimal scale J opt (n) as the one that maximizes the amplitude of the wavelet coefficients.

J opt (n) = arg max j=1,...,J [sup(Ω n 2 -j ) -inf(Ω n 2 -j )] (1)
An illustration of this resolution adaptation can be seen in Figure 3. In this example, the macro-blocks Ω n 2 -j are 16 × 16 pixels squares and we choose among 3 scales : j = 1, 2, 3. We easily that "low frequency" edges (hair and face) are represented with the coarsest scale (j = 3) whereas "high frequency" edges (scarf) are represented with the finest scale (j = 1). This fast and simple approach leads to better results when detecting edges directions since edges are then represented at their optimal resolution.

IMAGE SEGMENTATION

In part 3, we showed that edges can have very different characteristics, going from low frequency transitions to high frequency textures. However, other characteristics have to be taken into account to detect their directions : their size (long, short), their neighborhood (alone or close to each others) and their shape (curved or straight). To compute their directions, we choose to isolate straight parts of edges in square blocks of appropriate size, in order to give a unique edge direction for each block. Big blocks will then be defined for long and straight edges. On the other side, small blocks will be used for parts of the image where edges are concentrated and have different directions. Thus, this section describes a method that adapts the block sizes to isolate in each block at most one edge direction.

Quad-tree segmentation

The quad-tree segmentation is a well-known method that can perform this task. Our approach is similar to the quad-tree segmentation. We analyze the homogeneity of a criterion inside each macro-block created in section 3, of size (N × N ) by comparing it to a reference value (threshold) :

• if the macro-block is considered homogeneous compared to the threshold, it is not divided.

• if the macro-block is considered not homogeneous, it is divided into four blocks of size (N/2 × N/2), and the process is iterated on these 4 blocks individually.

• the process stops when no more division should be made (the minimum block size is reached or each block contains at most one edge direction).

In our case, the homogeneity criterion is the regularity of wavelet coefficients in a certain direction θ (the determination of θ is explained in Part 5). A tolerance factor α is also introduced to make the difference between homogeneous and non-homogeneous blocks. Let G ort be the integral over the block of the absolute values of the vertical and horizontal gradients. Let G dir be the integral over the block of the absolute values of gradients along the direction θ.

If we have :

G dir ≤ G ort α (2)
i.e. if the directional gradient is weak enough (α was experimentally set to a value determined around 5) compared to the orthogonal gradients, the block is considered to be regular in the direction θ, and the block is not divided.

On the other side, if we have :

G dir > G ort α (3)
then the block is divided into four sub-blocks of same size. Note that we try to keep blocks as big as possible since we will see in Part 5 that the precision we obtain on the angle value increases with the size of the block. However small blocks are needed to capture quick changes of orientations.

Example of segmentation

An example of segmentation performed with this algorithm is shown in Figure 4. We can verify that at most one contour direction is present inside each block. In this example, the maximum block size authorized was 16 × 16 pixels, and the smallest was 4 × 4.

Note that the segmentation is shown here over the original image for clarity, but this segmentation is actually obtained from the wavelet transform image with resolution adapted wavelet coefficients.

DIRECTION DETECTION METHOD

The method presented here consists in finding the dominant direction in square blocks, composed of wavelet coefficients.

To do so, we search for the Bresenham line which links the wavelet coefficients in the most regular way. The direction of the Bresenham line that maximizes the regularity of the wavelet coefficients is considered as the dominant direction of the block. The following paragraphs describe the extraction of Bresenham lines from a square block and the definition of the regularity criterion.

Pixels sequences

To design Bresenham lines inside a square block, a 2-D → 1-D transform is performed. The 2-D block is transformed into a 1-D sequence : S(θ) of consecutive pixels (except line jumps), containing all the pixels of one block and following the Bresenham line drawing algorithm along the direction θ. See Figure 5(a) for an example of such a transform. The Bresenham line design algorithm is chosen because it designs a digital line by minimizing pixel after pixel the geometric error compared to the continuous line of the same direction. This gives the insurance that the created sequence links neighboring pixels in the optimal way to represent the direction θ.

Moreover, it is interesting to notice that the number of digital lines that can be drawn in a square block increases with the size of the block. The angular resolution increases with the size of the block since we can discriminate a larger number of angles. Also, working on a larger number of pixels increases the robustness of the direction computation by reducing the influence of noise or parasite directions (aliasing). This is why the algorithm tries to work on large blocks whenever it is possible.

Definition of the regularity criteria along a direction

We describe here the computation of a homogeneity criteria of the pixels sequences. Let d(θ) be a 1-D vector containing the absolute values of the gradient along S(θ). For a sequence that is N 2 pixels long, we have :

d(θ) = N 2 -1 i=1 S(θ)[i + 1] -S(θ)[i] (4)
d(θ) can be seen as a measure of irregularity of the pixel sequence because it sums the absolute values of the gradient along the sequence. Thus, we look for the direction θ that minimizes d(θ) as the sequence S(θ) corresponding to the direction of the contour, is the one that contains the fewest discontinuities.

As described earlier, the sequences S(θ) are composed of wavelet coefficients that have different resolutions depending on the choice of J opt , performed in Part 3. By taking advantage of this resolution adaptation, we can improve the homogeneity criteria with an approach similar to the one performed in Part 3. By computing the 1D orthogonal wavelet transform of S(θ) (Haar transform for example), we can optimally emphasize the discontinuities inside S(θ), since we know their resolution (given by J opt ). More precisely, the scale J Haar , of the 1D orthogonal wavelet transform performed on S(θ) is chosen at:

J Haar = J opt -1 (5)
Note that no wavelet transform is performed for J Haar = 0, i.e. J opt = 1. Thanks to this resolution adaptation, sequences containing many discontinuities (sequences that do not follow the direction of the edge) will be more "penalized" when d W T (θ) is computed, which makes the discrimination of the optimal direction easier.

At the end, d(θ) is computed as follows:

d W T (θ) = N 2 -1 i=1 W T Haar J Haar ( S(θ)[i + 1]) -W T Haar J Haar ( S(θ)[i]) (6)
In Figure 6, d W T is plotted for a block containing a straight oblique edge. We observe a minimum of d W T at 78 degrees which corresponds to the direction of the contour. One can find in Figure 7 a complete scheme that summarizes the segmentation method as well as the direction detection method performed inside each macro-block.

Why chaining neighboring pixels?

A close concept can be found in the construction of bandlet bases introduced by Peyre and Mallat 89 to detect directions inside blocks. However, direction detection is performed in a different manner which is described in. [START_REF] Peyre | Bandlet toolbox, www[END_REF] Basically, instead of chaining neighbor pixels of the block along a Bresenham line, a rotation matrix is applied on the matrix containing the pixel position in the block. Let θ be the rotation angle. Then, the pixels are chained by order of proximity to the continuous lines of angle θ.

Although this method is theoretically the best way to create pixels sequences that follow a given direction, one can notice that using this method as a direction detector is not optimal since it tends to detect what we call the "basic" directions (i.e. π/4, π/2, 3π/4, π) more than others. This can be observed in Figure 8, where the direction detection is applied to a blurred circle. We expect that no directions will be privileged and that d W T (θ) will be flat, showing that all directions are equivalent. On the contrary we can observe that low peaks (a) Blurred Circle (b) Plot of d W T (θ) using the bandlet method to create S(θ) of d W T (θ) are obtained for the 4 basic directions, and more generally for values factor of kπ 2 n , for k = 1, 2, ..., 2 n . This indicates that pixels along these angles are seen as more regular as for other directions, which explains why basic directions tend to be detected too often. Our reason for this bias can be that for basic angles, the chained pixels inside S(θ) are geographically closer to each other than for other directions. This implies that inside a block (finite domain), more accurate chainings can be performed for basic directions. On the opposite, more inappropriate chainings have to be performed for other directions, which explains the higher values found in the computation of d W T (θ). One can observe in Figure 8 that our method also has a small bias but which is far weaker than the one inserted in the bandlet method. Note that for the basic angles, values of d W T (θ) are identical for both our method and the bandlet method since these angles will be represented by the chaining of neighboring pixels in the bandlet method as well.

RESULTS

Here are some results obtained using the whole process described in this paper. The results of the decomposition and the direction analysis are shown on Figure 9. This example contains many different directions. Our algorithm succeeds in following the directions of high frequency contours (stripes) as well as lower frequency ones (arm) thanks to the resolution adaptation that represents the image edges at the optimal resolution. The detected directions also seem to fit the actual contour directions thanks to the accuracy of the direction detection method. Finally quick changes of edges orientations can be handled thanks to the computation of an appropriate segmentation. Some remarks can however be made about this method:

• The block locations are fixed. We can only adapt the division inside the block, so it sometimes happens that even inside the smallest block size, there is still more than one direction.

• Weaker contours can be masked by well contrasted contours inside a block. Thus, the chosen direction will be the one of the strong contour without any further division.

• This method is quite demanding in computing resources since all the directions are tested per block, before choosing the optimal one. We could think of introducing statistical parameters, based for instance on the detected directions of neighbor blocks, to reduce the number of calculations.

A comparison with the orientation detection method of Perona (as described in [START_REF] Jiang | A new direction adaptive scheme for image interpolation[END_REF] ) can be seen in Figure 11, from the image part selected in Figure 10. We can observe that our flow is much more regular and closer to the real directions of the image content. The precision gain is obtained by working on blocks that are big enough to capture the actual direction of the contour, and that contain resolution adapted wavelet coefficients. This gives us more information to detect the edge orientation.

APPLICATIONS

Many applications can benefit from our multi-resolution directional description of images. Among them:

• Image enhancement that comprises deblurring, denoising, anti-aliaising

• Inpainting in particular in images that include long regularities

• Image compression that can be improved when compressing objects according to their geometrical characteristics

• Image interpolation (zoom) can also be improved by interpolating along the contour directions to avoid most of the classical interpolation artifacts (blur, ringing, staircase effect).

We show the interest of our directional decomposition by applying it to image interpolation (Figure 12). The figure shows the efficiency and the accuracy of our direction detection technique. The example shown is a four times interpolation performed from Figure 10. As expected, a reduction of the staircase and aliasing artifacts can be noticed in our interpolation thanks to the accuracy of the direction detection.

CONCLUSION

A complete method for the detection of edges directions has been presented in this paper. It provides a good description of any kind of edges thanks to the resolution adaptation. Moreover, it provides a high precision on the angle value since the directions of edges are captured inside blocks that are as big as possible. And finally, a novel non-biased direction detection method is explained based on the digital line drawing algorithm of Bresenham. We showed some applications on image interpolation, and our research now focuses on this point, but it is important to keep in mind that many other applications can benefit from such a directional analysis.

Figure 1 .

 1 Figure 1. Global process of the whole direction detection method.

Figure 2 .

 2 Figure 2. Example of a three level dyadic wavelet decomposition using the Mallat "'A trous"' algorithm. Resolution is decreasing from left to right. Approximation images are on top and Detail images are at the bottom.

Figure 3 .

 3 Figure 3. a) Detail Image D 2 -1 . b) Detail Image D 2 -2 . c) Detail Image D 2 -3 . d) Resolution adapted image.

Figure 4 .

 4 Figure 4. Left: Original image. Right: Adaptive segmentation performed with the described algorithm.

  (a) Sequence S(θ) created with the Bresenham algorithm (b) Sequence S(θ) created with the Bandlet algorithm

Figure 5 .

 5 Figure 5. Comparison of two different ways of creating sequences.

Figure 6 .

 6 Figure 6. Left: block to be analyzed. Right: Plot of dW T (θ).

Figure 7 .

 7 Figure 7. Complete algorithm of the direction detection method and the segmentation performed inside a macro-block

Figure 8 .

 8 Figure 8. Comparison of the bias introduced by the bandlet method (blue) and our method (red).

Figure 9 .

 9 Figure 9. a) Decomposition performed on the image. b) Associated directions.

Figure 10 .

 10 Figure 10. Selection of a part of the image Barbara to perform a direction analysis.

Figure 11 .

 11 Figure 11. Comparison of our detection method (a), with Perona's method (b).

Figure 12 .

 12 Figure 12. Comparison of different interpolation techniques.


Laurent Gosse
email: l.gosse@area.ba.cnr.it

TIME-SPLITTING SCHEMES AND MEASURE SOURCE TERMS FOR A QUASILINEAR RELAXING SYSTEM

Keywords: relaxation schemes, nonconservative products, conservation laws

Several singular limits are investigated in the context of a 2×2 system arising for instance in the modeling of chromatographic processes. In particular, we focus on the case where the relaxation term and a L 2 projection operator are concentrated on a discrete lattice by means of Dirac measures. This formulation allows to study more easily some timesplitting numerical schemes.

Introduction

Relaxation is a phenomenon which appears in a wide variety of physical situations. In gas dynamics, it occurs when the gas isn't in thermodynamic equilibrium. In elasticity, it is usually referred to as a fading memory mechanism. Hyperbolic conservation laws with relaxation are also served as discrete kinetic models. Relaxation approximations to scalar conservation laws can be considered for purely numerical purposes too; general references are e.g. 9,11,13,14,15,10,17,16,22,23,26,32 . In 23 , T.P. Liu studied the following 2×2 simple model which captures the basic features of this physical process:

∂ t u + ∂ x f (u, v) = 0, ∂ t v + ∂ x g(u, v) = 1 ε (V (u) -v), x ∈ R, t > 0 with ∂ v f (u, v) < 0. (1.1)
The numerical issue in such a system is to handle properly the relaxing process in its infinite strength limit ε → 0 on coarse computational grids, see 27,12 . This requirement is twofold as it includes some numerical stability properties and a consistency with the expected asymptotic behavior in the context of weak solutions. In order to derive numerical schemes meeting these criteria, attention has been driven onto several kinds of efficient treatments for the relaxation term: see 1,8,9,15,25 . Indeed, robust and efficient approximations can be derived based on "time-splitting" algorithms where one alternates at every time-step the resolution of the purely convective part and the handling of the reaction process by any O.D.E. solver.

It can be sometimes shown rigorously that these numerical schemes converge using e.g. the BV compactness framework together with appropriate entropy consistency properties. Our main goal here is to complete such a study using slightly different arguments. Roughly speaking, we are about to consider both the projection and the reaction term as being "localized" on a lattice; formally, this amounts to going from (1.1) to

       ∂ t u + ∂ x f (u, v) = n∈N * (P h (u) -u)δ(t -n∆t), ∂ t v + ∂ x g(u, v) = n∈N * ∆t ε (V (u) -v) + (P h (v) -v) δ(t -n∆t), (1.2)
where δ stands for the Dirac measure in t = 0, ∆t > 0 is a given time-step and P h denotes the standard L 2 projection operator on piecewise constant functions related to a mesh-size h > 0. Indeed, this new way to handle singular source terms in hyperbolic equations can be found in 2,7,8,33 in the context of kinetic equations or semilinear relaxation to the one-dimensional conservation law. Such an approach based on "measure source terms" allows in particular to recover a fix proposed in 10 to suppress spurious behavior in the stiff regime. Indeed, the idea proposed in 10 for reactive chemical flows consists in switching the order in which the reaction and projection steps appear inside the Dirac masses, as we shall see in §3.2, §3. [START_REF] Bouchut | Kružkov's inequalities for scalar conservation laws revisited[END_REF].

In this paper, we plan to work out only the simpler case of a quasilinear 2 × 2 system with relaxation previously considered in 27,32,19 , see (2.3). We shall mainly rely on an extensive use of some nonconservative products, 5,21,29 , in order to give a rigorous definition of the right-hand side, see Propositions 1 and 2. In return, our formulation entails direct stability and convergence proofs exploiting mainly estimates easily proved at the level of the original differential problem; this allows to bypass some lengthy computations.

Hence, this work is organized as follows: in §2, we establish some compactness properties and study the nonconservative singular system obtained as the relaxation and projection terms are concentrated inside Dirac masses, see (2.23). We introduce a notion of entropy solution well suited for this kind of problems, see Definition 2. Then, in §3, we present some applications to two types of time-splitting numerical schemes for which we prove convergence towards the "equilibrium equation" as the relaxation parameter diverges, see Theorems 4 and 5.

Study of a singular relaxing system

A quasilinear 2 × 2 model

We are interested in the Cauchy problem for the following balance laws:

   ∂ t (u + v) + ∂ x f (u) = 0, ∂ t v = µ (A(u) -v) , u(., 0) = u 0 , v(., 0) = v 0 . x ∈ R, t > 0 (2.3)
This system arises e.g. in chromatography, 11,26,30 . In this context, u, v ∈ [0, 1] stand for some species densities, one contained in a fluid flowing through a fixed bed and the other being absorbed by the material on the bed. This adsorption process is modeled by the right-hand side µR(u, v) = µ(A(u)v), µ ≫ 1 where 1/µ is the relaxation time. This model has been extensively studied within the theory of BV functions, 6,34 , by the authors of 31,32 (see also [START_REF] Klingenberg | Cauchy problem for hyperbolic conservation laws with a relaxation term[END_REF][START_REF] Kurganov | Stiff systems of hyperbolic conservation laws: convergence and error estimates[END_REF][START_REF] Natalini | Recent results on hyperbolic relaxation problems[END_REF][START_REF] Pember | Numerical methods for hyperbolic conservation laws with stiff relaxation I. Spurious solutions[END_REF] under the following hypotheses:

f (0) = 0, f ′ (u) ≥ 0; A(0) = 0, A(1) = 1, A ′ (u) ≥ 0.
(2.4)

We recall here some of their main results. Definition 1 We say that the pair (u, v) is an entropy solution of (2.3) if:

(i) u(., t) and v(., t) lie in L 1 ∩ BV (R) for any t ∈ R + and satisfy (2.3) in the sense of distributions;

(ii) there exists M ∈ R + such that for any t ≥ 0, s ≥ 0:

u(., t) -u(., s) L 1 (R) + v(., t) -v(., s) L 1 (R) ≤ M |t -s|;
(iii) there holds for any k, l ∈ R and any nonnegative test-function in D(R × R + * ):

∂ t |u -k| + |v -l| + ∂ x f (u) -f (k) ≤ µR(u, v) sgn(v -l) -sgn(u -k) . (2.5)
The relevance of this notion comes from the forthcoming theorems whose detailed proofs are to be found in 32 . We stress however that a more general definition of entropy solution to (2.3) has been proposed within the class of L ∞ functions, 26 . Theorem 1 Let u 0 , v 0 ∈ L 1 ∩ BV (R); under the hypotheses (2.4), there exists a unique entropy solution to (2.3) which satisfies furthermore for any t ∈ R + :

(i) µ R(u, v)(., t) L 1 (R) ≤ M for some M ∈ R + ;
(ii) Given another set of initial data ũ0 , ṽ0 in L 1 ∩ BV (R), there holds:

u(., t) -ũ(., t) L 1 (R) + v(., t) -ṽ(., t) L 1 (R) ≤ u 0 -ũ0 L 1 (R) + v 0 -ṽ0 L 1 (R) .
The theory of Kružkov, 18 , ensures that there exists a unique entropy solution w to the following "equilibrium" equation since A ′ ≥ 0:

∂ t (w + A(w)) + ∂ x f (w) = 0, x ∈ R, t > 0 w(., 0) = w 0 ∈ L 1 ∩ BV (R).
(2.6)

Theorem 2 Under the assumptions of Theorem 1 and if w 0 = u 0 , there exists a constant M ′ ∈ R + such that:

u(., t) -w(., t) L 1 (R) ≤ M ′ µ 1 3
, t ∈ R + .

Introduction of a "localized" singular system

In all the sequel, ∆t and h will stand for fixed positive parameters related to some cartesian computational discretization.

• We first introduce a projector onto piecewise constant functions; for any j ∈ Z,

we denote C j def = [(j - 1
2 )h, (j + 1 2 )h[. Then, we define:

P h : L 1 ∩ BV (R) → L 1 ∩ BV (R) ϕ → 1 h Cj ϕ(x).dx j∈Z (2.7)
• Let I n def = [n∆t, (n + 1)∆t[; we define two functions which are Lipschitz continuous with respect to time for ǫ > 0 small enough:

R + ∋ t → a ǫ (t) =    (n -1)∆t, t ∈ [(n -1)∆t, n∆t -ǫ[ n∆t + t -n∆t ǫ ∆t, t ∈ [n∆t -ǫ, n∆t[
(2.8) and:

R + ∋ t → b ǫ (t) =    n∆t + t -n∆t ǫ ∆t, t ∈ [n∆t, n∆t + ǫ[ (n + 1)∆t, t ∈ [n∆t + ǫ, (n + 1)∆t[ (2.9)
We can therefore consider the following system for any ǫ > 0, µ, ν in R + within the theory proposed in 32 :

   ∂ t u ǫ + ∂ x f (u ǫ ) = ν(P h (u ǫ ) -u ǫ )∂ t a ǫ -µR(u ǫ , v ǫ )∂ t b ǫ , ∂ t v ǫ = ν(P h (v ǫ ) -v ǫ )∂ t a ǫ + µR(u ǫ , v ǫ )∂ t b ǫ , u ǫ (., 0) = u 0 , v ǫ (., 0) = v 0 .
x ∈ R, t > 0 (2.10)

In the limit ǫ → 0, both a ǫ and b ǫ converge almost everywhere towards the piecewise constant function t → ∆t n∈N * Y (t -n∆t), where Y stands for the classical Heaviside distribution.

We give at once a relaxation estimate for (2.10) which is uniform in ǫ.

Lemma 1 Under the assumptions of Theorem 1, the entropy solution of (2.10)

satisfies the relaxation estimate µ R(u ǫ , v ǫ )∂ t b ǫ M loc (R×R + * ) ≤ C for some C ∈ R + . Proof.
We follow the classical ideas from 17,26,32 introducing a special entropy/entropy-flux pair for (2.10):

η(u ǫ , v ǫ ) = (u ǫ ) 2 2 +H(v ǫ ), H(v ǫ ) = v ǫ 0 A -1 (ξ).dξ, q(u ǫ ) = u ǫ ξ.f ′ (ξ).dξ.
We observe that under the hypotheses (2.4), the function H is strictly convex. Therefore, for any positive test-function, the following inequality holds:

∂ t η(u ǫ , v ǫ ) + ∂ x q(u ǫ ) ≤ -µ A(u ǫ ) -v ǫ u ǫ -A -1 (v ǫ ) ∂ t b ǫ +ν (P h (u ǫ ) -u ǫ )u ǫ + (P h (v ǫ ) -v ǫ )A -1 (v ǫ ) ∂ t a ǫ . (2.11)
We integrate (2.11) on R × [0, T ] for any T ∈ R + . Using Jensen's inequality, we can get rid of the terms involving the projection operator since ∂ t a ǫ ≥ 0:

R×[0,T ] (P h (u ǫ ) -u ǫ )u ǫ ∂ t a ǫ .dx.dt = h [0,T ] P h (u ǫ ) 2 -P h (u ǫ ) 2 ∂ t a ǫ .dt ≤ 0.
We observe now that since A ∈ C 1 (0, 1) is increasing according to (2.4), the derivative of its inverse mapping satisfies the following bound:

(A -1 ) ′ C 0 ≥ 1 A ′ C 0 def = C.
And we have for any w, v in [0, 1]:

A -1 (w) -A -1 (v) (w -v) ≥ C(w -v) 2 .
Hence the integral involving P h (v ǫ ) is treated the following way:

R×[0,T ] P h (v ǫ ) -v ǫ A -1 (v ǫ )∂ t a ǫ .dx.dt = - R×[0,T ] P h (v ǫ ) -v ǫ A -1 (P h (v ǫ )) -A -1 (v ǫ ) ∂ t a ǫ .dx.dt ≤ -C R×[0,T ] P h (v ǫ ) -v ǫ 2 ∂ t a ǫ .dx.dt ≤ 0.
Finally, choosing w = A(u), we have for C > 0, the lower bound of (A -1 ) ′ :

(A(u ǫ ) -v ǫ )(u ǫ -A -1 (v ǫ )) ≥ C(A(u ǫ ) -v ǫ ) 2 .
Thus we get the expected relaxation estimate (see also 17 ):

Cµ R×[0,T ] A(u ǫ ) -v ǫ 2 ∂ t b ǫ .dx.dt ≤ R η(u ǫ , v ǫ )(x, 0) -η(u ǫ , v ǫ ) (x, T ).dx. Since BV (R) ⊂ L ∞ (R), we are done. 2
Another important feature of (2.10) lies in the following compactness result which will be of constant use in the sequel of the paper.

Lemma 2 Under the assumptions of Theorem 1, let (u ǫ , v ǫ ) be a sequence of entropy solutions to (2.10)

. Then (u ǫ , v ǫ ) is relatively compact in L 1 loc (R × R + * ) as ǫ → 0.

Proof.

It follows the classical BV stability canvas and makes use of the so-called "quasi-monotonicity" property of the relaxation term, 1,26,31 .

• L 1 (R) bound: we multiply (2.10) by (sgn(u ǫ ), sgn(v ǫ )) T and integrate on x ∈ R. The point to be checked is the behavior of the source terms, but we see that:

R sgn(u ǫ )(P h (u ǫ ) -u ǫ )(x, t).dx ≤ j∈Z Cj |u ǫ (x, t)|.dx -u ǫ (., t) L 1 (R) ≤ 0.
Also, we notice that R(0, 0) = 0 and we can use the mean-value theorem.

Thanks to the sign assumption on A ′ in (2.4), we get:

R (sgn(v ǫ ) -sgn(u ǫ ))(A ′ (ξ)u ǫ -v ǫ ).dx ≤ 0.
Since ∂ t a ǫ and ∂ t b ǫ are nonnegative, we derive finally:

u ǫ (., t) L 1 (R) + v ǫ (., t) L 1 (R) ≤ u 0 L 1 (R) + v 0 L 1 (R) .
(2.12)

• BV(R) bound: Relying on 18 , we can differentiate the system (2.10) with respect to x, we multiply by (sgn(∂ x u ǫ ), sgn(∂ x v ǫ )) T and integrate on x ∈ R.

Using exactly the same arguments, we derive:

T V (u ǫ )(., t) + T V (v ǫ )(., t) ≤ T V (u 0 ) + T V (v 0 ). (2.13)
• Let t 0 ≥ 0 and t = t 0 + ξ, ξ > 0; we compute

                   R |u ǫ (x, t) -u ǫ (x, t 0 )|.dx ≤ R t t0 |∂ x f (u ǫ )|.dt.dx + R t t0 µ|R(u ǫ , v ǫ )|∂ t b ǫ .dt.dx + R t t0 ν|P h (u ǫ ) -u ǫ |∂ t a ǫ .dt.dx, R |v ǫ (x, t) -v ǫ (x, t 0 )|.dx ≤ R t t0 µ|R(u ǫ , v ǫ )|∂ t b ǫ .dt.dx + R t t0 ν|P h (v ǫ ) -v ǫ |∂ t a ǫ .dt.dx,
in conjunction with the bounds:

       µ R×[0,T ] |R(u ǫ , v ǫ )|∂ t b ǫ .dx.dt ≤ O(1), R |P h (u ǫ ) -u ǫ | + |P h (v ǫ ) -v ǫ | (x, t).dx ≤ h T V (u 0 ) + T V (v 0 ) .
And we derive:

sup ξ =0 R×[0,T ] |u ǫ (x, t + ξ) -u ǫ (x, t)| ξ + |v ǫ (x, t + ξ) -v ǫ (x, t)| ξ .dx.dt ≤ (T + ∆t). (νh + Lip(f ))(T V (u 0 ) + T V (v 0 )) + O(1) (2.14)
Therefore, we see that the sequence (u ǫ , v ǫ ) lies in BV loc (R × R + * ) and it remains to invoke Helly's compactness principle to conclude the proof. 2

We notice that because of the Dirac masses in time arising at the right-hand side of the equations, layers are likely to appear every time n∆t, n ∈ N * and

u ǫ , v ǫ ∈ C 0 (R + ; L 1 (R)), ǫ → 0,
as in (for instance) 28 in the case of initial data being not "well-prepared"; or it can be evidenced just by taking u 0 ≡ 0, v 0 (x) = sin(πx/h) inside (2.23).

A meaning for the ambiguous products

The preceding bounds for u ǫ imply some compactness for the terms lying in the right-hand side in the weak-⋆ topology of measures on R × R + * . More precisely, we plan to shed some light on the products emanating in (2.10) in the limit ǫ → 0. First, we state a stabilization result. Lemma 3 Under the assumptions of Theorem 1, the sequences of entropy solutions

(u ζ , v ζ ) to the Cauchy problem with initial data (u ζ 0 , v ζ 0 ) ∈ L 1 ∩ BV (R) for: ∂ t u ζ + ζ.∂ x f (u ζ ) = ν∆t(P h (u ζ ) -u ζ ), ∂ t v ζ = ν∆t(P h (v ζ ) -v ζ ), or ∂ t u ζ + ζ.∂ x f (u ζ ) = -µ∆tR(u ζ , v ζ ), ∂ t v ζ = µ∆tR(u ζ , v ζ ), are relatively compact in L 1 loc (R × R + * ) and belong to C 0 (R + ; L 1 (R)) as ζ → 0. Proof. It follows from the same arguments than the proof of Lemma 2. 2 Proposition 1 Under the hypotheses of Lemma 2, (u ǫ , v ǫ ) → (u, v) in L 1 loc (R×R + * )
as ǫ → 0 up to a subsequence. Moreover, there holds:

P h (u ǫ ) -u ǫ P h (v ǫ ) -v ǫ ∂ t a ǫ weak-⋆ M ⇀ n∈N * ∆t 1 0 P h (ū) - ū P h (v) -v (x, τ )dτ δ(t -n∆t) (2.15)
where ū, v satisfy the following differential equations for τ ∈ [0, 1]:

∂ τ ū v = ν∆t P h (ū) - ū P h (v) -v , (2.16
)

together with the initial data, x ∈ R, t = n∆t for n ∈ N * : ū(x, τ = 0) = u(x, t -0), v(x, τ = 0) = u(x, t -0).
(2.17)

Proof. We want to compute the value of the following expression for every φ ∈ C 0 c (R×R + * ), the space of continuous functions with compact support in R×R + * :

R×R + * (P h (u ǫ ) -u ǫ )∂ t a ǫ φ(x, t).dx.dt = n∈N * R n∆t n∆t-ǫ (P h (u ǫ ) -u ǫ ) ∆t ǫ φ(x, t).dx.dt.
(2.18)

We pick a n ∈ N * and define τ = 1+ t-n∆t ǫ . Thus, according to this "inner variable", the system (2.10) rewrites for τ ∈ [0, 1]:

∂ τ ūǫ + ǫ∂ x f (ū ǫ ) = ν∆t(P h (ū ǫ ) -ūǫ ), ∂ τ vǫ = ν∆t(P h (v ǫ ) -vǫ ),
together with the initial data:

ūǫ (x, τ = 0) = u ǫ (x, n∆t -ǫ), vǫ (x, τ = 0) = v ǫ (x, n∆t -ǫ).
We perform the change of variables t → τ in (2.18) to get:

n∈N * R n∆t n∆t-ǫ (P h (u ǫ ) -u ǫ ) ∆t ǫ φ(x, t).dx.dt = n∈N * R 1 0 (P h (ū ǫ ) -ūǫ )(x, τ ) ∆t ǫ φ(x, n∆t + ǫτ -ǫ).dx.ǫdτ.
Lemma 3 implies that ūǫ satisfies (2.15), (2.16) in the limit ǫ → 0 and we are done.

2 Remark 1 Thanks to the linear form of the projection term, it is possible to define directly the limit of (P h (u ǫ )u ǫ )∂ t a ǫ as ǫ → 0 as a distribution of order zero (i.e. a bounded measure) provided ∆t = O(h). Let φ ∈ D(R × R + * ), we can compute:

n∈N * R (P h (u) -u)φ(x, n∆t).dx = j,n∈Z×N * h P h (u)P h (φ) -P h (uφ) (jh, n∆t).

And we have

j∈Z |P h (u)P h (φ) -P h (uφ)|(jh, t) ≤ φ C 0 T V (u)(., t).
Concerning the relaxation term, the situation could be different because of its nonlinear structure. Nevertheless, we have the following result. Proposition 2 Under the hypotheses of Lemma 2, (u ǫ , v ǫ ) → (u, v) in L 1 loc (R×R + * ) as ǫ → 0 up to a subsequence. Moreover, there holds:

A(u ǫ ) -v ǫ ∂ t b ǫ weak-⋆ M ⇀ n∈N * ∆t 1 0 A(ū) -v (x, τ )dτ δ(t -n∆t) (2.19)
where ū, v satisfy the following differential equations for τ ∈ [0, 1]:

∂ τ ū v = µ∆t -A(ū) -v A(ū) -v , (2.20)
together with the initial data, x ∈ R, t = n∆t for n ∈ N * :

ū(x, τ = 0) = u(x, t -0), v(x, τ = 0) = v(x, t -0). (2.21) Proof. Let φ ∈ C 0 c (R × R + * )
, we intend now to pass to the limit ǫ → 0 in the following expression:

R×R + * A(u ǫ ) -v ǫ ∂ t b ǫ φ(x, t).dx.dt = n∈N * R n∆t+ǫ n∆t A(u ǫ ) -v ǫ ∆t ǫ φ(x, t).dx.dt. (2.22)
We pick a n ∈ N * and define τ = t-n∆t ǫ as an "inner variable" inside the system (2.10) which reads for τ ∈ [0, 1]:

∂ τ ūǫ + ǫ∂ x f (ū ǫ ) = -µ∆t A(ū ǫ ) -vǫ , ∂ τ vǫ = µ∆t A(ū ǫ ) -vǫ ,
together with the initial data:

ūǫ (x, τ = 0) = u ǫ (x, n∆t), vǫ (x, τ = 0) = v ǫ (x, n∆t).
In order to conclude the proof, we perform the change of variables t → τ in (2.22) and invoke Lemma 3 to derive (2. [START_REF] Kurganov | Stiff systems of hyperbolic conservation laws: convergence and error estimates[END_REF]), (2.20) in the limit ǫ → 0. 2

A contraction result "à la Kružkov"

We can now consider weak solutions of the following singular relaxing system for µ, ν in R + , x ∈ R, t > 0:

       ∂ t u + ∂ x f (u) = n∈N * ∆t ν(P h (u) -u) -µR(u, v) δ(t -n∆t), ∂ t v = n∈N * ∆t ν(P h (v) -v) + µR(u, v) δ(t -n∆t), u(., 0) = u 0 ∈ L 1 ∩ BV (R), v(., 0) = v 0 ∈ L 1 ∩ BV (R), (2.23)
where all the measure-valued source terms are always to be understood in the sense of distributions according to the results stated in Propositions 1 and 2. We first extend the notion of "entropy solution" to (2.23).

Definition 2

We say that the pair (u, v) is an entropy solution of (2.23) if:

(i) u and v lie in L 1 ∩ BV (R × [0, T ]) for any T ∈ R + and satisfy (2.23) in the sense of distributions;

(ii) there holds for any k, l ∈ R and any nonnegative test-function in D(R × R + * ):

∂ t |u -k| + |v -l| + ∂ x f (u) -f (k) ≤ n∈N * ∆t ν(P h (u) -u)sgn(u -k) + ν(P h (v) -v)sgn(v -l)+ µR(u, v) sgn(v -l) -sgn(u -k) δ(t -n∆t).
(2.24)

The main result of this section states that system (2.23), (2.24) is L 1 (R)-contractive.

Theorem 3 Let (u 0 , v 0 ) ∈ L 1 ∩ BV (R)
; under the hypotheses (2.4), there exists a unique entropy solution (u, v) to (2.23). Moreover, if (ũ 0 , ṽ0 ) stands for another set of initial data in L 1 ∩ BV (R) and (ũ, ṽ) for its associated entropy solution, there holds for any t ∈ R + :

u(., t) -ũ(., t) L 1 (R) + v(., t) -ṽ(., t) L 1 (R) ≤ u 0 -ũ0 L 1 (R) + v 0 -ṽ0 L 1 (R) .
Despite the fact that the solution of (2.23), (2.24) suffer discontinuities on the lines t = n∆t, n ∈ N * , its BV regularity allows one to define a set N ⊂ [0, T ] of measure zero containing 0 and T such that each point in the strip (x, t) ∈ R × [0, T ], t ∈ N is either an approximate jump or an approximate continuity location, 6 .

Proof. We need first to check the regularity criterion (i) in Definition 2. This is indeed a direct consequence of the uniform bounds (2.12), (2.13), (2.14) in the proof of Lemma 2. We start from the entropy inequality (2.5) satisfied by u ǫ , v ǫ for ǫ > 0:

∂ t |u ǫ -k| + |v ǫ -l| + ∂ x f (u ǫ ) -f (k) ≤ ν (P h (u ǫ ) -u ǫ )sgn(u ǫ -k) + (P h (v ǫ ) -v ǫ )sgn(v ǫ -l) ∂ t a ǫ + µR(u ǫ , v ǫ ) sgn(v ǫ -l) -sgn(u ǫ -k) ∂ t b ǫ .
(2.25)

Therefore, the inequality (2.24) comes in the limit of (2.25) by passing to the limit ǫ → 0 as in the proofs of Propositions 1 and 2.

Concerning the contraction property which implies uniqueness, we follow the ideas of 18,20,32 . Let us now introduce a nonnegative test function Ψ(x, t, y, s) = ψ(x, t).ζ(x-y).ζ(t-s) belonging to D((R×R + * ) 2 ) with ζ the standard approximation of the Dirac mass. We test the preceding entropy inequalities (2.25) for both u ǫ , v ǫ and ũǫ , ṽǫ with this particular function, we add and integrate on (R × R + * ) 2 . For ǫ > 0, the standard theory of Kružkov allows to let ζ concentrate to the Dirac measure since a ǫ , b ǫ are Lipschitz functions. We obtain:

∂ t |u ǫ -ũǫ | + |v ǫ -ṽǫ | + ∂ x f (u ǫ ) -f (ũ ǫ ) ≤ ν ((P h (u ǫ ) -u ǫ ) -(P h (ũ ǫ ) -ũǫ ))sgn(u ǫ -ũǫ )+ ((P h (v ǫ ) -v ǫ ) -(P h (ṽ ǫ ) -ṽǫ ))sgn(v ǫ -ṽǫ ) ∂ t a ǫ + µ(R(u ǫ , v ǫ ) -R(ũ ǫ , ṽǫ )) sgn(v ǫ -ṽǫ ) -sgn(u ǫ -ũǫ ) ∂ t b ǫ . (2.26)
We study mainly the zero-order terms. Using the mean-value theorem for the relaxation term leads to

R×R + * sgn(v ǫ -ṽǫ ) -sgn(u ǫ -ũǫ ) A(u ǫ ) -A(ũ ǫ ) -v ǫ + ṽǫ ∂ t b ǫ ψ(x, t).dx.dt = R×R + * A ′ (ξ) sgn(v ǫ -ṽǫ )(u ǫ -ũǫ ) -|u ǫ -ũǫ | ∂ t b ǫ ψ(x, t).dx.dt + R×R + * sgn(u ǫ -ũǫ )(v ǫ -ṽǫ ) -|v ǫ -ṽǫ | ∂ t b ǫ ψ(x, t).dx.dt,
for a ξ ∈ [min(u ǫ , ũǫ ), max(u ǫ , ũǫ )]. Since ∂ t b ǫ ≥ 0 and A ′ ≥ 0, this term gives a nonpositive contribution on the right-hand side of the entropy inequality (2.26). Concerning the projection operator, we see that

R×R + * sgn(u ǫ -ũǫ ) P h (u ǫ -ũǫ ) -(u ǫ -ũǫ ) ∂ t a ǫ ψ(x, t).dx.dt = R×R + * sgn(u ǫ -ũǫ ) -sgn(P h (u ǫ -ũǫ )) P h (u ǫ -ũǫ ) -(u ǫ -ũǫ ) ∂ t a ǫ ψ.dx.dt + R×R + * sgn P h (u ǫ -ũǫ ) P h (u ǫ -ũǫ ) -(u ǫ -ũǫ ) ∂ t a ǫ ψ(x,

t).dx.dt.

Since P h (u ǫũǫ ) is constant on each cell C j , the last integral vanishes. Once again, since ∂ t a ǫ ≥ 0, this term gives a nonpositive contribution in (2.26). We end up with:

∂ t |u ǫ -ũǫ | + |v ǫ -ṽǫ | + ∂ x f (u ǫ ) -f (ũ ǫ ) ≤ 0.
It remains now to select ψ as in 18,20,3 to complete the proof of Theorem 3. 2 Remark 2 We can notice that this contraction property gives back the BV (R) regularity for u(., t), v(., t) as soon as u 0 , v 0 belong to BV (R) since the problem (2.23) is translation invariant.

We notice finally that since P h is linear, any weak solution of (2.23) also satisfies:

   ∂ t (u + v) + ∂ x f (u) = n∈N * ν∆t P h (u + v) -(u + v) δ(t -n∆t), ∂ t v = n∈N * ∆t ν(P h (v) -v) + µR(u, v) δ(t -n∆t).
3. Study of the resulting "time-splitting" numerical schemes

A technical lemma

We point out quickly that the special structure of the "singular projection term" allows us to refine a bit the entropy inequality (2.24). Lemma 4 In the sense of Proposition 1, the following holds for any k ∈ R and any nonnegative continuous compactly supported test-function:

n∈N * ν∆t P h (u) -u sgn(u -k).δ(t -n∆t) ≤ n∈N * |P h (u) -k| -|u -k| 1 -exp(-ν∆t) .δ(t -n∆t). Proof. Let 0 ≤ φ ∈ C 0 c (R × R + * )
; we perform the following splitting: sgn(ūk) = sgn(P h (ū)k) + sgn(ūk)sgn(P h (ū)k) ,

P h (ū) -ū = (P h (ū) -k) -(ū -k).
Since we have sgn(x).y ≤ |y| for any x, y in R 2 , we get out of (2.16) for any k ∈ R:

∂ τ |ū -k| ≤ ν∆t |P h (ū) -k| -|ū -k| .
We observe that ∂ τ |P h (ū) -k| = 0 along the flow of (2.16) and this implies:

|ū -k|(x, 1) -|ū -k|(x, 0) ≤ 1 -exp(-ν∆t) . |P h (u) -k| -|u -k| (x, 0).
Therefore, we just have to recall the following from (2.15) to conclude the proof:

n∈N * ν∆t R 1 0 sgn(ū -k) P h (ū) -ū (x, τ ).dτ φ(x, n∆t).dx = n∈N * R 1 0 ∂ τ |ū -k|(x, τ ).dτ φ(x, n∆t).dx.

2

This means in particular that the following inequality holds as a consequence of (2.24) and this last lemma: (3.27) in the sense of Proposition 2. One can notice the similarity between this last inequality and the one derived in 31 §3; indeed the numerical approximation considered by these authors matches ours in the special case where ν → +∞ and a backward Euler solver is chosen instead of (2.20). This reduces to a time-splitting scheme with some kind of "straight lines" approximation of the quantity (2.19). In contrast, we recall that it has been proved in 3 §5, that one has:

∂ t |u -k| + |v -l| + ∂ x f (u) -f (k) ≤ n∈N * |P h (u) -k| -|u -k| + |P h (v) -l| -|v -l| 1 -exp(-ν∆t) +µ∆tR(u, v) sgn(v -l) -sgn(u -k) δ(t -n∆t),
n∈N * ν∆t P h (u) -u sgn(u -k) -sgn(P h (u) -k) .δ(t -n∆t) ≤ 0,
for any positive continuous test-function in the case where u ∈ C 0 (R + * ; L 1 (R)).

The classical time-splitting scheme

We plan to study the behavior of the system (2.23) in the limit ν → +∞. Roughly speaking, this reduces to insert a projection stage every time n∆t, n ∈ N * before igniting the relaxation mechanism. Proposition 3 Under the assumptions of Theorem 3, the sequence of entropy solutions of (2.23) converges strongly in L 1 loc (R × R + * ) as ν → +∞ towards the entropy solution of:

∂ t u + ∂ x f (u) = -n∈N * µ∆tR(u, v).δ(t -n∆t), ∂ t v = n∈N * µ∆tR(u, v).δ(t -n∆t), (3.28)
where the right-hand side is still defined by means of (2.19), (2.20), but with the piecewise constant initial data for t = n∆t, n ∈ N * :

ū(x, τ = 0) = P h (u)(x, t -0), v(x, τ = 0) = P h (v)(x, t -0). (3.29)
Moreover, the following entropy inequality holds true for any nonnegative test function in D(R × R + * ):

∂ t |u -k| + |v -l| + ∂ x f (u) -f (k) ≤ n∈N * |P h (u) -k| -|u -k| + |P h (v) -l| -|v -l| + µ∆t.R(u, v) sgn(v -l) -sgn(u -k) δ(t -n∆t), (3.30)
with k, l in R and according to (2.19), (2.20), (2.21).

Proof. We split it into several steps for the sake of clarity.

(i) Construction of the solution of (2.16) by a fixed point argument; for τ ∈ [0, 1], we consider any function w ∈ C 0 (0, 1; L 1 (R)) together with:

∂ τ ū = ν∆t(P h (w) -ū), ū(., 0) ∈ L 1 (R)
whose solution is obvious: ū(x, τ ) = exp(-ν∆tτ )ū(x, 0) + ν∆t exp(-ν∆tτ ) τ 0 exp(ν∆ts)P h (w)(x, s).ds.

This defines a linear mapping w → ū in C 0 (0, 1; L 1 (R)) for any choice of the initial datum. We want to establish a contraction property: for all w 1 , w 2 and any τ ∈ [0, 1], we observe that

R |ū 1 -ū2 |(τ ).dx ≤ ν∆t τ 0 exp(ν∆t(s -τ )) R |P h (w 1 ) -P h (w 2 )|(x, s).dx.ds ≤ (1 -exp(-ν∆t)) w 1 -w 2 C 0 (0,1;L 1 (R))
By Picard's fixed point theorem, we obtain a unique solution in C 0 (0, 1; L 1 (R)) for any ν ∈ R + to:

∂ τ ū = ν∆t(P h (ū) -ū), ū(., 0) ∈ L 1 ∩ BV (R)
Now, since ∂ τ P h (ū) = 0 along its flow, this differential equation therefore admits an explicit solution: ū(., τ ) = exp(-ν∆tτ )ū(., 0) + (1exp(-ν∆tτ ))P h (ū)(., 0).

In particular, we get:

ū -P h (ū) (., τ ) = exp(-ν∆tτ ) ū -P h (ū) (., 0). (3.31)
(ii) Limit as ν → +∞ of the measure (2.15); for any φ ∈ C 0 c (R × R + * ), we have:

n∈N * ν∆t R 1 0 P h (ū) -ū (x, τ ).dτ φ(x, n∆t).dx = n∈N * ν∆t R P h (u) -u (x, n∆t -0) 1 0 exp(-ντ ).dτ φ(x, n∆t).dx ν→+∞ → n∈N * R P h (u) -u (x, n∆t -0)φ(x, n∆t).dx,
which is a finite sum provided ∆t remains strictly positive.

(iii) Strong compactness as ν → +∞: the bound (2.14) seems to blow up despite the fact that (2.12), (2.13) still hold. Making use of (3.31) in the last step of the proof of Lemma 2, we notice that for any T ∈ R + and ξ > 0:

R×[t,t+ξ] 1+[T /∆t] n=1 ν∆t 1 0 ū -P h (ū) + v -P h (v) (x, τ )dτ .dx.ds ≤ ξ h ∆t (T + ∆t) T V (u 0 ) + T V (v 0 ) 1 -exp(-ν∆t) . (3.32)
Therefore, exploiting (2.15) and Lemma 1, we can deduce a fine estimate for the weak solutions of (2.23), (2.15), (2.16) which is uniform in both ν and µ:

sup ξ =0 R×[0,T ] |u(x, t + ξ) -u(x, t)| ξ + |v(x, t + ξ) -v(x, t)| ξ .dx.dt ≤ (T + ∆t) (h/∆t + Lip(f )) T V (u 0 ) + T V (v 0 ) + O(1) (3.33)
Together with (2.12), (2.13), this provides compactness in L 1 loc (R×R + * ) by means of Helly's theorem. And in the limit, we get the following system: (recall from (1.2))

   ∂ t u + ∂ x f (u) = n∈N * (P h (u) -u)(., n∆t -0) -µ∆tR(u, v) δ(t -n∆t), ∂ t v = n∈N * (P h (v) -v)(., n∆t -0) + µ∆tR(u, v) δ(t -n∆t).
Since we know that a ǫ (t) = b ǫ (tǫ), we must handle the projection step first and the initial data for (2.20) becomes (3.29). The preceding system then rewrites like (3.28).

(iv) Entropy inequality: Using the same arguments together with Lemma 4, we derive for any k ∈ R:

n∈N * ν∆t R 1 0 sgn ū -k P h (ū) -ū (x, τ ).dτ φ(x, n∆t).dx = n∈N * R |ū -k|(x, 1) -|ū -k|(x, 0) φ(x, n∆t).dx ≤ n∈N * R |P h (u) -k| -|u -k| (x, n∆t -0) 1 -exp(-ν∆t) φ(x, n∆t).dx ν→+∞ → n∈N * R |P h (u) -k| -|u -k| (x, n∆t -0)φ(x, n∆t).dx.
This way, we get (3.30) out of (2.24) and the contraction property of Theorem 3 is also preserved. 2

One can observe that as soon as the initial data of (3.28), (2.19), (2.20), (3.29) is P h -invariant and the CFL condition ∆t f ′ (u) L ∞ ≤ h holds, its entropy solution matches the piecewise constant approximation generated by a classical timesplitting Godunov scheme, see e.g. 1,31 and 9 .

Theorem 4 Under the assumptions (2.4), the CFL restriction ∆t f ′ (u) L ∞ = h and for v 0 = A(u 0 ), u 0 ∈ L 1 ∩ BV (R), the entropy solutions of (3.28), (3.30), (2.19), (2.20), (3.29) converge strongly in L 1 loc (R × R + * ) towards u, the entropy solution in the sense of Kružkov to:

∂ t u + A(u) + ∂ x f (u) = 0, u(x, 0) = u 0 .
as µ∆t → +∞, h → 0.

Proof. As we did before concerning Proposition 3, we are about to split the forthcoming proof for the sake of clarity.

(i) Study of the solution of (2.20); for τ ∈ [0, 1], we rewrite its second equation as:

∂ τ v = µ∆t( Ã(v) -v), Ã(v) = A ū(x, 0) + v(x, 0) -v(x, τ ) . (3.34)
We follow the classical approach, see e.g. 4 , and we introduce the following Banach space for some k ∈ R + to be fixed later:

X = ϕ ∈ C 0 (0, 1; L 1 (R)) such that sup t∈[0,1] exp(-k.t) ϕ(., t) L 1 (R) < +∞ .
For any w ∈ X, we consider the mapping w → v ∈ X,

v(x, τ ) = v(x, 0) exp(-µ∆tτ ) + τ 0 µ∆t exp(µ∆t(s -τ )) Ã(w)(x, s).ds,
which turns out to be a contraction for k > µ∆t(Lip(A) -1). By Picard's fixed point theorem, we get a unique solution to (3.34) v ∈ X for any µ ∈ R + . Moreover, it satisfies

∂ τ ( Ã(v) -v)(x, τ ) = -µ 1 + A ′ ū(x, 0) + v(x, 0) -v(x, τ ) ( Ã(v) -v)(x, τ ),
which leads to:

A(ū) -v (x, τ ) ≤ exp(-µ∆tτ ) A(ū) -v (x, 0).
(ii) Behavior of (3.30) as µ → +∞; the bound (3.33) can be simplified by means of the CFL condition:

sup ξ =0 R×[0,T ] |u(x, t + ξ) -u(x, t)| ξ + |v(x, t + ξ) -v(x, t)| ξ .dx.dt ≤ (T + ∆t) 2Lip(f ) T V (u 0 ) + T V (v 0 ) + O(1) . (3.35)
This provides strong L 1 loc compactness and for any positive continuous test function φ ∈ C 0 c (R × R + * ), we have:

n∈N * ∆t R µ 1 0 A(ū) -v sgn(v -l) -sgn(ū -k) (x, τ ).dτ φ(x, n∆t).dx ≤ 2 n∈N * R A(P h (u)) -P h (v) (x, n∆t) 1 0 µ∆t. exp(-µ∆tτ ).dτ φ(x, n∆t).dx ≤ O 1-exp(-µ∆t) µ∆t φ C 0 .
We used the preceding estimates in conjunction with the bound in Lemma 1. Therefore, this term vanishes in the limit µ∆t → +∞, h → 0. Moreover, by Jensen's inequality for convex functions, we also have thanks to (2.13) and the CFL condition,

n∈N * R |P h (u) -k| -|u -k| (x, n∆t -0)φ(x, n∆t).dx ≤ n∈N * R P h (u) -u (x, n∆t -0) P h (φ) -φ (x, n∆t).dx ≤ h n∈N * ∆tLip(f ) T V (u 0 ) + T V (v 0 ) Lip(φ) h→0 → 0.
(iii) It remains to pass to the limit in (3.30); we perform the following splitting for x ∈ R and t ∈ [n∆t, (n + 1)∆t[

|v -A(u)|(x, t) ≤ |v(x, t) -v(x, n∆t)| + |v -P h (v)|(x, n∆t)+ |A(P h (u)) -P h (v)|(x, n∆t)+ |A(P h (u)) -A(u)|(x, n∆t) + |A(u)(x, t) -A(u)(x, n∆t)|,
each term converging to zero in L 1 as µ∆t → +∞, h → 0 under the prescribed CFL condition. Therefore, we fix l = A(k) ∈ R and we observe that for any positive

ψ ∈ D(R × R + * ) R×R + * |v -l|∂ t ψ(x, t).dx.dt → R×R + * |A(u) -A(k)|∂ t ψ(x, t).dx.dt,
which leads to the integral form of Kružkov's entropy inequality:

R×R + * |u -k| + |A(u) -A(k)| ∂ t ψ(x, t) + f (u) -f (k) ∂ x ψ(x, t).dx.dt ≥ 0. 2

A modified time-splitting scheme

It is clear that the estimates of Lemmas 2 and 1 still hold for the slightly modified system (compare with (2.10)):

   ∂ t u ǫ + ∂ x f (u ǫ ) = -µR(u ǫ , v ǫ )∂ t a ǫ + ν(P h (u ǫ ) -u ǫ )∂ t b ǫ , ∂ t v ǫ = µR(u ǫ , v ǫ )∂ t a ǫ + ν(P h (v ǫ ) -v ǫ )∂ t b ǫ , u ǫ (., 0) = u 0 , v ǫ (., 0) = v 0 . x ∈ R, t > 0 (3.36)
The Cauchy problem for (3.36) still can be studied within the framework of 32 for any ǫ > 0. According to the same ideas, we send ǫ → 0 to get the following result. Proposition 4 Let (u 0 , v 0 ) ∈ L 1 ∩ BV (R); under the assumptions (2.4), the sequence (u ǫ , v ǫ ) of entropy solutions to (3.36) converges in L 1 loc (R × R + * ) towards the entropy solution in the sense of Definition 2 to

   ∂ t u + ∂ x f (u) = n∈N * ∆t -µR(u, v) + ν(P h (u) -u) δ(t -n∆t), ∂ t v = n∈N * ∆t µR(u, v) + ν(P h (v) -v) δ(t -n∆t), (3.37)
which satisfies for any k, l ∈ R and any nonnegative test function in D(R × R + * ):

∂ t |u -k| + |v -l| + ∂ x f (u) -f (k) ≤ n∈N * ∆t µR(u, v) sgn(v -l) -sgn(u -k) +ν(P h (u) -u)sgn(u -k) + ν(P h (v) -v)sgn(v -l) δ(t -n∆t), (3.38)
in the sense of Propositions 1 and 2. Given any other set of initial data (ũ 0 , ṽ0 ) in L 1 ∩ BV (R), this last inequality implies the contraction property for any t ∈ R + :

u(., t) -ũ(., t) L 1 (R) + v(., t) -ṽ(., t) L 1 (R) ≤ u 0 -ũ0 L 1 (R) + v 0 -ṽ0 L 1 (R) .
Proof. Similar estimates than the ones shown in the proof of Lemma 2 ensure the strong L 1 loc compactness. The inequality (3.38) comes from (2.25) together with Propositions 1 and 2. The contraction property in L 1 (R) is proved the same way than for Theorem 3. 2 Remark 3 It is important to notice that the entropy solutions of (3.37) do not coincide with the ones of (2.23) since the relaxation and the projection steps are not applied in the same order. This means in particular that one has to be careful when dealing with the right-hand sides as different processes are concentrated inside the Dirac masses. Hence there is no uniqueness problem for both systems (2.23), (3.37) relying on Propositions 1 and 2 together with the inequalities (2.24) and (3.38).

We plan now to carry out the same program as we did for the classical timesplitting scheme in the preceding section. By letting ν → +∞, we derive an equation whose entropy solution coincide with the piecewise constant approximation generated by a numerical scheme designed in the context of reactive Euler equations for which the reaction process operates before the projection stage, 10 . Proposition 5 Assume (2.4) and (u 0 , v 0 ) ∈ L 1 ∩ BV (R), the sequence of entropy solutions to (3.37) converges strongly in L 1 loc (R × R + * ) as ν → +∞ towards the entropy solution of: The final statement is concerned with the relaxation limit in (3.39). Theorem 5 Under the assumptions (2.4), the CFL restriction ∆t f ′ (u) L ∞ = h and for v 0 = A(u 0 ), u 0 ∈ L 1 ∩ BV (R), the entropy solutions of (3.39), (3.40), (2.19), (2.20), (2.21) converge strongly in L 1 loc (R × R + * ) towards u, the entropy solution in the sense of Kružkov to:

∂ t u + ∂ x f (u) =
∂ t u + A(u) + ∂ x f (u) = 0, u(x, 0) = u 0 .
as µ∆t → +∞, h → 0. The proofs of these last two statements are completely similar to the ones of Proposition 3 and Theorem 4 hence we skip them.

Conclusion

In this paper, we considered some "measure source terms" for a quasilinear relaxation system whose weak solutions coincide with piecewise constant approximations generated by commonly-used time-splitting numerical schemes. This allows to establish convergence results directly relying on properties of the underlying original system and bypassing some heavy computations required to show the stability of the numerical processes (see e.g. §2.5 in 7 ).

  n∈N * -µ∆tR(u, v) + (P h (u)u)(., n∆t + 0) δ(t -n∆t),∂ t v = n∈N * µ∆tR(u, v) + (P h (v)v)(., n∆t + 0) δ(t -n∆t), (3.39)where the right-hand sides are defined by means of (2.[START_REF] Kurganov | Stiff systems of hyperbolic conservation laws: convergence and error estimates[END_REF]),(2.20),(2.21). Moreover, the following entropy inequality holds true for any nonnegative test function in D(R × R + * ) with k, l in R:∂ t |u -k| + |v -l| + ∂ x f (u)f (k) ≤ n∈N * µ∆t.R(u, v) sgn(vl)sgn(uk) + |P h (u) -k| -|u -k| + |P h (v)-l| -|v -l| δ(t -n∆t),(3.40)

Acknowledgments: The author would like to thank Professor Wen-An Yong for several valuable discussions and suggestions. This work has been partially supported by the European Union TMR project HCL #ERBFMRXCT960033.


Eric Gobe

Deceptive Liberal Reforms: Institutional Adjustments and the Dynamics of Authoritarianism in Tunisia (1997-2005)

Currently no authoritarian regime south of the Mediterranean is likely to legitimate its rule and action independently of the dominant normative discourse praising democracy.

International institutions and organisations, notably the European Union, have made the rule of law and good governance the alpha and omega of their prescriptions and recommendations.

Consequently, authoritarian rulers now have to pay at least minimal tribute to a number of internationally established references 1 and cannot violate too obviously the principles governing the rule of law or completely eliminate political pluralism. Until the 1980s they still benefited from the rhetoric of unanimity inherited from the struggle for independence but various difficulties, in particular the confrontation with political Islam, prompted them to adopt institutional amendments supposed to 'provide a remedy for the distance that has appeared between the state and society', though without weakening their grip on power. 2 In the early 1990s, therefore a certain degree of pluralism emerged in the Maghreb countries, but it seems to be largely symbolic and does not allow an actual representation of political forces.

In Ben Ali's Tunisia, the institutional reforms introduced in the 1990s and at the beginning of the twenty-first century were presented by the regime as legal and political changes that sought to deepen pluralism within existing representative bodies. In reality, however, they only institutionalised a controlled pluralism that could not challenge the survival of the authoritarian regime. In fact, the changes were conceived so as to make a possible change in power unthinkable, while at the same time allowing the regime to talk about deepening democracy. Political systems north and south of the Mediterranean converge, insofar as the Maghreb countries in general and Tunisia in particular import legal devices that govern political institutions in democratic countries, to maintain the fiction of the rule of law while preventing the latter's translation into reality.

In other words, in Tunisia legal formalism participates in a discourse on broadening and deepening democracy and the rule of law. In reality, the recent constitutional and legislative reform in Tunisia paradoxically 'limits authoritarian practices while at the same time authorising them, indeed legitimating them if necessary.' 3

Constitutional Amendments to what End?

Recent amendments to the Tunisian constitution have been conceived and implemented by President Ben Ali in relation to the electoral calendar that determines the rhythm of the country's political life. The aim was to include in the constitution provisions to preserve and indeed to enlarge the prerogatives of the president of the republic, so as to ensure his 'reelection' while respecting legal forms.

The Constitutional Law of 27 October 1997: Strengthening Presidential Power and

Extending the Scope of the Referendum

The main provision of the constitutional law of 1997 seeks to extend the referendum to the revision of the constitution itself. According to the new article 76, 'the president of the republic can submit proposals for revision of the constitution to referendum.' Until that point, the constitution could be revised only through a vote in parliament on a proposition submitted by the president of the republic or by one third of the deputies. This amendment only took on its full significance in 2002, when it allowed the head of state to organise a referendum on his new proposals for constitutional reform that finally allowed him to stand for a fourth or even a fifth term. For an authoritarian regime, 'consulting the people in this or that form at this or that time is not a gratuitous act.' [START_REF] Badie | La politique comparée[END_REF] Organising a referendum on an issue as important as the president's term of office has a legitimising dimension, be this under Ben Ali or his predecessor, Habib Bourguiba; it takes on the value of a plebiscite.

The provision that broadens the scope of the referendum as a legislative procedure also strengthens the president's constitutional powers. [START_REF] Settembrini | Le point sur les aménagements du cadre juridique de la vie politique en Tunisie[END_REF] Previously, a referendum could be called only to reorganise the constitutional framework of the state and to ratify 'treaties having an impact on the working of the institutions.' According to the new article 47, 'the president of the republic can submit directly to referendum proposals for laws that have national importance or affect the superior interest of the country.' Since the head of state 'can alone determine this qualification, formulated in particularly general terms, it is legislation at large that is now covered by this procedure of semi-direct democracy.' 6 Another constitutional amendment further strengthens the president's powers by redefining the boundary between the domains of legislation and government decree. 7 The constitution of 1959 explicitly spelled out the prerogatives of the legislature without defining a regulatory domain, which meant that in theory, legislation was not confined to the narrow limits set by these prerogatives. Henceforth, the new article 35 removes all ambiguity, stipulating that 'matters other than those that are in the domain of law are in the purview of the regulatory power.' Moreover, the president of the republic holds the ultimate power to rule by decree (article 53) and has to protect it: he may have the Constitutional Council review any bill affecting the domain of rule by decree, and the council must reach a decision within ten days (article 35).

The amendments to article 8 constitutionalise a number of provisions of the law governing political parties. On the pretext of 'consecrating the role of parties in the constitution,' 8 the new provisions seek to ban any political group that is based, 'fundamentally in its principles, objectives, activity, or programs on a religion, a language, a race, a sex or a region' that do not respect 'the people's sovereignty, the values of the republic, human rights, and the principles governing personal status affairs'; or that fail to renounce 'any form of violence, fanaticism, racism, or discrimination.' Similar restrictions are found in the Algerian constitution as well as in the Moroccan party law of 2005. The latter explains that parties cannot be established on a 'religious, linguistic, ethnic, or regional basis that is discriminatory or contrary to human rights' (article 4), whereas article 42 of the Algerian constitution states that 'political parties cannot be based on a religious, linguistic racial, gender, corporatist, or regional foundation.' The phrasing is always imprecise and thus allows the regime to arbitrarily interpret the law. In fact, these provisions seek principally to exclude from the arena of political debate actors who have a popular base, especially Islamist groups, or at least those that are likely to contest the supremacy of the governing elites and to position themselves as a credible alternative force.

Following the lines of the 1997 amendments, those of 2002 seek above all to restorewithout saying so -the presidency for life that the 1976 constitution granted to the late Habib Bourguiba and that President Ben Ali abrogated in July 1988.

The Constitutional Reform of 1 June 2002: Restoring the Presidency for Life

The constitutional amendments examined by the Chamber of Deputies, sitting in extraordinary session on 27 February 2002, include a number of changes, most importantly to article 39. Hidden among numerous other changes, the amended article 39 de facto reestablishes the presidency for life by abolishing the earlier limitation of presidential terms to a total of three.

The other amendments (concerning 39 out of the 80 articles) fall into four categories. Some are simple enunciations of general principles or human rights. Thus the revised constitution proclaims that 'the republic guarantees the fundamental freedoms and human rights in their universal acceptation'; it also endorses 'the values of solidarity, mutual aid and tolerance among individuals, groups and generations' (new article 5), which is key to president Ben Ali's discourse of legitimation. The new article 15 stipulates that every 'citizen has a duty to protect the country, to safeguard its independence, its sovereignty, and the integrity of the national territory', allegedly seeking to 'root' the 'duty of loyalty to Tunisia.' 9

The provision allows the regime to persecute opponents who dare to criticise it in foreign media.

Other changes are deceptive and are no more than decoys. The new article 9 states that 'the inviolability of the home, the confidentiality of correspondence, and the protection of data are guaranteed,' except, of course, in cases defined by law. Moreover, 'police custody is subject to judicial supervision. Detention is permissible only on order of a judge,' 10 but the judiciary remains subservient to the executive branch of government.

The text proposes establishment of a two-round presidential election, but in the election of 1999 the two opposition candidates received only 0.52 per cent of the vote; [START_REF] Abdelhaq | Oppositions et élections en Tunisie[END_REF] opposition candidates running in the presidential election of 2004 obtained results that were only slightly less humiliating. [START_REF]See Vincent Geisser and Eric Gobe, 'Tunisie: consolidation autoritaire et processus électoraux,' L'Année du Maghreb 2004[END_REF] The 2002 reform broadened the powers of the Constitutional Council and constitutionalised its composition, which was previously governed by an organic law. The new text seems to strengthen the independence of the council. Whereas the organic law of 2 April 1996 provided for a single authority, the head of state, to appoint all nine members of the council, the new article 75 stipulates that four members are designated by the president of the republic, two by the president of the Chamber of Deputies (for a renewable period of three years), while three others sit ex officio: the first president of the Court of Cassation, the first president of the Administrative Tribunal, and the first president of the Court of Accounts. As in most democracies, members of the constitutional jurisdiction are barred from exercising responsibilities in government or parliament and 'cannot exercise functions of political or trade union leadership, or activities likely to affect their neutrality or independence.' Like the constitutional courts of some democratic regimes, the council is henceforth supposed to supervise and regulate elections. Thus it adjudicates complaints against the election of members of both houses of parliament, and it supervises referendums and announces their results (new article 72). [START_REF] Settembrini | [END_REF] The text of the revision incontestably provides the Constitutional Council with new powers insofar as its judgments have henceforth constraining for Authorities (new article 75). However, a key question remains: what is the role of a constitutional council in an authoritarian political system in which only the head of state may request it to issue a ruling? It is true that the reform of the council carries some promise, as 'the conditions governing the use of judicial instruments by [institutional] actors are not invariants.' [START_REF] Furthermore | [END_REF] However, for the time being, the authoritarian environment in which the council operates largely attenuates the features that remind us of constitutional courts in democratic regimes.

The third category of amendments allows the president to co-opt more members of his clientele by officially establishing an upper house. The new chamber is supposed to 'double representation […], so that the first chamber represents citizens directly, whereas the second represents them indirectly through the different components of society.' 15 According to the new article 19, the Chamber of Counsellors will comprise three sorts of members. The first group includes those who are chosen at the governorate level from among elected members of local councils. Each governorate sends one or two representatives to the new chamber, depending on the size of its population. Another third of members is elected at the national level by employers, farmers and salaried employees. 16 The third group is appointed by the president of the republic from 'among leading national figures', to allow Tunisia to benefit from 'the experience and the competence of its people.' 17 The procedure basically prevents any representation of oppositional forces in the upper house, since the Democratic Constitutional Rally (DCR) has a quasi-monopoly on representation on local councils and controls most of the professional organisations. In the first elections to the council on 3 July 2005, all counsellors elected from among local office holders were members of the DCR. Nonetheless, an unforeseen glitch jammed the presidential machine: the seats reserved for 'employees' were not filled because of a boycott by the Tunisian General Labour Union (Union générale tunisienne du travail, UGTT). The union refused to participate in the elections because the office of the president tried to interfere directly in the choice of 'its' candidates. In May 2005 the leadership of the UGTT voted against the union's participation in the elections. [START_REF] Geisser | Des fissures dans la "Maison Tunisie"? Le régime de Ben Ali face aux mobilisations protestataires[END_REF] As for the designation of the last third, it indeed illustrated extensive nepotism as, to use a phrase coined by Abdelwahab Hani, numerous palace 'cronies' were appointed 'crony-counsellors'. [START_REF] Abdelwahab | s analysis, 'Qui trouve-t-on dans la liste clientéliste du président?[END_REF] Prominent among them are DCR lawyers who skim off more benefits than all other members of the profession. For them, the second chamber was a godsend, rewarding the most zealous.' [START_REF]These 'crony-counsellors,' among whom we can cite the lawyers Habib Achour, Foued Haouat, Chakib Dhaouadi, Mohamed Samir Abdallah[END_REF] The third group also included representatives of Tunisia's 'true-false NGOs' [START_REF] Elyes | president of the 'official' association 'Jeunes médecins sans frontières[END_REF] and 'recycles' several former ministers and other dignitaries. 22 Consistent with the 1997 reform, the fourth category of constitutional amendments accentuates the preponderance of the president of the republic. Certain provisions strengthen his constitutional powers to the detriment of the parliament. The new article 32 assigns to the president alone the power to ratify international treaties, while the new article 28 states that 'the proposals for laws presented by members of the Chamber of Deputies are not admissible when their adoption would result in a reduction of public resources or an increase in taxes, or new expenses.' Article 41 also grants the head of state 'legal immunity', while in office, for decisions made during the exercise of his duties.

In debates at committee stage most DCR deputies showed particular zeal when it came to proposing amendments to strengthen the powers of the president. The parliamentary committee examining the reform project expressed its desire to raise the age limit for candidates for president from 70 to 75. Not surprisingly, this proposal was immediately accepted by the regime. Combined with the provisions of the new article 39, this amendment allowed President Ben Ali (born 3 September 1936) to seek a fifth term in office in 2009.

Thus the heart of the constitutional reform of 1 June 2002 concerns articles 39 and 40 of the constitution, that is, those that govern the election of the head of state.

The amendments were put to a referendum in order to give additional legitimacy to reform presented by President Ben Ali as an 'historic achievement.' The results of the referendum of 27 May 2002 confirm that the direct consultation of the people is conceived by the Tunisian regime as a plebiscite for the president and his policies as, according to official figures, the proposed amendments were approved by 99.52 per cent of the voters, with a turnout of 99.59 per cent. [START_REF] Gobe | Tunisie 2002: un référendum pour quoi faire?[END_REF] Apart from their unrealistic character, these figures are clear indicators of an authoritarian institutionalisation of political expression. The electoral reforms and new rules governing political parties adopted between 1992 and 2003 further confirm this trend.

Electoral Reforms and the Funding of Political Parties: Features of a

Graciously Granted Pluralism

Recent changes affecting elections and the funding of political parties have continued to 'defuse' [START_REF] Tozy | Représentation/Intercession. Les enjeux de pouvoir dans les "champs politiques désamorcés" au Maroc[END_REF] Tunisian politics. With the exception of the Progressive Democratic Party (PDP), the Democratic Forum for Labour and Liberties (le Forum démocratique pour le travail et les libertés, FDTL), [START_REF] Led | the FDTL was legalised in October 2002[END_REF] and more recently and to a lesser extent, the Ettajdid movement (formerly the Communist Party), the recognised political groups are 'organs of legitimation for the dominant party' rather than opposition groups. [START_REF] Djaziri | La problématique partisane dans les systèmes politiques du Maghreb. Relance des études comparatives[END_REF] President Ben Ali has not only sought to avoid political conflict and to construct a malleable opposition that can legitimise his rule; he has succeeded in doing so.

The Clientelist Regulation of Party Funding

If Habib Bourguiba paid lip service to pluralism and kept the opposition at a distance, the current president has made the opposition his satellite and now keeps it dependent on him. [START_REF] Bourguiba | dues paid by party members, the sale of newspapers, and fund-raising rallies provided party funding. Today[END_REF] That is the meaning of the 21 July 1997 law governing the funding of political parties. On the condition that they are represented in parliament, parties receive an annual allocation divided into a fixed amount (60,000 Tunisian dinars -DT) and a variable amount calculated on the basis of their number of deputies (5,000 DT per deputy). Opposition deputies mainly challenged the provision in the law that excludes from public funding recognised parties without deputies even though they participate in elections: 'When we look at the title of the proposed law, it seems clearly inappropriate. Its content concerns [only] parties […] represented in the Chamber of Deputies […]. If it is necessary to specify which parties are concerned, it would be appropriate to speak of the 'proposed law on the financing of the political parties represented in the Chamber of Deputies. […] In addition to its arbitrary character overall, this restriction only accentuates the marginalisation of a certain number of organisations. We do not understand what governs such a provision and what condemns certain parties to renunciation and impotence.' [START_REF] Khaskhoussi | Ard wa munaqasha mashru' yata'allaq bi-t-tamwil al-'umumi li-l-ahzab as-siyasiya, 'adad 37, atthulatha[END_REF] Moreover, political movements receive a set allowance for their newspapers. The law is phrased in such a way that the regime has great latitude in distributing these public funds.

One article provides for an 'annual subsidy, whose amount is fixed by decree and is allocated to political parties as a contribution to the cost of paper and printing for the publication of their newspapers. This subsidy will be paid in four instalments, on condition that publication is regular.' [START_REF] As-Sabah | [END_REF] The latter provision is of considerable importance, because opposition parties find it difficult to ensure regular publication of their newspapers.

Elections constitute a way of compensating the most 'deserving' opponents. In the legislative elections of 1999 and 2004, no alternative means of funding were available to opposition groups. The distribution of seats among these parties is governed by clientelist considerations. The most recent changes to the electoral law seek to keep the opposition parties dependent on the regime while at the same time fuelling the illusion of a decisive advance on the long path to democracy.

An Opposition Quota in Parliament

The mode of election -majority vote for party lists -has remained a constant feature of Tunisian electoral law. Obsessed by the threat that the Tunisian nation might explode, the leaders of the Néo-Destour party, the ancestor of the DCR, chose this mode of election after independence in order to obtain a homogeneous assembly. [START_REF] Camau | Contrôle politique et régulations électorales en Tunisie, Collection Maghreb contemporain[END_REF] It has always allowed the regime party to avoid having its predominance questioned and has transformed elections into mere plebiscites on the ruling regime and its policies. The most recent electoral reforms remain consistent with this logic.

An initial reform in 1994 introduced a quota of nineteen seats allocated to opposition parties, to be distributed according to their election results. It created the appearance of pluralism without fundamentally changing the situation. The allocation of these seats depends more on the goodwill of the regime than on the voters. The electoral law thus avoided any rivalry between the DCR and the opposition parties; instead, it forced the latter to compete with each other.

The adoption in October 1998 of an organic law completing certain provisions of the electoral law simply continued the 1994 reform. The increase in the number of seats reserved for the opposition to 20 per cent did not change the rules of the game. 31 In parliamentary debates, some critical voices denounced the absence of consultation regarding a reform that seemed to be largely handled by princely fiat: 'The new parliamentary pluralism is in reality the direct result of the political will of the head of state. It is not the product of political and ideological pluralism that is anchored in society and the public sphere.' [START_REF]Presentation and discussion of the organic bill modifying and supplementing certain provisions of the electoral code[END_REF] The most virulent charge against this proposed law came from a member of the Movement of Democratic Socialists (MDS), [START_REF]Le syndrome autoritaire. Politique en Tunisie de Bourguiba à Ben Ali[END_REF] Ahmed Khaskhoussi, who put his finger on the contradiction in a law that claims to deepen democracy:

From a logical point of view, it is not normal that only 80 per cent of the seats are assigned to [the party] that won 97 per cent of the vote, just as the one that won only 3 per cent should not be assigned 20 per cent of the seats. Such a system is unjust. In fact, injustice consists in giving human beings less than what their rights entitle them to. Injustice also consists in attributing to an individual more than (s)he deserves. I think our society must not be passive and dependent, living on gifts and donations.

Khaskoussi also denounced the logic of a 'pluralism of the clientelist type':

If the leaders of the parties are neither representative nor legitimate, […] the future candidates who will automatically represent citizens, no matter what the results of the

elections are, will know in turn that they have been placed there or appointed. That is why they will not only turn their backs on the citizens and public opinion, but worse, they will look to those who give them their jobs. What will be most important for them is to keep their positions and to survive. As a Tunisian man of letters put it metaphorically, 'they eat the bread and walk on the table.' Still more, they will appear to observers to be ornaments in a decor without beauty. [START_REF] Khaskhoussi | Presentation and discussion of the organic bill modifying and supplementing certain provisions of the electoral code[END_REF] The legislative reform does not challenge the DCR's hegemonic position. In reality, it preserves the absolute monopoly of the regime party on the seats assigned by constituency and keeps the opposition in a marginal position. [START_REF] Chebbi | [END_REF] The 2003 reform of the electoral law further reduces political competition. Presented as a consequence of the constitutional reform of 2002 creating the Chamber of Councellors, [START_REF]The reform of the electoral law affects 48 articles and introduces 48 others[END_REF] it afforded the regime the opportunity to reduce free expression abroad after it had already restricted it at home: article 62.3 of the amended electoral law 'prohibits any person from making use during the electoral period of a radio or television station that is private or foreign, or broadcasts from abroad for the purpose of calling upon people to vote or to abstain from voting in favour of a candidate, or from using the aforementioned television and radio stations for the purpose of electoral propaganda.' [START_REF]qanun al-asasi al-muta'alliq bi-tanqih wa ittimam al-majalla al-intikhabiyya, 'adad 37, at-thulatha', 22 jwilia 2003[END_REF] In fact, Tunisian authorities seek to limit the impact of satellite television stations that broadcast in Arabic. These media worry the regime because they are popular among Tunisians, especially the poorer strata that do not speak French. [START_REF] Geisser | Le président Ben Ali en campagne contre "les médias sataniques[END_REF] During the May 2002 referendum, satellite television, notably al-Jazeera, provided a sounding board for opposition leaders and militant supporters of human rights. [START_REF] Shoukat | Tunisia… Elections without satellite TV[END_REF] It provided a forum for leaders of outlawed groups like Moncef Marzouki and more outspoken representatives of legal parties, like the president of the PDP, Nejib Chebbi, and offered them an opportunity to call for a boycott of the referendum. Moreover, the outlawed human rights organisation, the National Council for Freedoms in Tunisia (NCFT), pointed out that the law did not refer to the electoral campaign but to the electoral 'period', which considerably extends the length of time concerned. [START_REF] Ncft | Le régime tunisien criminalise la liberté d'expression à l'étranger après l'avoir annihilé à l'intérieur et empêche la presse étrangère de couvrir les élections[END_REF] The former lasts about two weeks, while the latter begins with the official call for elections and ends with the publication of the results, thus lasting some three months. [START_REF] Hédi | Speech by the minister of the interior and local development[END_REF] Part of the opposition reacted very strongly; some deputies voted against the text proposed to them. Normally, opposition members of parliament simply abstain when they disagree with the regime. This time, five representatives of the Ettajdid Movement and Mokhtar Jalali, a marginalised member of the Unionist Democratic Union (UDU), voted against the reform, while six other deputies abstained. [START_REF]The UDU is one of the regime's client parties[END_REF] Mohamed Harmel, the general secretary of the Ettajdid Movement, described the ban on statements from abroad as a 'flagrant infringement of free expression.' To justify the measure, the government had the minister of the interior and local development, Hédi M'henni, play on nationalist feelings and imply that opponents of this provision were not 'fervent patriots':

[article 62.3] protects national sovereignty by preventing foreign radio and television from interfering in our internal affairs […]. This article should be a source of pride

because what is important is equality among the candidates, the dignity and independence of the country with regard to other countries. The electoral question should be treated in terms of Tunisian moral values to make it a purely Tunisian affair.

That is our objective: emancipated countries object to the curiosity of foreigners […]. In Canada, politicians and members of parliament spent hardly a quarter of an hour

discussing such a provision. [START_REF]The minister of the interior emphasised that an 'old democracy' like Canada had included a similar provision in its electoral law. See the speech given by the minister of the interior and local development, Hédi M'Henni[END_REF] Ultimately, the reform of the electoral law maintains the single-round majority vote for party lists and continues to entrust the supervision of the elections to the ministry of the interior and its local branches. [START_REF]PDP : to incriminate declarations made to foreign media is an attack on freedom of expression[END_REF] It therefore confirms the crushing domination of the regime and its party and makes opposition representatives indebted to presidential favours. The 1999 and 2003 constitutional amendments regarding presidential elections further illustrate a pluralism graciously granted by the head of state.

Constitutional Amendments Concerned with Purely Formal Competition in

Presidential Elections

In order to introduce a purely formal competition in the presidential elections of 1999 and 2004 and to co-opt 'consensual' opponents, the regime proceeded to an 'exceptional' reform of the constitution that enabled the president of the republic to choose acceptable competitors.

On neither occasion did President Ben Ali take the risk of introducing new conditions for candidacy in the presidential election. Paragraph 3 of article 40 of the constitution requires presidential candidates to obtain the sponsorship of thirty elected officials (deputies or presidents of local councils), a condition that no opposition party has been able to fulfil in the history of independent Tunisia. The approval of ad hoc constitutional laws allowed President Ben Ali to temporarily lift this requirement without losing control of the procedure that governed the selection of his competitors.

In 1999, extremely restrictive conditions were imposed: only the leaders of political parties (the secretary general or the president) could stand for election, and only if they had held this position for at least five years at the time they officially declared candidacy. Moreover, they had to be less than seventy years old, and their parties had to have a least one deputy. [START_REF]The PUP emerged from a split in Ahmed Ben Salah's Popular Unity Movement, and is one of the political parties that serve to legitimise the regime[END_REF] These conditions allowed the president to choose two acceptable competitors: Mohamed Belhaj Amor, president of the Popular Unity Party (PUP), and Abderrahmane Tlili, secretary general of the UDU. [START_REF](Al-qira'a al-ula li-mashru' al-qanun ad-dusturi al-muta'alliq bi-ahkam istithna'iyya lil-fiqra ath-thalitha min al-fasl 40 min ad-dustur, Mudawalat majlis an-nuwwab[END_REF] In 2003, the right to stand for election was no longer limited solely to leaders of political parties. The new text states that on the day he declares, a candidate for election must have been a member of the executive leadership of his party for at least five years without interruption, and that his party must be represented in the Chamber of Deputies.

In accord with their status as client-parties, the opposition groups in parliament, with the exception of the Ettajdid Movement, expressed relative satisfaction with the ad hoc amendments. Thus Mustapha Bouaouaja, a PUP deputy, declared that 'The opening up of the competition for the highest constitutional office […] constitutes an additional step in a process in which we have participated with a faith that allows us to imagine a bright future for a country in which the choice and the will of the people will be respected.' [START_REF] Bouaouaja | (Al-qira'a al-ula limashru' al-qanun ad-dusturi al-muta'alliq bi-ahkam istithna'iyya li-l-fiqra ath-thalitha min alfasl 40 min ad-dustur, Mudawalat majlis an-nuwwab, 'adad 17[END_REF] Major criticism came from the Ettajdid Movement (five deputies) and from Mokhtar Jalali. Denouncing the need for sponsorship, Jalali attacked the DCR and the police state:

The impossibility for any [opposition] candidate in the presidential elections to collect a sufficient number of sponsorships from deputies and presidents of local councils is not only an indication of the opposition's weakness, but also of the fact that the opposition cannot develop in a climate marked by the quasi-absolute domination of a single party that makes use of the state apparatus, including the official or semi-official media,

controls the administration and the various institutions of the country […]. The opposition is incapable of attracting supporters and exerting influence, no matter what its program and its slogans, in a context […] of continual surveillance where the police

spies on everything it does. [START_REF] Jalali | (Al-qira'a aththaniya li-mashru' al-qanun ad-dusturi al-muta'alliq bi-ahkam istithna'iyya li-l-fiqra aththalitha min al-fasl 40 min ad-dustur, Mudawalat majlis an[END_REF] The two other legalised opposition parties not represented in the Chamber of Deputies vigorously denounced the law. The day before the parliamentary vote, Nejib Chebbi announced in a press conference that he had sent a letter to all deputies urging them to reject the law and to vote for an alternative proposal that would open the way to pluralism, free political participation and a peaceful transition of power. The head of the PDP proposed to follow the example of Portugal, where a candidate only needs to be sponsored by 7,500 citizens. As for Mustafa Ben Jaafar, the president of the FDTL, he drew the bitter conclusion that the palace in Carthage had become 'an air-tight lockup with zero risk' and predicted another 'simulacrum of an election.' [START_REF] Beaugé | Réforme de l'éligibilité à la présidence en Tunisie[END_REF] In sum, the analysis of recent institutional change shows the extent to which legal formalism and constitutionalism participate in the reproduction and the dynamics of authoritarianism. Official Tunisian discourse on democracy and representation is situated in the framework of a repressive conception of law in which the rulers seek to 'protect themselves against possible "excesses" committed by ordinary citizens and by organised groups.' [START_REF] Geisser | Tunisie : des élections pour quoi faire? Enjeux et "sens" du fait électoral de Bourguiba à Ben Ali[END_REF] The laws that allegedly seek to improve representation are conceived in such a way as to prevent opposition actors from taking advantage of their seemingly liberal provisions. In the words of Michel Camau, legislation governing political institutions in Tunisia is characterised by 'a distortion between its conceptual universe, which is that of constitutionalism, and the weak legal tenor of its organisational content'. [START_REF] Camau | Caractère et rôle du constitutionalisme dans les États maghrébins[END_REF] Consequently this legislation, contrary to that of countries north of the Mediterranean, does not perform 'a function of effective protection and guarantee' and does not establish a system of 'brakes that are effective and normalised.' [START_REF] Camau | Pouvoirs et institutions au Maghreb[END_REF] Insofar as it does not constitute 'a formula for mediation and representation of social conflicts by the state,' but 'a system of representing the state in its relationship to society,' [START_REF] Camau | Caractère et rôle du constitutionalisme dans les États maghrébins[END_REF] it performs chiefly a function of legitimising the state and the elites that run it. It appears all the more plastic and unstable as the political calendar and the survival of the authoritarian regime dictate the rhythm and the nature of institutional changes. These legal gymnastics fool hardly anyone, inside or outside the country, and convince only those who want to be convinced. Tunisia's European partners, however, are more than willing to accept this kind of practice so as not to disturb the economic and security co-operation between the northern and southern sides of the Mediterranean.