Filipa Caetano

François Dubois

Benjamin Graille
email: benjamin.graille@universite-paris-saclay.fr

A RESULT OF CONVERGENCE FOR A MONO-DIMENSIONAL TWO-VELOCITIES LATTICE BOLTZMANN SCHEME

Keywords: Mathematics Subject Classification. 35L60, 65M08, 65M12, 76P99 Lattice Boltzmann scheme, relaxation method, hyperbolic system

We consider a mono-dimensional two-velocities scheme used to approximate the solutions of a scalar hyperbolic conservative partial differential equation. We prove the convergence of the discrete solution towards the unique entropy solution by first estimating the supremum norm and the total variation of the discrete solution, and second by constructing a discrete kinetic entropy-entropy flux pair being given a continuous entropy-entropy flux pair of the hyperbolic system. We finally illustrate our results with numerical simulations of the advection equation and the Burgers equation.

INTRODUCTION

The lattice Boltzmann method is a numerical method which is largely used to simulate fluid dynamics equations, such as Navier Stokes, heat, acoustics equations, multi-phase and multi-component fluids (see Succi [START_REF] Succi | Lattice Boltzmann[END_REF], Lallemand and Luo [START_REF] Lallemand | Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance, and stability[END_REF], and e.g. [START_REF] Krüger | The lattice Boltzmann method[END_REF]). Its origin is in a discretized velocities version of the continuous Boltzmann equation (see Broadwell [START_REF] Broadwell | Shock structure in a simple discrete velocity gas[END_REF] and Gatignol [START_REF] Gatignol | Théorie cinétique des gaz à répartition discrète de vitesses[END_REF]), with a specific collision kernel. The algorithm of the lattice Boltzmann method reads then as a fully discretized Boltzmann equation on a lattice.

The lattice Boltzmann method supposes that particles progress on a discrete Cartesian lattice with a finite set of speeds. In one time step, each velocity allows particles to jump from one vertex of the lattice to another one. One iteration of the method can be described in two steps: a relaxation step, which is local to each vertex and which corresponds to the collision of the particles, followed by a transport step, which corresponds to the evolution of the particles on the lattice.

Despite the fact that this method is widely used, the mathematical numerical analysis is far from being complete. In [START_REF] Rheinländer | Stability and multiscale analysis of an advective lattice Boltzmann scheme[END_REF], a stability analysis is proposed by Rheinländer for a twovelocities lattice Boltzmann scheme for the linear advection equation. In [START_REF] Junk | Convergence of lattice Boltzmann methods for Navier-Stokes flows in periodic and bounded domains[END_REF], Junk and Yang studied the convergence of approximation of smooth solutions for the incompressible Navier-Stokes equations. Concerning the linear one-dimensional convection-diffusion equation, Dellacherie proves the convergence in ∞ norm of the solutions of a two-velocities lattice Boltzmann scheme [START_REF] Dellacherie | Construction and analysis of lattice Boltzmann methods applied to a 1D convection-diffusion equation[END_REF]. We remark that in previous works studying lattice Boltzmann methods, links between lattice Boltzmann and finite difference methods were done (see Junk [START_REF] Junk | A Finite Difference Interpretation of the Lattice Boltzmann Method[END_REF] and [START_REF] Graille | Approximation of mono-dimensional hyperbolic systems: A lattice Boltzmann scheme as a relaxation method[END_REF]). Concerning non-linear models, in [START_REF] Boghosian | Entropic lattice Boltzmann model for Burgers's equation[END_REF], Boghosian, Love, and Yepez developed a twovelocities entropic scheme for the viscous Burgers equation and studied its properties.

In this contribution we consider a two-velocities lattice Boltzmann scheme in one dimension, known in the domain as D 1 Q 2 , in order to approach a scalar conservation law. The purpose of our paper is to prove the convergence of this lattice Boltzmann scheme towards the unique entropy solution of the hyperbolic equation as the mesh size tends to zero, when the relaxation parameter of the scheme lies between 0 and 1.

For this purpose, we make a link with the relaxation system of Jin and Xin [START_REF] Jin | The relaxation schemes for systems of conservation laws in arbitrary space dimensions[END_REF] and we use, in the context of our scheme, some techniques used to study the convergence of relaxation schemes based on the Jin and Xin approximation. Nevertheless, we point out that the lattice Boltzmann scheme is not constructed as a relaxation scheme, and it does not correspond to a natural discretization of the Jin and Xin system.

The convergence of the solutions of the Jin and Xin system towards the equilibrium, described by the scalar conservation law, was obtained independently by Natalini [START_REF] Natalini | Convergence to equilibrium for the relaxation approximations of conservation laws[END_REF] and by Serre [START_REF]Relaxations semi-linéaire et cinétique des systèmes de lois de conservation[END_REF]. Concerning the numerical approximation of the scalar conservation law, Aregba-Driollet and Natalini in [START_REF] Aregba-Driollet | Convergence of relaxation schemes for conservation laws[END_REF] and [START_REF]A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws[END_REF], and Lattanzio and Serre in [START_REF] Lattanzio | Convergence of a relaxation scheme for hyperbolic systems of conservation laws[END_REF], proved the convergence of finite volumes relaxation schemes based on the relaxation approximation of Jin and Xin. The relaxation schemes investigated in these works depend on two parameters (the relaxation rate and the mesh step Δ ) and the convergence of the schemes is proven by passing to the limit both parameters. In [START_REF] Aregba-Driollet | Convergence of relaxation schemes for conservation laws[END_REF][START_REF]A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws[END_REF], uniform ∞ and total variation estimates are obtained to pass to the limit separately in both parameters. In [START_REF] Lattanzio | Convergence of a relaxation scheme for hyperbolic systems of conservation laws[END_REF], compensated compactness techniques are used to study the limit of a relaxation scheme as both parameters go simultaneously to 0.

The lattice Boltzmann scheme we investigate only involves one parameter Δ . Although this scheme is not a relaxation scheme, it can be linked to this kind of schemes, with a particular choice of the relaxation rate proportional to Δ . As we can see in the figure 1, the convergence of the LB scheme corresponds to a diagonal limit (a limit on and on Δ at the same time).

The aim of our work is to prove, by using finite volumes methods techniques such as total variation and ∞ bounds inspired by [START_REF] Aregba-Driollet | Convergence of relaxation schemes for conservation laws[END_REF], [START_REF]A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws[END_REF], and [START_REF] Lattanzio | Convergence of a relaxation scheme for hyperbolic systems of conservation laws[END_REF], that the solutions of the D 1 Q 2 scheme converge towards the numerical solution of a scalar hyperbolic conservation law, under a stability condition. To the knowledge of the authors, this result is the first one of convergence for the solution of a lattice Boltzmann scheme towards the solution of a nonlinear conservation equation.

The plan of the paper is as follows. In section 2 we describe the D 1 Q 2 scheme and recall its construction, by following the framework of d'Humières [START_REF] Humières | Generalized Lattice-Boltzmann equations[END_REF]. In section 3 we establish ∞ and total variation discrete estimates. By using these estimates, in section 4 we prove the convergence of the D 1 Q 2 scheme towards a weak solution of the scalar conservation law, in the case where the relaxation parameter of the scheme lies between 0 and 1. In section 5, by following Serre [START_REF]Relaxations semi-linéaire et cinétique des systèmes de lois de conservation[END_REF], Bouchut [START_REF] Bouchut | Entropy satisfying flux vector splittings and kinetic BGK models[END_REF], and [START_REF]Stable lattice Boltzmann schemes with a dual entropy approach for monodimensional nonlinear waves[END_REF], we introduce numerical entropies for the D 1 Q 2 scheme. Based on a 1 estimation of the equilibrium gap independent of the mesh step established in section 3, we prove in section 6 the convergence of the D 1 Q 2 scheme towards the unique entropy solution of the scalar conservation law. Finally, section 7 presents several F. Caetano, F. Dubois, and B. Graille numerical tests that illustrate the convergence of the scheme. We also present some tests that precise the numerical convergence rate of the scheme, however without theoretical results.

DESCRIPTION OF THE

D 1 Q 2 SCHEME
We consider the following mono-dimensional scalar conservation law

(1) ( , ) + ( )( , ) = 0, > 0, ∈ ℝ,
where the flux is a C 1 function on ℝ, with the initial condition

(2) (0, ) = 0 ( ), ∈ ℝ.
It is well known that the Cauchy problem (1-2) possesses a unique entropy solution which belongs to ∞ (]0, [×ℝ), for all > 0, and such that ( , ⋅) ∈ (ℝ), ∀ > 0, provided that the initial data 0 ∈ ∞ (ℝ) ∩ (ℝ) (see e.g. [START_REF] Serre | Systèmes de lois de conservation[END_REF], [START_REF] Godlewski | Hyperbolic systems of conservation laws, Mathématiques & Applications[END_REF]). In this contribution, a two-velocities lattice Boltzmann scheme is used to approximate the solution of this Cauchy problem.

The framework of d'Humières.

In the contribution, we use the notations proposed by d'Humières in [START_REF] Humières | Generalized Lattice-Boltzmann equations[END_REF] by considering L = Δ ℤ, a regular lattice in one dimension of space with typical mesh size Δ . The time step Δ is determined after the specification of the velocity scale by the relation:

Δ = Δ .
For the scheme denoted by D 1 Q 2 , we introduce = {-, } the set of the two velocities. The aim of the D 1 Q 2 scheme is to compute a particle distributions vector = ( -, + ) T on the lattice L at discrete values of time: it is a numerical scheme that is formally a discretization in time, space and velocity, where only a finite number of velocities is considered (two, in our contribution), of the Boltzmann equation

( , , ) + ( , , ) = Q( )
(even if it cannot be used to simulate this partial differential equation), with a specific collision operator Q( ), whose effect is to relax the particle distributions towards its equilibrium value eq . The scheme splits into two phases for each time iteration: first, the relaxation phase that is local in space (corresponding to the consideration of the collision operator Q), and second, the transport phase for which an exact characteristic method is used.

In the framework proposed by d'Humières [START_REF] Humières | Generalized Lattice-Boltzmann equations[END_REF], the relaxation phase reads as a linear relaxation towards the equilibrium that is diagonal into a peculiar base. The vectors of this base are called "moments", the terminology being taken from the kinetic theory. Moreover, the equilibrium is a priori a nonlinear function of the conservative variables.

In the following, we denote by = ( , ) T the moments defined for each space point ∈ L and for each time by

= -+ + , = --+ + .
The matrix of the moments such that = satisfies

= 1 1 - , -1 = 1∕2 -1∕(2 ) 1∕2 1∕(2 ) .
Let us now describe one time step of the scheme. The starting point is the particle distributions vector ( , ) in ∈ L at time , the moments being computed by [START_REF] Bellotti | Finite difference formulation of any lattice boltzmann scheme[END_REF] ( , ) = ( , ).

F. Caetano, F. Dubois, and B. Graille

The relaxation phase then reads

(4) ⋆ ( , ) = ( , ), ⋆ ( , ) = ( , ) + ( eq ( , ) -( , )),
where is the constant relaxation parameter and eq the equilibrium of the second moment, which is considered to be a function of . As a consequence, the first moment is conserved during the relaxation phase. The relaxation parameter is usually taken in (0, 2], for stability reasons.

The particle distributions after the relaxation phase are then computed by ( 5)

⋆ ( , ) = -1 ⋆ ( , ).
The transport phase finally reads

(6) ( , + Δ ) = ⋆ ( -Δ , ), ∈ {-, +}.
In order to be consistent with Eq. ( 1), we impose that eq = ( ) [START_REF] Graille | Approximation of mono-dimensional hyperbolic systems: A lattice Boltzmann scheme as a relaxation method[END_REF].

A finite volume formalism.

In order to study the convergence of the scheme, we rewrite it into a finite volume formalism in this section. We first introduce usual finite volume notations. We note ( ) ∈ℤ the sequence of the discrete points in space that make up the lattice L and ( ) ∈ℕ the sequence of the discrete times, with

= Δ , ∈ ℤ, = Δ , ∈ ℕ.
The scheme can be described either in terms of the particle distributions variables or in terms of the moments. It computes ( -, , +, ) ∈ℤ, ∈ℕ and ( , ) ∈ℤ, ∈ℕ , which are respectively approximations of the local averages of the particle distributions and of the moments in each volume

[ , +1 ]×[ , +1 ].
According to (3), the particle distributions and the moments are linked by the relations

= -, + +, , = --, + +, , -, = 1 2 -1 2 , +, = 1 2 + 1 2 , ∈ ℤ, ∈ ℕ.
We denote by (

+ 1 ∕2
-, ,

+ 1 ∕2 +, ) (resp. ( + 1 ∕2 , + 1 ∕2
)) the approximated particle distributions functions (resp. moments) at time after the relaxation step. By using these notations, according to (4) the relaxation step of the scheme reads (7)

+ 1 ∕2 = , + 1 ∕2 = (1 -) + ( ), ∈ ℤ, ∈ ℕ.
According to (5), the particle distributions after the relaxation step are then defined by ( 8)

+ 1 ∕2 -, = 1 2 -1 2 + 1 ∕2 , + 1 ∕2 +, = 1 2 + 1 2 + 1 ∕2 , ∈ ℤ, ∈ ℕ,
and the transport phase (6) now reads (9)

+1 +, = + 1 ∕2 +, -1 , +1 -, = + 1 ∕2 -, +1 , ∈ ℤ, ∈ ℕ.
We can then describe a complete step of the scheme in terms of the approximated distribution functions ( -, , +, ) by

⎧ ⎪ ⎨ ⎪ ⎩ +1 -, = (1 -2 ) -, +1 + 2 +, +1 -2 ( -, +1 + +, +1 ), +1 +, = 2 -, -1 + (1 -2 ) +, -1 + 2 ( -, -1 + +, -1 ), ∈ ℤ, ∈ ℕ,
and in terms of the approximated moments ( , ) by ( 10)

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ +1 = 1 2 ( +1 + -1 ) -1 2 ( +1 --1 ) + 2 ( ( -1 ) --1 ) -( ( +1 ) -+1 ) , +1 = 1 2 ( +1 + -1 ) -2 ( +1 --1 ) + 2 ( ( -1 ) --1 ) + ( ( +1 ) -+1 ) ,
for ∈ ℤ, ∈ ℕ, the discrete initial data ( 0 , 0 ) ∈ℤ being taken as an approximation of ( 0 , ( 0 )) that will be specified below. Since = Δ Δ , equation ( 10) can be rewritten as ( 11)

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ +1 - Δ + +1 --1 2Δ - +1 -2 + -1 2Δ = 2 Δ ( -1 ) --1 -( +1 ) -+1 , +1 - Δ + 2 +1 --1 2Δ - +1 -2 + -1 2Δ = 2Δ ( -1 ) --1 + ( +1 ) -+1 .
Let us now make a link between the D 1 Q 2 scheme and a relaxation approximation of the scalar conservation law. For given Λ and > 0, let us consider the Jin and Xin relaxation system [START_REF] Jin | The relaxation schemes for systems of conservation laws in arbitrary space dimensions[END_REF]:

(12) ⎧ ⎪ ⎨ ⎪ ⎩ + = 0, + Λ 2 = 1 ( ( ) -).
The D 1 Q 2 scheme can be linked to a discretization of the relaxation system [START_REF]Stable lattice Boltzmann schemes with a dual entropy approach for monodimensional nonlinear waves[END_REF], by a splitting between the hyperbolic part-discretized with a Lax-Friedrichs scheme-and the relaxation part-discretized with an explicit Euler method-where first Λ = and second the relaxation parameter , the time step Δ , and the Jin and Xin relaxation rate are linked by the relation = Δ ∕ . Relaxation schemes corresponding to a discretization of [START_REF]Stable lattice Boltzmann schemes with a dual entropy approach for monodimensional nonlinear waves[END_REF] for fixed relaxation rate were widely studied in the past ( [START_REF] Natalini | Convergence to equilibrium for the relaxation approximations of conservation laws[END_REF], [START_REF] Aregba-Driollet | Convergence of relaxation schemes for conservation laws[END_REF] and [START_REF] Lattanzio | Convergence of a relaxation scheme for hyperbolic systems of conservation laws[END_REF]). In the first two papers the authors study the convergence of a discretization of [START_REF]Stable lattice Boltzmann schemes with a dual entropy approach for monodimensional nonlinear waves[END_REF], as the mesh step goes to 0, then as the relaxation rate goes to 0. In [START_REF] Lattanzio | Convergence of a relaxation scheme for hyperbolic systems of conservation laws[END_REF], the convergence of a relaxation scheme is studied as both parameters, the mesh step and the relaxation rate , go to 0 independently.

In the D 1 Q 2 scheme, both parameters (mesh step and relaxation rate ) are linked via the new parameter . Regarding the convergence of the D 1 Q 2 scheme can thus also be viewed as studying the limit of the scheme as both the space mesh and the relaxation rate = Δ ∕ go to 0 at the same time as the relaxation parameter is constant.

We emphasise however that the D 1 Q 2 scheme is not constructed as a discretization of the relaxation system [START_REF]Stable lattice Boltzmann schemes with a dual entropy approach for monodimensional nonlinear waves[END_REF]. In comparison with the previously mentioned works, in addition to the differences concerning the role of the parameters, it introduces a different (and not natural in a finite volume framework) discretization of the source term, the discretization of ( 12) F. Caetano, F. Dubois, and B. Graille considered in [START_REF] Lattanzio | Convergence of a relaxation scheme for hyperbolic systems of conservation laws[END_REF] being :

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ +1 - Δ + +1 --1 2Δ -Λ +1 -2 + -1 2Δ = 0 +1 - Δ + Λ 2 +1 --1 2Δ -Λ +1 -2 + -1 2Δ = 1 ( +1 ) - +1 .
2.3. Comments on the choice of the formalism. As the lattice Boltzmann method was introduced as a pure algorithm and not as a specific discretization of a partial differential equation, the interpretation of the discret quantities (here and for instance) as node or cell informations is left up to the user.

Note that recent works manage to rewrite any lattice Boltzmann schemes as a multi-step algorithm only on the conserved variables, that is, for instance in our case, without using the non-conserved variable [START_REF] Bellotti | Finite difference formulation of any lattice boltzmann scheme[END_REF]. It yields for the

D 1 Q 2 scheme +1 = 2- 2 ( +1 + -1 ) -(1 -) -1 - 2 ( ( +1 ) -( -1 )).
In our present contribution, we do not use this property because it is not really adapted to this work where the dynamics on the particule distribution functions has to be investigated. However, for smooth solutions, a multi-step finite difference framework can then be used to prove that the algorithm is consistent with the target hyperbolic equation and moreover that the equivalent equation of the scheme-in the sense of the finite difference schemes-is the same that the one obtained by the Taylor expansion method [START_REF] Dubois | Equivalent partial differential equations of a lattice Boltzmann scheme[END_REF][START_REF] Graille | Approximation of mono-dimensional hyperbolic systems: A lattice Boltzmann scheme as a relaxation method[END_REF]. This equivalent equation reads

(13) + ( ) = Δ 1 -1 2 ( 2 -( ′ ( )) 2 ) + O(Δ 2 ).
It points out that, if the relaxation parameter lies in (0, 2), the right member acts as a smoothing second-order operator for large enough scheme velocity . However, as we are interested in discontinuous solutions that can be involved in the nonlinear hyperbolic equations, we retain a cell-centered finite volume interpretation of the discret quantities. The finite volume formalism of the D 1 Q 2 scheme allowed us to use specific techniques of finite volume schemes for hyperbolic conservation laws, whose solutions might be discontinuous, to prove the convergence of the discrete solution to the unique entropy solution of the scalar conservation law. At our knowledge, we can not use the same techniques for a general lattice Boltzmann scheme and we do not claim that the lattice Boltzmann schemes are finite volume schemes, in the sense that they are not built as a finite volume discretisation of a particular partial differential equation. We just interpret the discrete quantities in the finite volume formalism in order to use the same objects and techniques of these methods to prove our convergence result.

TOTAL VARIATION AND ∞ ESTIMATIONS

In this section we establish ∞ and estimates for the numerical scheme [START_REF] Humières | Generalized Lattice-Boltzmann equations[END_REF]. We begin this section by recalling the definitions of the functional spaces that we will use in the sequel and by establishing the assumptions we make in order to get these estimates.

Notations and definitions.

Let us introduce some notations useful in the sequel. If = ( ) ∈ℤ ∈ ℝ ℤ , we denote by ∶= (( ) + 1 ∕2 ) ∈ℤ the sequence of ℝ ℤ defined by

( ) + 1 ∕2 = +1 -, ∈ ℤ.
F. Caetano, F. Dubois, and B. Graille If = ( ) ∈ℕ ∈ ℝ ℕ , we denote by ∶= (( ) + 1 ∕2 ) ∈ℕ the sequence of ℝ ℕ defined by

( ) + 1 ∕2 = +1 -, ∈ ℕ.
In order to investigate the convergence of the numerical solution, we recall some classical normed sub-spaces of ℝ ℤ and the corresponding norms.

Definition 1 ( 1 (ℝ ℤ ) space, ‖ ⋅ ‖ Δ , 1 norm). Let = ( ) ∈ℤ ∈ ℝ ℤ and Δ > 0 being given. We define the sequential one norm by

‖ ‖ Δ , 1 = Δ ∑ ∈ℤ | |,
and the associated space 1 (ℝ ℤ ) as

1 (ℝ ) = ∈ ℝ ℤ ∶ ‖ ‖ Δ , 1 < +∞ .
Definition 2 (sequential total variation). Let = ( ) ∈ℤ ∈ ℝ ℤ . We define the total variation in space by

( ) = ∑ ∈ℤ |( ) + 1 ∕2 |.
We recall below the classical Banach space of functions with bounded variation and the compactness Helly's theorem that we will use in the proof of our main result. To more details and to the proofs of these results, we address e.g. to [START_REF] Giusti | Minimal surfaces and functions of bounded variation[END_REF] and to [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]. Definition 3 (Bounded variation space). Let Ω be an open set of ℝ . For ∈ 1 (Ω), the total variation of over Ω denoted by Ω ( ) is defined by

Ω ( ) = sup ∈C 1 0 (Ω) ‖ ‖ ∞ ≤1 ∫ Ω div .
We denote by (Ω) the subspace of 1 (Ω) of functions with bounded total variation over Ω:

(Ω) = ∈ 1 (Ω) ∶ Ω ( ) < +∞ . Proposition (Functional space). The space 1 (Ω) ∩ (Ω) is a Banach space for the norm

‖ ‖ = ‖ ‖ 1 (Ω) + Ω (
). Theorem (Helly's sequential theorem). Let Ω ⊂ ℝ be a bounded open set with a Lipschitz boundary and let ( ) ∈ℕ be a sequence which is bounded in 1 (Ω) ∩ (Ω). Then there exists a sub-sequence denoted by ( ( ) ) ∈ℕ and a function ∈ 1 (Ω) ∩

(Ω) such that

( ) ⟶ in 1 (Ω) and a.e. ∈ Ω, Ω ( ) ≤ lim inf ∈ℕ Ω ( ( ) ).
In the sequel, we use the following notations for the sequences in

ℝ ℤ ± = ±, ∈ℤ , = ∈ℤ , = ∈ℤ , ( ) = ( ) ∈ℤ ,
for ∈ ℕ, ± ∈ {+, -}, and we use the following notations for the sequences in

ℝ ℕ ±, = ±, ∈ℕ , = ∈ℕ , = ∈ℕ , ( ) = ( ) ∈ℕ , for ∈ ℤ, ± ∈ {+, -}.
Let us also introduce the functions

(14) ℎ + ( ) = + ( ) 2 , ℎ -( ) = -( ) 2 , ∈ ℝ.
F. Caetano, F. Dubois, and B. Graille

The functions ℎ + and ℎ -correspond to the equilibrium of the particle distribution functions + and -. From relations ( 7) and ( 8), the relaxation phase can be rewritten as ( 15)

+ 1 ∕2 ±, = (1 -) ±, + ℎ ± ( ), ∈ ℤ, ∈ ℕ.
3.2. Assumptions. We suppose throughout the paper that the initial condition 0 belongs to ∞ (ℝ) ∩ (ℝ). We then define

= ess inf 0 , = ess sup 0 , = max | ′ ( )|, for ≤ ≤ .
and we make the following main assumptions concerning the numerical scheme [START_REF] Humières | Generalized Lattice-Boltzmann equations[END_REF].

Assumption 1. The relaxation parameter lies in (0, 1].

Often in the applications, the value of the relaxation parameter is chosen larger than 1 and often close to 2. In the case of the linear D 1 Q 2 , the scheme remains numerically stable in a 2 -sense for ∈ [0, 2] and the first-order numerical diffusion term is proportional to 1∕ -1∕2: the choice = 2 minimises then the error and the convergence rate is equal to 2 in that case (see [START_REF] Graille | Approximation of mono-dimensional hyperbolic systems: A lattice Boltzmann scheme as a relaxation method[END_REF]). For general lattice Boltzmann schemes, the "optimal" choices for the relaxation parameters are more complicated and are motivated by a combination of stability and accuracy reasons. However, these optimal parameters are generally larger than 1. The reason why we impose that lies in (0, 1] is that the scheme has monotonicity properties in this case. And these properties are essential for dealing with weak-solutions according to our technique of estimates. Assumption 2. The velocity of the scheme satisfies = Δ ∕Δ ≥ .

This assumption is known as the sub-characteristic condition. It states that the solutions of the equilibrium equation propagate with a characteristic speed smaller than that of the numerical scheme.

Assumption 3. The initial state is given by

0 = 1 Δ ∫ +1 0 ( ) d , 0 = ( 0 ), ∈ ℤ.
This last assumption means that the initial values of the non-conserved moment are taken to the equilibrium. This choice is done more often than not as this second moment has to be a perturbation of the equilibrium state [START_REF] Lallemand | Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance, and stability[END_REF].

Preliminary lemmas.

We begin by proving two lemmas that will be used throughout the proofs of the estimates and convergence of the numerical solutions. The first one concerns the total variation of the numerical initial data. Lemma 1. The total variation of the discretized conserved moment at initial time 0 = ( 0 ) ∈ℤ is controlled by the total variation of the initial function 0 :

( 0 ) = ∑ ∈ℤ |( 0 ) + 1 ∕2 | ≤ ℝ ( 0 ).
Proof. This result is commonly used in the context of finite volumes approximation of conservation laws; we propose a precise proof in appendix A.

■

The second lemma concerns some properties of the functions ℎ + and ℎ -defined by [START_REF] Giusti | Minimal surfaces and functions of bounded variation[END_REF].

Lemma 2.

Under assumption 2, the functions ℎ + and ℎ -are non decreasing over [ , ].

Moreover, for all ∈ ℝ, ℎ + ( ) + ℎ -( ) = and (ℎ + ) ′ ( ) + (ℎ -) ′ ( ) = 1.

Proof. The functions ℎ + and ℎ -are C 1 (ℝ) and their derivative is non negative over [ , ],

under assumption 2. The second result of the lemma is a trivial consequence of the definition of the functions ℎ ± . ■ 3.4. Uniform bound estimates. In this section we establish a maximum principle for the numerical solutions of the scheme [START_REF] Humières | Generalized Lattice-Boltzmann equations[END_REF]. The technique of the proof is the same as in [START_REF] Aregba-Driollet | Convergence of relaxation schemes for conservation laws[END_REF] or [START_REF] Lattanzio | Convergence of a relaxation scheme for hyperbolic systems of conservation laws[END_REF]. For the clearness of our work we present it here.

Proposition 3 (maximum principle). Under assumptions 1, 2 and 3, we have

∈ [ , ], ±, ∈ [ℎ ± ( ), ℎ ± ( )], ∈ ℤ, ∈ ℕ. Proof. A recursive reasoning is done. Since 0 ∈ [ , ], . . ∈ ℝ, we have that 0 ∈ [ , ]
for ∈ ℤ. By lemma 2, and since 0 = ( 0 ), we get

0 ±, = ℎ ± ( 0 ) ∈ [ℎ ± ( ), ℎ ± ( )], ∈ ℤ.
We then assume that the three inclusions are true for a certain ∈ ℕ. As ∈ (0, 1], Eq. ( 15) implies that

+ 1 ∕2 -,
and

+ 1 ∕2 +,
are respectively convex linear combinations of -, and ℎ -( ), and of +, and ℎ + ( ), so that

+ 1 ∕2 -, ∈ [ℎ -( ), ℎ -( )]
and

+ 1 ∕2 +, ∈ [ℎ + ( ), ℎ + ( )], for ∈ ℤ.
The transport phase just shift the distribution functions, so that the same inclusions yield for +1 -, and +1 +, for ∈ ℤ. Finally, we have

+1 = +1 -, + +1 +, ∈ [ℎ -( ) + ℎ + ( ), ℎ -( ) + ℎ + ( )] = [ , ],
∈ ℤ, as ℎ -( ) + ℎ + ( ) = for all ∈ ℝ, by lemma 2.

■ 3.5. Total variation estimates. We establish now estimates on the total variation in space and in time of the numerical solutions. This kind of estimates use now classical tools in the framework of finite volume schemes for hyperbolic conservation laws (see for instance [START_REF] Crandall | Monotone difference approximation for scalar conservation laws[END_REF]), and are similar but slightly different to the ones of [START_REF] Aregba-Driollet | Convergence of relaxation schemes for conservation laws[END_REF] and of [START_REF] Caetano | Sur certains problèmes de linéarisation et de couplage pour les systèmes hyperboliques non linéaires[END_REF]. For the completeness of our work we present the proofs here.

Proposition 4 (Spatial total variation estimations). Under assumptions 1, 2 and 3, the particle distributions functions satisfy the total variation decreasing estimate

( +1 + ) + ( +1 -) ≤ ( + ) + ( -), ∈ ℕ.
Moreover, we have

( + ) + ( -) ≤ ℝ ( 0 ), ∈ ℕ, (16)
( ) ≤ ℝ ( 0 ), ∈ ℕ, (17)
( ) ≤ ℝ ( 0 ), ∈ ℕ. ( 18
)
Proof. First we remark that we have ( ) ≤ ( -) + ( + ), as = -, + +, . We first evaluate the total variation of the approximated particle distributions functions after the relaxation phase. We deduce from [START_REF] Godlewski | Hyperbolic systems of conservation laws, Mathématiques & Applications[END_REF] that

( + 1 ∕2 ± ) + 1 ∕2 = (1 -)( ± ) + 1 ∕2 + ℎ ± ( +1 ) -ℎ ± ( ) .
F. Caetano, F. Dubois, and B. Graille Moreover we have [START_REF] Lattanzio | Convergence of a relaxation scheme for hyperbolic systems of conservation laws[END_REF] ∑

∈ℤ | | | ( ) + 1 ∕2 | | | ≤ 2 ℝ ( 0 ), ∑ ∈ℤ | | | ( ) + 1 ∕2 | | | ≤ 2 ℝ ( 0 ), ∈ ℕ.
Proof. Suppose that ∑ < +∞.

Since = + + -, then we also have

∑ ∈ℤ | | | ( ) -1 ∕2 | | | < +∞.
From ( 9) and ( 15), we have

+1 ±, = + 1 ∕2
±, ∓1 = (1 -) ±, ∓1 + ℎ ± ( ∓1 ). We obtain then

( ±, ) + 1 ∕2 = (1 -)( ±, ∓1 ) -1 ∕2 + ℎ ± ( ∓1 ) -ℎ ± ( - 1
∓1 ) = (1 -)( ±, ∓1 ) -1 ∕2 + ± ′ ( -1 ∕2 ∓1 ) 2 ( ∓1 ) -1 ∕2
, where we used that, for ∈ ℤ,

ℎ ± ( ) -ℎ ± ( -1 ) = (ℎ ± ) ′ ( -1 ∕2 )( ) -1 ∕2 = ± ′ ( -1 ∕2 ) 2 ( ) -1 ∕2 ,
with -1 ∕2 lying between and -1 .

Assumptions 1 and 2 yield

| | | ( ±, ) + 1 ∕2 | | | ≤ (1 -) | | | ( ±, ∓1 ) -1 ∕2 | | | + ± ′ ( -1 ∕2 ∓1 ) 2 , ∀ ∈ ℕ.
Now, we have, by using assumption 3, .6. Estimation of the equilibrium gap. We aim to prove that, at the discrete level, the equilibrium gap ( ) -tends to 0, as the discretization step tends to 0. The purpose of the next proposition is then to estimate the quantity ( ) -.

( +, ) 1 ∕2 = | 1 +, - 0 +, | = |(1 -) 0 +, -1 + ℎ + ( 0 -1 ) - 0 +, | = |ℎ + ( 0 -1 ) -ℎ + ( 0 )| ≤ | 0 -1 - 0 |,
Proposition 6 (estimation of the discrete equilibrium gap). Under assumptions 1, 2 and 3, the equilibrium gap is bounded, that is

‖ ( ) -‖ Δ , 1 ≤ 2 Δ ℝ ( 0 ), ∈ ℕ.
Proof. We have to estimate

‖ ( ) -‖ Δ , 1 = Δ ∑ ∈ℤ | ( ) -|.
For = 0, we have ‖ ( 0 ) -0 ‖ Δ , 1 = 0. Then, we use the relations

( +1 ) -( ) = ′ ( + 1 ∕2 )( +1 -)
with + 1 ∕2 ∈ ( , ) and + 1 ∕2 = (1 -) + ( ). We have

( +1 ) - +1 = ( +1 ) -( ) + ( ) - + 1 ∕2 + + 1 ∕2 - +1 = (1 -) ( ) - + ′ ( + 1 ∕2 )( +1 -) -( +1 - + 1 ∕2 ) = (1 -) ( ) - + ′ ( + 1 ∕2 ) + 1 ∕2 +, -1 + + 1 ∕2 -, +1 - + 1 ∕2 +, - + 1 ∕2 -, - + 1 ∕2 +, -1 - + 1 ∕2 -, +1 - + 1 ∕2 +, + + 1 ∕2 -, = (1 -) ( ) - + ′ ( + 1 ∕2 ) - + 1 ∕2
+, -1 -

+ 1 ∕2 +, + ′ ( + 1 ∕2 ) + + 1 ∕2 -, +1 - + 1 ∕2 -, .
By assumptions 1 and 2, we obtain

| ( +1 ) - +1 | ≤ |1 -| | ( ) -| + 2 | + 1 ∕2 +, -1 - + 1 ∕2 +, | + | + 1 ∕2 -, +1 - + 1 ∕2
-, | . As a consequence of proposition 4, summing the above inequalities over ∈ ℤ yields

‖ ( +1 ) -+1 ‖ Δ , 1 ≤ |1 -|‖ ( ) -‖ Δ , 1 + 2 Δ ( 0 )
as (

+ 1 ∕2 + ) + ( + 1 ∕2 - ) ≤ ( 0 ). A recursive reasoning then implies that ‖ ( ) -‖ Δ , 1 ≤ 2 Δ ( 0 ) ∑ =0 |1 -| ≤ 2 Δ ( 0 ) ≤ 2 Δ ℝ ( 0 ), ∈ ℕ,
as ∈ (0, 1] by assumption 1 and using lemma 1.

CONVERGENCE OF THE NUMERICAL SCHEME

This section is devoted to proving the convergence of the numerical solution towards a weak solution of the nonlinear Cauchy problem (1-2). This result is obtained as a consequence of the spatial and temporal estimates proved in section 3. The compactness of the numerical sequences is used in the space of the functions with bounded variation and is obtained as a consequence of Helly's theorem. F. Caetano, F. Dubois, and B. Graille As usual in the context of finite volume schemes, we seek an approximated solution of the form

Δ ,Δ ( , ) = ∑ ∈ℤ ∑ ∈ℕ 1 [ , +1 ) 1 [ , +1 ) , Δ ,Δ ( , ) = ∑ ∈ℤ ∑ ∈ℕ 1 [ , +1 ) 1 [ , +1 ) .
We begin by proving the convergence of the sequence ( Δ ,Δ , Δ ,Δ ), as the space-meshing Δ and the time-step Δ tend to 0, towards a function ( ̄ , ̄ ), where ̄ is a weak solution of the Cauchy problem (1-2) and ̄ = ( ̄ ). The main tools of the proof are classical in the context of finite volume schemes [START_REF] Godlewski | Hyperbolic systems of conservation laws, Mathématiques & Applications[END_REF], even if the treatment of the source term is specific to the D 1 Q 2 scheme. We propose the proof in appendix B.

Theorem 7 (convergence towards a weak solution). Under assumptions 1, 2 and 3, there exist sub-sequences of Δ ,Δ , Δ ,Δ (still denoted by Δ ,Δ , Δ ,Δ ) and functions ̄ , ̄ with

̄ , ̄ ∈ ∞ (ℝ + ×ℝ) ∩ ([0, ]×ℝ),
for all > 0, such that

( Δ ,Δ , Δ ,Δ ) ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → Δ ,Δ →0 ( ̄ , ̄ ) 1 ([0, +∞[×ℝ) × 1 ([0, +∞[×ℝ).
We have in addition that ̄ is a weak solution of (1-2) in [0, +∞[×ℝ, and that ̄ = ( ̄ ).

ENTROPIES AND NUMERICAL ENTROPY ESTIMATES

In this section we aim to establish discrete entropy estimates for the numerical scheme [START_REF] Humières | Generalized Lattice-Boltzmann equations[END_REF]. To do so, we will use the relaxation entropies introduced in [START_REF]Relaxations semi-linéaire et cinétique des systèmes de lois de conservation[END_REF] in order to construct numerical entropies for the scheme. We will also make a link between these relaxation entropies and a kinetic decomposition of the dual entropy for the nonlinear conservation law, introduced in [START_REF] Bouchut | Entropy satisfying flux vector splittings and kinetic BGK models[END_REF][START_REF]Stable lattice Boltzmann schemes with a dual entropy approach for monodimensional nonlinear waves[END_REF]. 5.1. Entropy. Let us consider an entropy-entropy flux pair ( , ), with strictly convex, for the hyperbolic scalar conservation law (1),

′ ( ) ′ ( ) = ′ ( ), ′′ ( ) > 0, ∈ [ , ].
For given , we introduce the homogeneous Jin and Xin relaxation system [START_REF] Jin | The relaxation schemes for systems of conservation laws in arbitrary space dimensions[END_REF]:

+ = 0, + 2 = 0, (24)
which we can write in an equivalent way in the characteristic variables ( + , -):

(25) + + + = 0,
---= 0. We remark that [START_REF]A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws[END_REF] is the homogeneous part of [START_REF]Stable lattice Boltzmann schemes with a dual entropy approach for monodimensional nonlinear waves[END_REF], with Λ = , where is the velocity scale in the D 1 Q 2 scheme; we consider here this choice in order to motivate the construction of numerical entropies for our scheme.

Following Serre [START_REF]Relaxations semi-linéaire et cinétique des systèmes de lois de conservation[END_REF], we define an entropy-entropy flux couple ( , ) for system (24) as follows. Let us first define the couple ( + , -) of the kinetic entropies by

± ( ) = ± 2 ((ℎ ± ) -1 ( )), ∈ [ℎ ± ( ), ℎ ± ( )],
and the couple ( ̃ , ̃ ) by

( ̃ , ̃ )( -, + ) = + ( + ) + -( -), + ( + ) --( -) , ± ∈ [ℎ ± ( ), ℎ ± ( )].
We remark then that the couple ( ̃ , ̃ ) is an entropy-entropy flux pair for system (25) that lies in the set

[ℎ + ( ), ℎ + ( )] × [ℎ -( ), ℎ -( )].
Let us now define the couple ( , ) by

( , ) = + + 2 + - - 2 , ( ,
) = + + 2 -- - 2 , for ( , ) such that ( + )∕(2 ) ∈ [ℎ + ( ), ℎ + ( )], ( -)∕(2 ) ∈ [ℎ -( ), ℎ -( )].
Then, the couple ( , ) is an entropy-entropy flux pair for the system [START_REF]A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws[END_REF], which satisfies

( , )( , ( )) = ( , )( ), ∀ ∈ [ , ].
Let us now state some properties of the entropies + and -that are useful to establish the entropy estimates for the numerical solutions.

Due to the assumption (2), we can easily check the following lemma.

Lemma 8. Under assumption (2), the functions + and -satisfy

±′ ( ) = ′ ((ℎ ± ) -1 ( )), ±′′ ( ) > 0, ∈ [ℎ ± ( ), ℎ ± ( )].

Furthermore we have

+′ (ℎ + ( )) = -′ (ℎ -( )) = ′ ( ), ∈ [ , ].
Proof. The result of the lemma follows immediately from the definition of ± and from the relation

′ ′ = ′ . Since ℎ ± ′ ( ) = ± ′ ( ) 2
, for ∈ ℝ, we have

±′ ( ) = ′ ± ′ 2 ((ℎ ± ) -1 ( )) 1 (ℎ ± ) ′ ((ℎ ± ) -1 ( )) = ′ ((ℎ ± ) -1 ( )).

Dual entropy.

In this paragraph we make a link between the relaxation entropies + , - defined previously with the kinetic decomposition of the dual entropy introduced by Bouchut in [START_REF] Bouchut | Entropy satisfying flux vector splittings and kinetic BGK models[END_REF] and by one of the authors in [START_REF]Stable lattice Boltzmann schemes with a dual entropy approach for monodimensional nonlinear waves[END_REF] for the nonlinear conservation law [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF].

Let us introduce the entropy variable for the scalar conservation law:

= ′ ( ).
We define the dual entropy ⟼ ⋆ ( ) according to ⋆ ( ) = sup -( ) .

We then have

⋆ ( ) = ( ′ ) -1 ( ) -( ′ ) -1 ( ) = -( ), d ⋆ d ( ) = ( ′ ) -1 ( ) = .
The dual entropy flux ⟼ ⋆ ( ) is then defined according to

⋆ ( ) = ( ′ ) -1 ( ) -( ′ ) -1 ( ) = ( ) -( ).
We have the following differential

d ⋆ d ( ) = ( ′ ) -1 ( ) = ( ).
Following [START_REF]Stable lattice Boltzmann schemes with a dual entropy approach for monodimensional nonlinear waves[END_REF], let us introduce two convex functions ⟼ ⋆ ± ( ) that satisfy the kinetic decomposition of the dual entropy We can then define a kinetic hyperbolic system ( 27)

⎧ ⎪ ⎨ ⎪ ⎩ + + + = 1 eq + ( + + -) -+ , -- -= 1 eq -( + + -) --.
We introduce the duals ⟼ ± ( ) of the functions ⋆ ± defined in [START_REF] Serre | Systèmes de lois de conservation[END_REF]:

± ( ) = sup - ⋆ ± ( ) . Then ′ ± ( ) = d ⋆ ± d -1 ( ).
We know that such a framework is able to put in evidence a "H-theorem" [START_REF]Stable lattice Boltzmann schemes with a dual entropy approach for monodimensional nonlinear waves[END_REF]. Just multiply each equation of ( 27) by ′ ± ( ± ); then

+ ( + ) + -( -) + + ( + ) --( -) ≤ 0.
The natural question is to make a link between this framework and the tools introduced in this contribution, id est to make a link between ℎ ± and eq ± and between ± and ± . We have the following proposition.

Proposition 9 (Duality and entropy decomposition). We have

eq ± ( ) = ℎ ± ( ), ∈ [ , ], ± ( ) = ± ( ), ∈ [ℎ ± ( ), ℎ ± ( )].
Proof. We first have

⋆ ± ( ) = ⋆ ( ) ± ⋆ ( ) 2 , d ⋆ ± d ( ) = ± ( ) 2 = ℎ ± ( ) with = ′ ( ).
Then, we can identify eq ± = ℎ ± . At the optimum value that define ± ( ), we have

= ⋆ ± ′ ( ) = ℎ ± ( ) with ′ ( ) = . Then ± ( ) = - 1 2 ⋆ ( ) ± 1 ⋆ ( ) = - 1 2 -( ) ± 1 ( ) -( ) = - 1 2 ± 1 ( ) + 1 2 ( ) ± 1 ( ) = ( -ℎ ± ( )) + 1 2 ± 1 (ℎ ± ) -1 ( ) = ± ( ),
and the proof is established. In this section, we construct a numerical entropy and the corresponding numerical entropy-flux for the numerical scheme and we prove the dissipation of this numerical entropy. Inspired by [START_REF] Lattanzio | Convergence of a relaxation scheme for hyperbolic systems of conservation laws[END_REF], let us define the numerical entropies for the numerical scheme by

+ 1 ∕2 = + ( + 1 ∕2 +, ) + -( + 1 ∕2 -, ), ∈ ℤ, ∈ ℕ, (28)
+ 1 ∕2 + 1 ∕2 = + ( + 1 ∕2 +, ) --( + 1 ∕2 -, +1 ), ∈ ℤ, ∈ ℕ,
where = Δ ∕Δ . Note that the numerical entropies are defined for each time step after the relaxation phase and before the transport phase. This is essential in order to obtained the estimates. Let us now define the numerical entropy production

(29) E = + 1 ∕2 - -1 ∕2 Δ + -1 ∕2 + 1 ∕2 - -1 ∕2 -1 ∕2 Δ , ∈ ℤ, ∈ ℕ * .
We begin by proving that the entropy production has a sign.

Proposition 10. Under assumptions 1 and 2, we have

E ≤ 0, ∈ ℤ, ∈ ℕ * .
Proof. We develop E . Since we have = Δ ∕Δ , we obtain

E = + ( + 1 ∕2 +, ) -+ ( -1 ∕2 +, -1 ) Δ + -( + 1 ∕2 -, ) --( -1 ∕2
-, +1 ) Δ .

By using one time step of the scheme, we have

+ 1 ∕2 ±, = -1 ∕2 ±, ∓1 ± 2 ( -1 ∕2 ) - -1 ∕2 = -1 ∕2 ±, ∓1 ± 2 (1 -) ( + 1 ∕2 ) - + 1 ∕2 ,
the convexity of ± implies that

± ( + 1 ∕2 ±, ) ∓ 2 (1 -) ( + 1 ∕2 ) - + 1 ∕2 ±′ ( + 1 ∕2 ±, ) ≤ ± ( -1 ∕2 ±, ∓1 ).
We have thus that

± ( + 1 ∕2 ±, ) -± ( -1 ∕2 ±, ∓1 ) Δ ≤ ± 2 (1 -) ( + 1 ∕2 ) - + 1 ∕2 ±′ ( + 1 ∕2 ±, ).
Hence we get

(30) E ≤ 2 (1 -) ( + 1 ∕2 ) - + 1 ∕2 +′ ( + 1 ∕2 +, ) --′ ( + 1 ∕2
-, ) .

Now from

+ 1 ∕2 ±, = ℎ ± ( + 1 ∕2 ) ± 1 2 + 1 ∕2 -( + 1 ∕2 ) it follows that ±′ ( + 1 ∕2 ±, ) = ±′ (ℎ ± ( + 1 ∕2 )) ± 1 2 + 1 ∕2 -( + 1 ∕2 ) ±′′ ( ±, + 1 ∕2 ),
where ±, + 1 ∕2 lie between

+ 1 ∕2 ±,
and ℎ ± ( + 1 ∕2 ). Combining with (30) and using Lemma 8, we conclude that

E ≤ - 4 2 (1 -) ( + 1 ∕2 ) - + 1 ∕2 2 +′′ ( +, + 1 ∕2 ) + -′′ ( -, + 1 ∕2 ) ≤ 0,
and the proof is established.

CONVERGENCE TOWARDS THE ENTROPIC SOLUTION

In this section we establish the final convergence result, by using the discrete entropy estimates obtained at the previous section. We will prove that the weak solution of the nonlinear Cauchy problem (1-2) given by Theorem 7, obtained as the limit ̄ of the numerical scheme, is indeed the unique entropic solution of the Cauchy problem (1-2).

Theorem 11 (convergence result). Let be a weak solution of the Cauchy problem [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF][START_REF] Aregba-Driollet | Convergence of relaxation schemes for conservation laws[END_REF] given by Theorem 7. Then we have that is the unique entropic solution of [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF][START_REF] Aregba-Driollet | Convergence of relaxation schemes for conservation laws[END_REF].

The main tools of the proof are classical in the context of the finite volume schemes [START_REF] Godlewski | Hyperbolic systems of conservation laws, Mathématiques & Applications[END_REF], but the choice of the numerical entropies [START_REF] Succi | Lattice Boltzmann[END_REF] being specific to the D 1 Q 2 scheme, we present it here.

Proof. Let ( , ) be an entropy-entropy-flux pair for (1), with strictly convex and let E be defined by (29). Let us also consider ∈ D(]0, +∞[×ℝ), ≥ 0 and put

= ( , ), ∈ ℤ, ∈ ℕ, Δ ,Δ ( , ) = ∑ ∈ℕ ∑ ∈ℤ 1 [ , +1 ) ( )1 [ , +1 ) ( ) .
The result of Proposition 10 implies that

Δ Δ E ≤ 0, ∈ ℤ, ∈ ℕ * ,
and by summing over ∈ ℕ * and over ∈ ℤ, we get

Δ Δ ∑ ∈ℤ ∑ ∈ℕ * ⎛ ⎜ ⎜ ⎝ + 1 ∕2 - -1 ∕2 Δ + -1 ∕2 + 1 ∕2 - -1 ∕2 -1 ∕2 Δ ⎞ ⎟ ⎟ ⎠ ≤ 0.
We now do a discrete integration by parts. We obtain

Δ Δ ∑ ∈ℤ ∑ ∈ℕ + 1 ∕2 -+1 Δ + ∑ ∈ℤ ∑ ∈ℕ * -1 ∕2 + 1 ∕2 -+1 Δ ≤ 0.
By using the definition (28) of the numerical entropies + 1 ∕2 and

-1 ∕2 + 1 ∕2
, the above inequality writes as

∑ ∈ℤ ∑ ∈ℕ ∫ +1 ∫ +1 + ( + 1 ∕2 +, ) + -( + 1 ∕2 -, ) ( , ) -( +1 , ) Δ d d + ∑ ∈ℤ ∑ ∈ℕ * ∫ +1 ∫ +1 + ( + 1 ∕2 +, ) --( + 1 ∕2 -, +1 ) ( , ) -( , +1 ) Δ d d ≤ 0.
F. Caetano, F. Dubois, and B. Graille

We now use that

+1 +, = + 1 ∕2 +, -1 , +1 -, = + 1 ∕2
-, +1 , for all ∈ ℤ and ∈ ℕ. We obtain

∑ ∈ℤ ∑ ∈ℕ ∫ +1 ∫ +1 + ( + 1 ∕2 +, ) + -( + 1 ∕2 -, ) ( , ) -( +1 , ) Δ d d + ∑ ∈ℤ ∑ ∈ℕ * ∫ +1 ∫ +1 + ( + 1 ∕2 +, ) --( + 1 ∕2 -, +1 ) ( , ) -( , +1 ) Δ d d = ∫ +∞ 0 ∫ ℝ + ( + Δ ,Δ ( + Δ , + Δ )) + -( - Δ ,Δ ( + Δ , -Δ )) × Δ ,Δ ( , ) -Δ ,Δ ( + Δ , ) Δ d d + ∫ +∞ Δ ∫ ℝ + ( + Δ ,Δ ( + Δ , + Δ )) --( - Δ ,Δ ( + Δ , )) × Δ ,Δ ( , ) -Δ ,Δ ( , + Δ ) Δ d d ≤ 0.
Following the definition of ( + , -) and since ( + Δ ,Δ , - Δ ,Δ ) tends to (ℎ + ( ), ℎ -( )) when Δ and Δ vanish, by passing to the limit as Δ , Δ → 0, we obtain

∫ +∞ 0 ∫ ℝ ( ) + ( ) d d ≥ 0,
and the result of the theorem follows.

■

NUMERICAL ILLUSTRATIONS

In this section, numerical simulations are given in order to illustrate the theoretical results of the previous sections. Two models are investigated: the advection equation with a constant velocity and the Burgers equation, both considering regular then discontinuous initial conditions. In particular, the numerical rate of convergence and the entropy production are computed in these cases.

Two initial conditions are systematically chosen: first a regular function

( ) 0 ( ) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 0 if ≤ -, 1 2 + ( -)(3 2 -( -) 2 ) 4 3 if -≤ ≤ + , 1 if + ≤ ≤ + , 1 2 - ( -)(3 2 -( -) 2 ) 4 3 if -≤ ≤ + , 0 if + ≤ ,
and second a discontinuous function

( ) 0 ( ) = ⎧ ⎪ ⎨ ⎪ ⎩ 0 if ≤ , 1 if ≤ ≤ , 0 if ≤ .
Numerically, we took = 1∕4, = 3∕4, = 1∕10. Note that the second case ( ) can be obtained from the first case ( ) in the limit goes to 0.

Concerning the choice of the relaxation parameter, asumption 1 only imposes that has to be a constant value lying in (0, 1]. We have therefore chosen several values ∈ {0.1, 0.2, 0.5, 0.7, 0.9, 1.0} to illustrate the asymptotic convergence of the solution.

( , ) + ( , ) = 0, ∈ ℝ, > 0, (0, ) = 0 ( ), ∈ ℝ,
where the velocity is taken equal to 0.75. The exact solution is well known for both initial conditions ( ) and ( ) and reads ( , ) = 0 ( -).

x, 1

Convergence of the second moment s = 0.1, p = 0.99 s = 0.2, p = 0.99 s = 0.5, p = 1.00 s = 0.7, p = 1.00 s = 0.9, p = 1.00 s = 1.0, p = 1.00 Fig. 2. Initial condition ( ). Error in norm 1 of the numerical solution of the advection equation (31) according to the space steps Δ for several relaxation parameter values ∈ {0.1, 0.2, 0.5, 0.7, 0.9, 1.0}. is the convergence rate. The convergence rates of the numerical solution can be read in Figure 2 for the regular solution and in Figure 3 for the discontinuous solution: the error on the first moment and on the second moment converges towards 0 with a convergence rate equal to 1 for regular solutions and equal to 0.5 for discontinuous solutions. As expected, the error is even larger when the relaxation parameter is small, this numerical results staying true for lying in [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF][START_REF] Aregba-Driollet | Convergence of relaxation schemes for conservation laws[END_REF] as observed in [START_REF] Graille | Approximation of mono-dimensional hyperbolic systems: A lattice Boltzmann scheme as a relaxation method[END_REF] for the first moment in other test cases. We now present the results on the numerical entropy production. As the system (31) is linear, the exact entropy production is zero. In Figure 4, the local in space numerical entropy production E , ∈ ℤ, is shown for the regular initial condition ( ). This production is negative as proved in Proposition 10, located where the solution is not constant, and is smaller for small space step Δ and for large relaxation parameter . In Figure 5, the entropy production is shown for the discontinuous initial condition ( ). This production is also negative and located on the discontinuities. The 1 -norm is decreasing when the space step Δ is lower or when the relaxation parameter is larger.

x, 1

Convergence of the second moment s = 0.1, p = 0.99 s = 0.2, p = 0.99 s = 0.5, p = 1.00 s = 0.7, p = 1.00 s = 0.9, p = 1.00 s = 1.0, p = 1.00 Fig. 6. Initial condition ( ). Error in norm 1 of the numerical solution of the Burgers equation (32) according to the space steps Δ for several relaxation parameter values ∈ {0.1, 0.2, 0.5, 0.7, 0.9, 1.0}. is the convergence rate.

x, 1

Convergence of the second moment s = 0.1, p = 0.83 s = 0.2, p = 0.83 s = 0.5, p = 0.84 s = 0.7, p = 0.85 s = 0.9, p = 0.85 s = 1.0, p = 0.86 Fig. 7. Initial condition ( ). Error in norm 1 of the numerical solution of the Burgers equation (32) according to the space steps Δ for several relaxation parameter values ∈ {0.1, 0.2, 0.5, 0.7, 0.9, 1.0}. is the convergence rate.

The exact solution for both initial conditions can be computed: for the regular function ( ) with the theory of the characteristics until the shock appears and for the discontinuous function ( ) with the combination of a shock and a rarefaction wave. The convergence rates of the numerical solution can be read in Figure 6 for the regular solution and in Figure 7 for the discontinuous solution: the error on the first moment and on the second moment converges towards 0. For regular solutions, the convergence rate is equal to 1 and, for discontinuous solutions, it is about 0.8. As expected, the error is even larger when the relaxation parameter is small, this numerical results staying true for lying in [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF][START_REF] Aregba-Driollet | Convergence of relaxation schemes for conservation laws[END_REF] as observed in [START_REF] Graille | Approximation of mono-dimensional hyperbolic systems: A lattice Boltzmann scheme as a relaxation method[END_REF] for the first moment in other test cases. Note that we cannot explain the better convergence rate of this nonlinear equation for discontinuous solutions as previously shown in [START_REF] Graille | Approximation of mono-dimensional hyperbolic systems: A lattice Boltzmann scheme as a relaxation method[END_REF].

We now present the results on the numerical entropy production. For the system (32), the exact entropy production is zero but on the decreasing discontinuities where it is a Dirac measure multiplied by a negative value. In Figure 8, the local in space numerical entropy production E , ∈ ℤ, is shown for the regular initial condition ( ). This production is negative as proved in Proposition 10, located where the solution is not constant, and is smaller for small space step Δ and for large relaxation parameter . In Figure 9, the entropy production is shown for the discontinuous initial condition ( ). This production is also negative and essentially located on the discontinuity (a smaller contribution is located on the rarefaction wave, this contribution being decreasing when the space step Δ is smaller or when the relaxation parameter is larger).

7.3. Concluding remarks. These numerical illustrations confirm the theoretical results of the convergence of the numerical solution for the 1 -norm for both moments and . Moreover, the discrete entropy production is non-positive and seems to converge towards the exact entropy production.

Concerning the numerical convergence rate, we observe that it is equal to 1 for a smooth initial condition, as long as the solution remains regular enough. By putting the two equations (1) and ( 13) (the target equation and the equivalent equation) side by side, this value is compatible with a truncation error of order 1 in time. In the case of a solution with regularity breaks, we notice a lower convergence rate. One of the authors already observed these results without rigorous proof in [START_REF] Graille | Approximation of mono-dimensional hyperbolic systems: A lattice Boltzmann scheme as a relaxation method[END_REF].

CONCLUSION

In this paper, we prove, by using techniques based on finite volume methods-in particular on relaxation methods-, a convergence result for a non-linear hyperbolic one-dimensional scalar conservation law. More precisely, we prove the convergence of the numerical solutions of the D 1 Q 2 scheme towards the unique entropy solution of the scalar conservation law. We prove in addition a numerical entropy estimation. Our results are based on the convexity properties of the scheme.

Future works could be dedicated to extend the present results to a wide range of relaxation parameters. Moreover, other popular lattice Boltzmann schemes, like a scheme with three velocities in one space dimension, should also be studied. Of course, general results concerning the convergence of more complex lattice Boltzmann schemes remains an open question. From a numerical point of view, we noticed different convergence rates according to whether the solution of the conservation law is discontinuous or not. The convergence rate of the scheme is also a subject to be investigated.

We multiply both sides of (11) by Δ Δ , we sum over ∈ ℕ and ∈ ℤ, do a discrete integration by parts, and pass to the limit as Δ , Δ → 0. The first term on the left-hand side reads By a similar reasoning we get for the second term on the left-hand side we conclude, as a consequence of Proposition 6, that this last term tends towards 0 when Δ and Δ go to 0. We conclude then that ̄ is a weak solution of (1).

Δ Δ ∑ ≥0 ∑ ∈ℤ -1 -+1 2Δ = ∫ +∞ 0 ∫ ℝ Δ ,Δ ( , ) Δ ,Δ ( , -Δ ) -Δ ,Δ ( , +Δ ) 2Δ d d ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → Δ ,Δ →0 -∫ +∞ 0 ∫ ℝ ̄ ( , ) (

Δ 0 Fig. 1 .

 01 Fig. 1. Representation of the schemes parameters in the plane ( , Δ ).

  which, combined with[START_REF] Natalini | Convergence to equilibrium for the relaxation approximations of conservation laws[END_REF], implies[START_REF] Lallemand | Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance, and stability[END_REF].Since | | ≤ | + | + | -| and | | ≤ (| + | + | -|),we also obtain[START_REF] Lattanzio | Convergence of a relaxation scheme for hyperbolic systems of conservation laws[END_REF]. ■ F. Caetano, F. Dubois, and B. Graille

3

 3

■

■

■F. 5 . 3 .

 53 Caetano, F. Dubois, and B. Graille Numerical entropies estimates.

■F.

  Caetano, F. Dubois, and B. Graille

7. 1 .

 1 Constant velocity advection equation. The first studied model corresponds to the advection equation at velocity (31)

1

 1 Convergence of the first moment s = 0.1, p = 0.98 s = 0.2, p = 0.99 s = 0.5, p = 1.00 s = 0.7, p = 1.00 s = 0.9, p = 1.00 s = 1.0, p = 1

1 1 Fig. 3 .

 113 Fig.3. Initial condition ( ). Error in norm 1 of the numerical solution of the advection equation (31) according to the space steps Δ for several relaxation parameter values ∈ {0.1, 0.2, 0.5, 0.7, 0.9, 1.0}. is the convergence rate.

nFig. 4 .nnFig. 5 .

 45 Fig. 4. Local in space numerical entropy production of the advection equation with initial condition ( )for varying space steps Δ at left and for varying relation parameters at right. The numerical and exact solutions at = 0.1 are shown bellow and the corresponding entropy productions above.

7. 2 . 1 2 2 (

 212 Burgers equation. The second studied model corresponds to the Burgers equation (32) ( , ) + ( , )) = 0, ∈ ℝ, > 0, (0, ) = 0 ( ), ∈ ℝ.

1

 1 Convergence of the first moment s = 0.1, p = 0.97 s = 0.2, p = 0.99 s = 0.5, p = 1.00 s = 0.7, p = 1.00 s = 0.9, p = 1.00 s = 1.0, p = 1

1

 1 Convergence of the first moment s = 0.1, p = 0.78 s = 0.2, p = 0.79 s = 0.5, p = 0.82 s = 0.7, p = 0.83 s = 0.9, p = 0.84 s = 1.0, p = 0

nnFig. 8 .

 8 Fig. 8. Local in space numerical entropy production of the Burgers equation with initial condition ( )for varying space steps Δ at left and for varying relation parameters at right. The numerical and exact solutions at = 0.1 are shown bellow and the corresponding entropy productions above.

nnFig. 9 .

 9 Fig. 9. Local in space numerical entropy production of the Burgers equation with initial condition ( )for varying space steps Δ at left and for varying relation parameters at right. The numerical and exact solutions at = 0.1 are shown bellow and the corresponding entropy productions above.

Δ

  ,Δ ( , ) Δ ,Δ ( -Δ , ) -Δ ,Δ ( , ) Δ d d -∫ ℝ Δ ,Δ (0, ) Δ ,Δ (0, ) d ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → Δ ,Δ →0 -∫ +∞ 0 ∫ ℝ ̄ ( , ) ( , ) d d -∫ ℝ 0 ( ) (0, ) d .

  , ) d d = -∫ +∞ 0 ∫ ℝ ( ̄ ( , )) ( , ) d d .Concerning the third term, it can be written as∫ +∞ 0 ∫ ℝ Δ ,Δ ( , ) Δ ,Δ ( , -Δ ) -Δ ,Δ ( , ) 2Δ -Δ ,Δ ( , ) -Δ ,Δ ( , +Δ ) 2Δ d d ,which vanishes as Δ , Δ → 0. Let us now treat the right-hand side of[START_REF] Dubois | Equivalent partial differential equations of a lattice Boltzmann scheme[END_REF]. After integration by parts, we getΔ ) -Δ ,Δ Δ ,Δ ( , +Δ ) -Δ ,Δ ( , -Δ ) 2Δ d d . Let > 0, > 0, such that supp( ) ⊆ [0, ] × [-, ]. Δ ) -Δ ,Δ Δ ,Δ ( , +Δ ) -Δ ,Δ ( , ∞ ‖ ( Δ ,Δ ) -Δ ,Δ ‖ ∞ ([0, ]; 1 (ℝ) ,

F. Caetano, F. Dubois, and B. Graille

ACKNOWLEDGMENTS

The authors are grateful to Frédéric Lagoutière and to Jean-François Babadjian for their explanations concerning bounded variation functions. The simulations have been obtained with the python package pylbm (https://pylbm.readthedocs.io/en/latest/).

And we have, by performing a first order Taylor expansion,

where + 1 ∕2 ∈ ( , ) is such that ( +1 ) -( ) = ′ ( + 1 ∕2 )( +1 -). We then have, using that (ℎ ± ) ′ ≥ 0 over [ , ] and that ∈ (0, 1],

Since (ℎ + ) ′ + (ℎ -) ′ = 1, summing the above two inequalities over ∈ ℤ, for ± ∈ {+, -}, yields

and finally,

Concerning the transport phase, the total variation of the particle distributions functions after this phase is unchanged as they are defined as a translation of the particle distributions before the phase. We have thus

which allow us to conclude that

We also have for the total variation of the conserved moment

). The definition of the initial numerical data in assumption 3 implies that

.

By summing these two equalities over ∈ ℤ, for ± ∈ {+, -}, we get ( 0 -) + ( 0 + ) = ( 0 ), which combined with [START_REF] Junk | Convergence of lattice Boltzmann methods for Navier-Stokes flows in periodic and bounded domains[END_REF] and with [START_REF] Krüger | The lattice Boltzmann method[END_REF] gives ( 16) and [START_REF] Jin | The relaxation schemes for systems of conservation laws in arbitrary space dimensions[END_REF]

we also obtain [START_REF] Junk | A Finite Difference Interpretation of the Lattice Boltzmann Method[END_REF].

■

In order to control the total variation of the numerical approximation in time and space variables, we prove now uniform total time variation estimates for the approximated solutions. To do so, we estimate the quantity

Proposition 5 (total variation in time estimates). Under assumptions 1, 2 and 3, the particle distributions functions satisfy the estimate

Using the definition of 0 given in assumption 3, we have that

We will then prove that

Suppose first that 0 ∈ C 1 (ℝ). In this case we have that

We now prove (33) for general 0 ∈ (ℝ). To do so, we use the following result (see [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]): for ∈ (ℝ), there exists a sequence ( ) ∈ ∞ (ℝ) such that → in 1 (ℝ) and ℝ ( ) → ℝ ( ). We consider then such a sequence ( ) that converges towards 0 . We have, for all > 0,

F. Caetano, F. Dubois, and B. Graille since ∈ ∞ (ℝ). We have now, on the one hand, that

On the other hand, ℝ ( ) → ℝ ( 0 ), so that, by passing (34) to the limit as → ∞, we obtain

By letting now → ∞, we obtain

that ends the proof.

APPENDIX B. PROOF OF THEOREM 7.

We have to prove that the families of functions Δ ,Δ and Δ ,Δ are bounded in ∞ (Ω) ∩ (Ω) for all bounded open subset Ω of ℝ + ×ℝ. Proposition 3 implies that both Δ ,Δ and

for some > 0 and > 0, and let , ∈ ℕ such that ( -1)Δ ≤ < Δ , ( -1)Δ ≤ < Δ . We have then that

.

From Proposition 4 we obtain that

which is bounded as Δ → 0.

On the other hand, as a consequence of Proposition 5, we have

which is also bounded as Δ → 0. We obtain for Δ ,Δ a similar result. These estimations imply that the set {( Δ ,Δ , Δ ,Δ )} Δ ,Δ remains bounded in ∞ (Ω) ∩ (Ω), and this is valid for any bounded set Ω ⊆ ℝ + × ℝ. The compactness of (Ω) ∩ ∞ (Ω) in 1 (Ω) (Helly's theorem) implies that there exists a sub-sequence of ( Δ ,Δ , Δ ,Δ ) and functions ( ̄ , ̄ ) satisfying the conditions of the Theorem.

Proposition 6 now implies that

Since ( Δ ,Δ , Δ ,Δ ) ⟶ ( ̄ , ̄ ), we get ( ̄ ) = ̄ , a.e. ∈ ℝ, for all > 0.

In order to prove that ̄ is a weak solution of (1-2), let us consider ∈ C ∞ 0 (ℝ + × ℝ) and put = ( , ), Δ ,Δ ( , ) =

F. Caetano, F. Dubois, and B. Graille


Cyrielle Mathias

Nicolas Boulbes

Garcia-Fermet Thomas

Jean-Luc Guadelli

Anne-Sophie Lartigot-Campin

Anne-Marie Moigne

Agnès Testu

Jose Utge-Buil

Amélie Vialet

Dusting off the old stones. Correlating old collections with new excavations: The case of "layer 2" of Coupe-Gorge (Montmaurin, France)

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Dusting off the old stones.

Mathias C., Boulbes N., Garcia-Fermet T., Guadelli J.-L., Lartigot-Campin A.-S., Moigne A.-M., Testu A., Utge J., Vialet A.

Cammas collections (MNP)

  -Bonnoure, E., Boulbes, N., Guérin, C., Pernaud, J., Tavoso, A., Cammas, R., 2010. Le contexte géomorphologique et faunique de l´homme de Montmaurin (Haute-Garonne). Préhistoires Méditerranéennnes, 1, p.3-85. / Gaillard C. 1979. L'industrie lithique du Paléolithique inférieur et moyen de la grotte de Coupe-Gorge à Montmaurin (Haute-Garonne), thèse de doctorat de l'Université de Marseille, Laboratoire de Paléontologie humaine et de Préhistoire, 586 p. / Girard M., Renault-Miskovsky J. 1979. Analyse pollinique de la grotte de Coupe-Gorge à Montmaurin (Haute-Garonne), Bulletin de l'Association française pour l'Etude du Quaternaire, 4, p. 175-189. Montmaurin karstic system (Haute-Garonne, France) was explored early in the XXth century in the frame of quarrying activities. Several sites were thus excavated (Middle Pleistocene >> Holocene). A new project (dir. A. Vialet) is conducted since 2018 in order to date and understand the environmental and cultural context of occupations. New fieldworks are focused on the upper levels of Coupe-Gorge cave, especially layers from Unit 2, which were previously attributed by L. Méroc to Upper Paleolithic (Chatelperronian, Aurignacian at the basis & Magdalenian on the top) with possible perturbations.

►►

  Cold & dry continental climate ►Mixed occupations by both Humans and Carnivores in several layers ► Upper Pleistocene = MIS 3? (59-27 ka) ► Possible ancient Homo sapiens humerus (DNA preliminary results from coprolites* & sediment**) ► Mostly reindeers, foxes and bears ► Consistency between old and new excavations ► Low impact of carnivores (even if present) & human actions ► ≠ spatial location of carnivores species and burnt bones (hearths) within the site; presence of coprolites (Crocuta) and deciduous teeth = occupations by carnivores Layer 2.4 Maybe an ancient Homo sapiens left humerus, but falls within the variation of Neanderthals and Homo heidelbergensis ► Carnivores traces (on fresh bone)


Keywords: Bandpass NGD, microstrip circuit, temperature effect characterization, S-parameters analysis, robustness study, experimentation, electrothermal analysis

This paper explores an original study of bandpass (BP) negative group delay (NGD) robustness applied to ring-stub passive circuit. The proof of concept (PoC) circuit is constituted by a ring associated to open-end stub implemented in microstrip technology. An innovative experimental setup of temperature room containing the NGD PoC connected to a vector network analyzer is described. Then, the electrothermal data of Sparameters are measured by varying the ambient or room temperature range from 20°C to 60°C, i.e., 40°C maximal variation. The empirical results of the group delay (GD), transmission and reflection coefficient mappings versus the couple (temperature, frequency) highlights how the temperature affects the BP NGD responses. An innovative electrothermal calibration technique by taking into account the interconnection cable influence is developed. The electrothermal robustness analysis is carried out by variations of the NGD center frequency, cut-off frequencies and value in function of the temperature.

I. INTRODUCTION

HE PRACTICAL EXISTENCE of the intriguing negative group delay (NGD) effect was initially experimented by physicists with optical system [START_REF] Chu | Linear Pulse Propagation in an Absorbing Medium[END_REF][START_REF] Ségard | Observation of Negative Velocity Pulse Propagation[END_REF][START_REF] Macke | Propagation of light-pulses at a negative groupvelocity[END_REF][START_REF] Munday | Observation of Negative Group Delays within a Coaxial Photonic Crystal Using an Impulse Response Method[END_REF]. At the beginning, the observation of the NGD effect was made by using negative group velocity (NGV) dispersive media [START_REF] Chu | Linear Pulse Propagation in an Absorbing Medium[END_REF][START_REF] Ségard | Observation of Negative Velocity Pulse Propagation[END_REF]. The analytical study of the NGD effect is necessary for the basic physical interpretation. Then, the NGD function existence at the microwave wavelength becomes an attractive topic for electronic design and research engineers [START_REF] Eleftheriades | Transmission Line for Negative Refractive Index Media and Associated Implementations without Excess Resonators[END_REF][START_REF] Siddiqui | Periodically Loaded Transmission Line With Effective Negative Refractive Index and Negative Group Velocity[END_REF][START_REF] Siddiqui | Time-Domain Measurement of Negative-Index Transmission-Line Metamaterials[END_REF][START_REF] Kokkinos | Periodic finitedifference time-domain analysis of loaded transmission-line negativerefractive-index metamaterials[END_REF][START_REF] Markley | Quad-Band Negative-Refractive-Index Transmission-Line Unit Cell with Reduced Group Delay[END_REF][START_REF] Monti | Negative Group Velocity in a Split Ring Resonator-Coupled Microstrip Line[END_REF][START_REF] Nesimoglu | A Tunable Metamaterial Resonator Using Varactor Diodes to Facilitate the Design of Reconfigurable Microwave Circuits[END_REF][START_REF] Barroso | Negative group velocity in resistive lossy left-handed transmission lines[END_REF].

The remarkable period of the NGD function development in the microwave area has been found during the design of Fayu Wan, Hongyu Du and Blaise Ravelo are with the Nanjing University of Information Science & Technology (NUIST), Nanjing 210044, Jiangsu, China (E-mail: {fayu.wan, blaise.ravelo}@nuist.edu.cn, 614920361 @qq.com).

Yang Liu is with Altran, 78140 Vélizy-Villacoublay, France. (E-mail: liuyang2101@gmail.com). metamaterial artificial transmission line (TL) exhibiting negative refractive index effect [START_REF] Eleftheriades | Transmission Line for Negative Refractive Index Media and Associated Implementations without Excess Resonators[END_REF][START_REF] Siddiqui | Periodically Loaded Transmission Line With Effective Negative Refractive Index and Negative Group Velocity[END_REF][START_REF] Siddiqui | Time-Domain Measurement of Negative-Index Transmission-Line Metamaterials[END_REF][START_REF] Kokkinos | Periodic finitedifference time-domain analysis of loaded transmission-line negativerefractive-index metamaterials[END_REF]. The metamaterial based NGD circuits were designed with periodical structures [START_REF] Siddiqui | Periodically Loaded Transmission Line With Effective Negative Refractive Index and Negative Group Velocity[END_REF][START_REF] Kokkinos | Periodic finitedifference time-domain analysis of loaded transmission-line negativerefractive-index metamaterials[END_REF]. Diverse microwave NGD devices operating as quad-band circuits [START_REF] Siddiqui | Time-Domain Measurement of Negative-Index Transmission-Line Metamaterials[END_REF], reconfigurable circuits with tunable metamaterial resonators [START_REF] Nesimoglu | A Tunable Metamaterial Resonator Using Varactor Diodes to Facilitate the Design of Reconfigurable Microwave Circuits[END_REF], and resistive lossy left-handed TLs [START_REF] Barroso | Negative group velocity in resistive lossy left-handed transmission lines[END_REF] were proposed. It was emphasized that some NGD circuits are susceptible to propagate pulse signals with superluminal effect [START_REF] Cao | Negative group delay and pulse compression in superluminal pulse propagation[END_REF][START_REF] Macke | Optimal superluminal systems[END_REF][START_REF] Ségard | Two-pulse interference and superluminality[END_REF][START_REF] Kitano | Negative group delay and superluminal propagation: An electronic circuit approach[END_REF]. Moreover, the superluminal effect was experimented with different systems with optimal behavior [START_REF] Macke | Optimal superluminal systems[END_REF], two interference pulses [START_REF] Ségard | Two-pulse interference and superluminality[END_REF], and active circuit approach [START_REF] Kitano | Negative group delay and superluminal propagation: An electronic circuit approach[END_REF]. Nevertheless, the existence of the NGD function and superluminal effect is not in contradiction with the causality [START_REF] Mitchell | Negative group delay and ''fronts'' in a causal system: An experiment with very low frequency bandpass amplifiers[END_REF][START_REF] Mitchell | Causality and Negative Group-delays in a Simple Bandpass Amplifier[END_REF]. Such counterintuitive and extraordinary effect observation raises curious questions on the meaning and interpretation of the NGD function in the electronic and microwave engineering. Therefore, a simple theorization of NGD function based on the analogy with the filter function was initiated [START_REF] Ravelo | Similitude between the NGD function and filter gain behaviours[END_REF].

Innovative NGD circuits operating with low-pass and bandpass behaviors were designed. Thanks to the ambitious research work made by some research group around the world, various topologies of typical bandpass NGD circuits were investigated [START_REF] Wu | Maximally flat negative group-delay circuit: A microwave transversal filter approach[END_REF][START_REF] Zhang | Unconditionally Stable Non-Foster Element Using Active Transversal-Filter-Based Negative Group Delay Circuit[END_REF][START_REF] Qiu | Absorptive bandstop filter with prescribed negative group delay and bandwidth[END_REF][START_REF] Wang | A Negative Group Delay Microwave Circuit Based on Signal Interference Techniques[END_REF][START_REF] Liu | Compact transmission-type negative group delay circuit with low attenuation[END_REF][START_REF] Shao | A compact transmission line self-matched negative group delay microwave circuit[END_REF][START_REF] Chaudhary | Miniaturized dual-band negative group delay circuit using dual-plane defected structures[END_REF][START_REF] Shao | A Compact Dual-Band Negative Group Delay Microwave Circuit[END_REF][START_REF] Wan | O=O Shape Low-Loss Negative Group Delay Microstrip Circuit[END_REF][START_REF] Zhou | S-Matrix and Bandpass Negative Group Delay Innovative Theory of Ti-Geometrical Shape Microstrip Structure[END_REF]. The main challenges on the NGD microwave circuit of the last two decades were on the loss compensation [START_REF] Wu | Maximally flat negative group-delay circuit: A microwave transversal filter approach[END_REF][START_REF] Zhang | Unconditionally Stable Non-Foster Element Using Active Transversal-Filter-Based Negative Group Delay Circuit[END_REF], circuit operation understanding [START_REF] Qiu | Absorptive bandstop filter with prescribed negative group delay and bandwidth[END_REF][START_REF] Wang | A Negative Group Delay Microwave Circuit Based on Signal Interference Techniques[END_REF], the size reduction [START_REF] Liu | Compact transmission-type negative group delay circuit with low attenuation[END_REF][START_REF] Shao | A compact transmission line self-matched negative group delay microwave circuit[END_REF][START_REF] Chaudhary | Miniaturized dual-band negative group delay circuit using dual-plane defected structures[END_REF][START_REF] Shao | A Compact Dual-Band Negative Group Delay Microwave Circuit[END_REF], and the implementation of lowattenuation passive circuits [START_REF] Wan | O=O Shape Low-Loss Negative Group Delay Microstrip Circuit[END_REF][START_REF] Zhou | S-Matrix and Bandpass Negative Group Delay Innovative Theory of Ti-Geometrical Shape Microstrip Structure[END_REF] susceptible to operate in the gigahertz frequency range. An active NGD circuit designed with microwave transversal filter approach was proposed [START_REF] Wu | Maximally flat negative group-delay circuit: A microwave transversal filter approach[END_REF]. To highlight the NGD design feasibility of some microstrip Lala Rajaoarisoa is with IMT Lille Douai, Univ. Lille, Unité de Recherche en Informatique et Automatique, F-59000 Lille, France. (E-mail: lala.rajaoarisoa@imt-lille-douai.fr).

Sébastien Lalléchère is with Université Clermont Auvergne, Institut Pascal, SIGMA Clermont, France. (E-mail: sebastien.lallechere@uca.fr).

Wenceslas Rahajandraibe is with the Aix-Marseille University, CNRS, University of Toulon, IM2NP UMR7334, Marseille, France (E-mail: wenceslas.rahajandraibe@im2np.fr).

Preeti Thakur, and Atul Thakur are with the Amity University Haryana, Gurgaon, 122413 India (E-mail: pthakur@ggn.amity.edu, athakur1@ggn.amity.edu).

Glauco Fontgalland is with Federal University of Campina Grande, Applied Electromagnetic and Microwave Lab., Campina Grande/PB, 58429, Brazil. (Email: fontgalland@dee.ufcg.edu.br). T topologies, an absorptive bandstop filter based [START_REF] Zhang | Unconditionally Stable Non-Foster Element Using Active Transversal-Filter-Based Negative Group Delay Circuit[END_REF] and signal interference [START_REF] Qiu | Absorptive bandstop filter with prescribed negative group delay and bandwidth[END_REF] based NGD circuit were introduced. Furthermore, different NGD compact circuits based on microstrip TLs were investigated [START_REF] Liu | Compact transmission-type negative group delay circuit with low attenuation[END_REF][START_REF] Shao | A compact transmission line self-matched negative group delay microwave circuit[END_REF][START_REF] Chaudhary | Miniaturized dual-band negative group delay circuit using dual-plane defected structures[END_REF][START_REF] Shao | A Compact Dual-Band Negative Group Delay Microwave Circuit[END_REF]. Then, innovative geometrical shape topologies of low loss NGD passive microstrip circuits were proposed [START_REF] Wan | O=O Shape Low-Loss Negative Group Delay Microstrip Circuit[END_REF][START_REF] Zhou | S-Matrix and Bandpass Negative Group Delay Innovative Theory of Ti-Geometrical Shape Microstrip Structure[END_REF]. Some tentative applications of NGD function targeted for the GD equalization were initiated [START_REF] Ahn | Group Delay Equalized UWB InGaP/GaAs HBT MMIC Amplifier using Negative Group Delay Circuits[END_REF][START_REF] Shao | A Full-Passband Linear-Phase Band-Pass Filter Equalized with Negative Group Delay Circuits[END_REF].

Behind the progress of the microwave circuit design, further curious questions remain to be answered. For example, we would like to study the temperature effect on the NGD microstrip circuit as investigated in [START_REF] Ravelo | Microstrip dielectric substrate material characterization with temperature effect[END_REF][START_REF] Ravelo | Multiphysics Model of Microstrip Structure Under High Voltage Pulse Excitation[END_REF][START_REF] Xu | Multiphysics Tensorial Network Analysis Applied to PCB Interconnect Fatigue Under Thermal Cycle Aggression[END_REF]. The paper is organized in four main sections as follows:

• Section II introduces the design of the proof of concept (PoC) of the BP NGD circuit constituted by the combined ring and stub microstrip structure. • Section III investigates the experimental setup on the BP NGD circuit under test (CUT) of the empirical robustness study. The guideline of the innovative electrothermal test operation process with the calibration technique of the temperature room will be explained. • Section IV discusses on the observable results of the robustness study. The electrothermal empirical results illustrating the NGD performances versus the ambient temperature will be examined which show the variation of the BP NGD responses. • Then, Section V is the final conclusion.

II. DESIGN DESCRIPTION OF THE RING-STUB BP NGD CIRCUIT

This section describes the design and implementation of the PoC prototype. The BP NGD circuit is designed by modification of parallel line topology [START_REF] Wan | O=O Shape Low-Loss Negative Group Delay Microstrip Circuit[END_REF]. Then, the performance was optimized in function of the positioning and the physical length of an open-ended stub.

A. PoC of Microstrip Circuit Design

The considered PoC is a two-port distributed passive circuit. As illustrated by the 2-D design shown by Fig. 1(a), the circuit is constituted by two microstrip circular lines presenting physical inner radius, R1, outer radius, R2 and width, w. The two circular lines are connected in parallel. The access lines have physical length, lport and width, wport.

The top circular line is connected to open ended straight stub having physical length, l1, and width, w1, at its mid-point. The physical parameters of this PoC prototype are indicated in Table I. The fabricated circuit prototype photographed in Fig. 1(b) is implemented on Cu-metalized FR4 dielectric substrate in microstrip technology. The substrate physical characteristics are addressed by Table II.

B. BP NGD Function Specifications

The analysis of the ring-stub passive symmetric circuit is based on the 2-D frequency-dependent S-matrix:

  11 21 21 11 ( ) ( ) () ( ) ( ) S j S j Sj S j S j  =      (1)
with the angular frequency, ω. The main parameters are basically the transmission coefficient magnitude, and the reflection coefficient, respectively: (3) Then, the essential parameter is the GD defined by:

  21 ( ) arg ( ) / GD S j = -     . (4)
Fig. 2(a) represents the GD response of BP NGD ideal circuit. This BP NGD function is characterized by the cut-off angular frequencies, ω1<ω2, with:

1 2 ( ) 0 GD       . ( 5
)
The associated NGD bandwidth (BW) is given by:

21  = -    . ( 6
)
The response is also characterized by a center angular frequency 1 0 2     which can be defined by: These specifications will serve to the BP NGD electrothermal characterization in the rest of the paper.

0 0 ( ) 0 GD GD =  . ( 7
To investigate the electrothermal NGD behavior of the circuit, an innovative experimental setup described in the following section was considered.

III. EXPERIMENTAL SETUP OF THE ROBUSTNESS STUDY APPLIED TO THE RING-STUB CIRCUIT

The present section describes the robustness study empirical principle. The PoC of the robustness is based on the electrothermal analysis of the ring-stub NGD circuit. The experimental setup will be introduced in the following subsection.

A. Principle of the Experimental Setup for the NGD Electrothermal Analysis

The electrothermal empirical study principle of microwave circuit is fundamentally aimed to determine the transmission and reflection coefficients of S-matrix proposed in equation [START_REF] Chu | Linear Pulse Propagation in an Absorbing Medium[END_REF] with the evolution of ambient temperature, T:

  11 21 21 11 ( , ) ( , ) ( , ) ( , ) ( , )

S jf T S jf T S jf T S jf T S jf T

 =   . ( 10
)
It means that the empirical S-parameter data should be expressed in function of couple ( 2 , ) j j f T =  . To measure the S-parameters, we proposed the innovative experimental setup highlighted by illustrative diagram in Fig. 3. The vector network analyzer (VNA) is facing the side of the test box. After calibration, two ports are respectively connected with a 1mlong cable. The cable enters the test box through the hole on the side of the test box. At the same time, the two cables are relatively parallel and do not cross for a certain distance to avoid mutual interference. After putting it into the test chamber, fix both ends of the cable on the iron frame inside the test chamber with adhesive tape, and fix the thermometer. After placing, close the door of the test chamber to start the test. Of course, the VNA apparatus is needed to record the Sparameters. The thermal room specifications will be described in the following subsection.

B. Electrothermal Test and the Thermal Room Specification

Fig. 4 shows the photograph of the electrothermal test setup. The employed VNA is referenced Agilent® AV3672B-S, which operates from 10 MHz to 26.5 GHz. We use two identical SMA cables 1meter-length. Then, the main innovative part of this research work is the consideration of the thermal room with volume 1 m × 1 m × 1 m. This later one is referenced ESPEC® [START_REF]Thermal room specifications[END_REF]. During the experimentation, the circuit under test (CUT) is placed in the room as indicated in Fig. 4. Then, it is connected to the VNA as previously depicted in Fig. 3. Prior to running the test, the VNA was calibrated in SOLT configuration at the ambient temperature, Ta=20°C. We underline that the present study is limited to the case of temperature, T, higher than T a . Then, we realize that such a calibration remains valid to measure the circuit with the variation of the room temperature, ΔT=T-Ta<100°C. Moreover, the influence of the room temperature on the interconnection cables and CUT connectors is negligible.

C. Guideline of the Thermal Room Test Operation

The thermal room test operation can be summarized by the following steps: For the present case of study, the users must save 11 groups of ring-stub circuit Sparameter data family. To verify the feasibility of the robustness analysis of the NGD circuit under ambient temperature stress, practical investigation will be presented in the next section.

•

IV. INVESTIGATION ON THE RESULTS OF THE BP NGD ELECTROTHERMAL EXPERIMENTATION

The present section deals with the examination of the ringstub circuit electrothermal test results. The BP NGD validation based on the comparison between simulation and measurement is presented. Then, the BP NGD responses will be discussed in function of the ambient temperature variation.

A. Preliminary Validation of the BP NGD Behavior of the Ring-Stub Circuit

To validate the BP NGD behavior of the CUT, comparisons between the simulated ("Sim.") and measured ("Meas.") results were carried out in the frequency band from 2.35 GHz to 2.5 GHz. The simulated results were obtained from the commercial tool of electronic and RF/microwave circuit simulator ADS® from Keysight Technologies®. The ring-stub circuit operates as dual-band NGD function. The simulated, and measured GDs and magnitudes of the reflection and transmission coefficients are plotted in Fig. 5(a) and in Fig. 5(b), respectively. It can be pointed out that these comparative results present a very good agreement. In addition, the simulated and measured GD responses are plotted in Fig. 5(c). These results confirm that the ring stub CUT behaves as a BP NGD function. Table III presents the comparison between the simulated and experimented results of BP NGD specifications. As expected, the CUT exhibits NGD center frequency of approximately f0=2.42 GHz and NGD value, GD(fo), of -1.9 ns. It can be seen from Fig. 5(b) that in the test frequency band, the attenuation remains better than -4 dB and the reflection coefficient is better than -11 dB. The slight frequency shifts of NGD center frequency are mainly due to the fabrication inaccuracies, substrate effective permittivity tolerance, and losses versus the numerical computation accuracy. The electrothermal test results of the ring-stub circuit were generated with respect to the guideline indicated in Subsection III-C. The experimented data results obtained after the postprocessing will be analyzed in the two following subsections.

B. Interconnection Cable Calibration Including the Temperature Stress Effect

Before the NGD characterization of the ring-stub circuit, the interconnection cable was separately calibrated at each sample of the room temperature variation, ΔT, was progressively increased from 0°C to 40°C step 1°C. During the test, the VNA was calibrated from 2. The associated GD determined from the transmission coefficient is given by:

  21 arg ( , ) ( , ) 2 cable cable S jf T GD f T f - =   . ( 12
)
Consequently, the obtained electrothermal mappings of the employed interconnection cable GD, transmission and reflection coefficients are displayed in Fig. 6(a), Fig. 6(b) and, Fig. 6(c), respectively. Significant variations of the interconnection cable S-parameter magnitudes can be observed.

The NGD center frequency, f0, of the ring-stub circuit under temperature stress is considered to carry out the calibration. Accordingly, we consider the GD(f0), S21(f0), and S11(f0) empirical data of the tested cable summarized in Table IV. The influence of the interconnection cable can be quantified from this table. It can be emphasized from this empirical result that the cable presents the variations of:

• Interconnect cable GD presenting medium value of about mean(GDcable(fo))≈9.1 ns and variation max(GDcable(fo))min(GDcable(fo))<0.1 ns, • Transmission coefficient having absolute variations of about max(S21cable(fo))-min(S21cable(fo))<0.2 dB, • And reflection coefficient with maximal value max(S11cable(fo)) lower than -23 dB. By taking into account the results of the cable calibration, the BP NGD characterization of the ring-stub circuit including the electrothermal robustness analysis will be discussed in the following paragraph.

C. BP NGD Characterization After Processing at Ambient Temperature on the Ring-Stub Circuit

The heart of the thermal stress robustness investigation is explored in the present subsection. The robustness study of the BP NGD CUT is focused around the NGD frequency band. We choose the operation frequency band between 2.35 GHz and 2.5 GHz. The following paragraph explains the calibration approach by considering the cable electrothermal effect.

1) NGD Circuit Calibration Approach by Taking Into Account the Cable Electrothermal Effect Response

The electrothermal observable data processing is elaborated from the measurement of the total circuit composed of the interconnection cable characterized in the previous subsection connected to the CUT. Accordingly, the total circuit Sparameters can be denoted by:

) 13
As the interconnection cable with S-parameters given by equation ( 11) and the tested circuit with S-parameters given by equation ( 10) are very well matched in the NGD frequency band, we can deduce the CUT transmission coefficient from the following relation: ( , ) ( , ) ( , )

total cable S jf T S jf T S jf T  . ( 15
)
It can be derived from equation ( 14) that the ring-stub CUT measured GD and transmission coefficient magnitude in decibel are respectively expressed by: ( , ) ( , ) ( , )

total cable GD f T GD f T GD f T - (16) 21 21 21
( , ) ( , ) ( , )

dB dB dB total cable S f T S f T S f T - . ( 17
)
Based on the proposed approach, the empirical results are examined in the following paragraph.

2) Electrothermal Data Mapping Processing of the Ring-Stub CUT

The thermal stress measurement of the ring-stub circuit was carried out by including the interconnect cable. The experimented data represented by S-parameter equation [START_REF] Monti | Negative Group Velocity in a Split Ring Resonator-Coupled Microstrip Line[END_REF] were recorded after 11 minutes of the temperature change. After the thermal stress, the BP NGD characterization of the ring NGD circuit is performed by using previously suggested equations ( 13), [START_REF] Ségard | Two-pulse interference and superluminality[END_REF], and [START_REF] Kitano | Negative group delay and superluminal propagation: An electronic circuit approach[END_REF]. Similar to the analysis of the previous subsection, the analyses were based on the mappings of the ring-stub CUT GD, transmission and reflection coefficients displayed by Fig. 8(a), Fig. 8(b), and Fig. 8(c), respectively. As expected, an interesting electrothermal effect is curiously observed. It can be seen in Fig. 8(a) that the ringstub circuit conserve the BP NGD behavior despite the thermal stress. However, the NGD value and the NGD center frequency are decreasing inversely to the temperature variation. Moreover, it can be underlined from Fig. 8(a) that the NGD bandwidth is slightly increasing. Furthermore, in the NGD frequency range, despite the temperature stress, at the NGD center frequency, the attenuation depicted by Fig. 8(b) remains better than 3 dB and the ring-stub circuit access matching illustrated by Fig. 8(c) remains better than -11 dB.

In addition, similar to the normal operation of BP NGD circuits available in the literature, the transmission coefficient magnitude presents an electrothermal behavioral effect shown by Fig. 8(b) similar to the GD of Fig. 8(a). Based on the explored raw data, the robustness study related to the BP NGD specifications of the ring-stub NGD circuit will be examined in the following paragraph.

3) Robustness related to the BP NGD specifications

In the area of electronic and microwave engineering, the robustness study must be referenced by observable data. For the present study, we are presenting a BP NGD behavior of a microstrip circuit against the thermal stress. For the best of the authors' knowledge, this is the first study of electrothermal BP characterization applied to microstrip circuit.

D. Discussion on the Application of BP NGD Robustness Study of the Ring-Stub CUT

These previous empirical results enable to qualify the BP NGD robustness of the ring-stub circuit. The main robustness risk related to such results should be specified in function of the electronic system applications. For example, for the case of IEEE 802.11a/b wireless transceiver system, the standard communication channel bandwidths are higher than 20 MHz. Therefore, we can expect that the NGD circuit robustness can induce tolerable operation within the ambient temperature variation, ΔT, less than 40°C.

V. CONCLUSION

An original study of BP NGD microstrip circuit, never being achieved before, is investigated. The study is based on the test of ring-stub microstrip circuit. After the recall on the BP NGD specifications, the NGD design of the proof of concept is described. Then, the electrothermal test guideline is explained with the consideration of the experimental setup of the thermal room and the VNA calibration.

The robustness of the ring-stub circuit is validated by the empirical test under temperature fluctuation with maximal amplitude of about 40°C. The variation of the BP NGD characteristics as the NGD center frequency, NGD value, transmission and reflection coefficients in function of the temperature are discussed.

The proposed study is useful for the future application of the BP NGD circuit robustness and also reliability in the electronic system susceptible to operate in the wide range of ambient temperature variation.

  Manuscript received xxx xx, 2021; revised xxx xx, 2021; accepted xxx xx, 2021. Date of publication xxx xx, 2021. (Corresponding author: Dr. Yang Liu) This research work was supported in part by NSFC under Grant 61971230, and in part by Jiangsu Specially Appointed Professor program and Six Major Talents Summit of Jiangsu Province (2019-DZXX-022) and in part by the Startup Foundation for Introducing Talent of NUIST.

Fig. 1 .

 1 Fig. 1. (a) 2-D design and (b) photo of the ring-stub circuit under study.

Fig. 2 .

 2 Fig. 2. BP NGD circuit ideal responses: (a) GD, (b) S21 and (c) S11.

Fig. 3 .

 3 Fig. 3. Illustrative diagram of NGD electrothermal test.

Fig. 4 .

 4 Fig. 4. Photo of the considered electrothermal experimental setup.

3 .

 3 Step A: Installation with respect to the diagram shown by Fig. This step requires the familiarization to the use of VNA and the control of the temperature (alternating humidity and heat) with the thermal test chamber. • Step B: Connect the extension cable and put it into the high and low temperature (alternating humidity and heat) test chamber without test circuit to ensure that the two cables entering the temperature chamber are equal length and parallel. Close the thermal room and connect the NGD circuit to be tested on the no-load cable. • Step C: VNA calibration with respect to the range of the frequency of the CUT. • Step D: Measure the S-parameters of the cable with the temperature change when no load. • Step E: Adjust the room temperature and wait for the temperature stabilization. • Step F: Adjust the temperature box, the range of each is 4°C within the time of 5 minutes. After reaching a rise, keep the temperature stable during 1 minute. Then, derive the S-parameter data from the VNA. • Step G: The previous step must be repeated until the temperature variation reaches 40°C.

Fig. 5 .

 5 Fig. 5. (a) Reflection and (b) transmission coefficients and (c) GD responses of the ring-stub CUT prototype shown by Figs. 1 in the second bandwidth at ΔT=0.

Fig. 6 .

 6 Fig. 6. Mappings of (a) GD, (b) S 21 and (c) S 11 for the interconnection cable versus frequency and temperature effect.

Fig. 7 .

 7 Fig. 7. (a) GD, (b) S21 and (c) S11 of the interconnection cable at NGD center frequencies versus temperature effect.

Fig. 8 .

 8 Fig. 8. Mappings of experimented (a) GD, S 21 and (c) S 11 of ring-stub circuit versus (f,T).

  As aforementioned by Figs. 2, the BP NGD specifications are quantified by the NGD center frequency, NGD value, NGD bandwidth, attenuation and reflection coefficient of the CUT. Based on the data depicted in Figs. 8, we extracted the NGD center frequency, NGD value, and NGD bandwidth versus stress temperature which are plotted in Fig. 9(a), Fig. 9(b) and Fig. 9(c), respectively. Then, the variations of the transmission coefficient, reflection coefficient, and insertion loss flatness versus stress temperature are shown in Fig. 10(a), Fig. 10(b), and Fig. 10(c).The overall BP NGD specifications are characterized by the NGD center frequency, f0, BP NGD characteristics of the ringstub circuit under temperature variation from 0°C to 40°C with step of 4°C.

Fig. 10 .

 10 Fig. 10. (a) Transmission and (b) reflection coefficients at the NGD center frequency, and (c) attenuation flatness versus temperature of the ring-stub circuit after test processing value.

TABLE I GEOMETRICAL

 I PARAMETERS OF THE CIRCUIT POC

	Circuit	Description	Parameter	Value
	Whole circuit	Total length Total width	L w	58.2 mm 35 mm
	Access line	Length Width	lport wport	5 mm 2.8 mm
	Stub	Length Width	w1 l1	2.2 mm 28 mm
	Ring	Inner radius Outer radius	R1 R2	9.5 mm 11.5 mm
		TABLE II
	PHYSICAL PARAMETERS OF THE POC CIRCUIT SUBSTRATE
	Structure	Description	Parameters	Values
	Substrate	Relative permittivity Loss tangent	εr tan(δ)	4.5 0.02
		Thickness	h	1.6 mm
	Metallization conductor	Copper conductivity Thickness	σ t	58 MS/m 35 µm

  3 GHz and 2.6 GHz to generate the thermal and frequency dependent S-parameters of the employed interconnection cable which can be denoted:

		S	cable	( , ) jf T	 11 21 cable cable S S  21 11 ( , ) ( , ) ( , ) ( , ) cable cable jf T S jf T jf T S jf T  = 	.	(11)

TABLE IV GD

 IV , TRANSMISSION AND REFLECTION COEFFICIENTS AT THE NGD CENTRE FREQUENCY OF THE INTERCONNECTED CABLE

	ΔT (°C)	f0 (GHz)	GD(f0) (ns)	S21(f0) (dB)	S11(f0) (dB)
	0	2.434	9.070	-1.725	-27.329
	4	2.434	9.064	-1.658	-27.360
	8	2.432	9.052	-1.792	-24.292
	12	2.429	9.086	-1.676	-28.924
	16	2.427	9.082	-1.715	-29.243
	20	2.426	9.086	-1.684	-29.135
	24	2.425	9.088	-1.794	-32.331
	28	2.423	9.084	-1.738	-29.512
	32	2.421	9.090	-1.711	-29.587
	36	2.418	9.109	-1.709	-26.062
	40	2.416	9.115	-1.704	-25.918