Jean-Marc Marc Girault
email: jean-marc.girault@univ-tours.fr

Zheng

Zheng Zhao

Synchronous detector as a new paradigm for automatic microemboli detection

Keywords: Doppler, ultrasound, microemboli, synchronous detection, cyclo-stationarity

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INTRODUCTION

Early detection of cerebral microemboli is of great importance since cerebral vascular accidents represent more than two thirds of all ischemic strokes ( [START_REF] Feigin | Stroke Epidemiology in the Developing World[END_REF]).

Cerebral embolism is the result of migration of insoluble microparticles (pieces of fat for example) in the blood circulation. When these microemboli block an arteriol, ischemia occurs ( [START_REF] Siebler | Cerebral Microembolism and the Risk of Ischemia in Asymptomatic High-Grade Internal Carotid Artery Stenosis[END_REF]). Cerebral embolism covers a wide variety of clinical entities revealed by cerebral angiography ( [START_REF] Markus | Microscopic air embolism during cerebral angiography and strategies for its avoidance[END_REF]), prosthetic heart valve insertion ( [START_REF] Georgiadis | Prevalence and characteristics of intracranial microemboli signals in patients with different types of prosthetic cardiac valves[END_REF]), and carotid surgery ( [START_REF] Ackerstaff | Carotid endarterectomy and intraoperative emboli detection[END_REF]), to name but a few.

Transcranial Doppler (TCD) is currently used to detect microemboli ( [START_REF] Aaslid | Non invasive transcranial Doppler ultrasound recording of flow velocity in basal cerebral arteries[END_REF][START_REF] Spencer | Detection of middle cerebral artery emboli during carotid endarterectomy using transcranial Doppler ultrasonography[END_REF]). This real time ultrasound system, also used to assess blood flow velocity for intracranial arteries, is a low-cost and non-invasive modality.

In most studies, commercial TCD detectors seem to be sufficient to detect most microembolic events ( [START_REF] Aaslid | Transcranial Doppler, chapter : Developments and principles of transcranial Doppler[END_REF]). However, clinical practitioners sometimes observe previously undetected microembolic signatures that are present on the spectrogram displayed on the TCD screen during clinical examination (see Fig. 1c). These undetected microemboli that are not visible on observation of the amplitude of the Doppler signal (see Fig. 1a) are sometimes unaudible. The main explanations for the inability of existing detectors to detect the smallest microemboli may be either: a threshold level adjusted by the user that is too high (see Fig. 1b) and microembolic signals of lower magnitude than that of the blood Doppler signal located at the systolic phase, i.e. the maximum amplitude level during the cardiac cycle (see Fig. 1b).

To solve the problem of inability of commercial systems to detect, a search for the implementation of reliable and robust methods was initiated. Without attempting to draw up an exhaustive list of existing methods, it should be noted that a wide variety of methods have been proposed during the last ten years such as those based on detection of sudden changes ( [START_REF] Girault | Microembolic detection: an ultrasound Doppler signal processing view point[END_REF]), matching pursuit ( [START_REF] Devuyst | The matching pursuit: A new method of characterizing microembolic signals?[END_REF]), expert systems ( [START_REF] Fan | Automated embolus identification using a rule based expert system[END_REF]), wavelets ( [START_REF] Aydin | The use of the wavelet transform to describe embolic signals[END_REF]), spectral correlation ( [START_REF] Girault | Spectral Correlation of the embolic blood Doppler signal[END_REF]), fuzzy logic ( [START_REF] Kouamé | Adaptive AR and Neurofuzzy Approaches: Access to Cerebral Particle Signatures[END_REF][START_REF] Xu | An automated feature extraction and emboli detection system based on the PCA and fuzzy sets[END_REF]), GARCH models ( [START_REF] Girault | Automatic Detection of microemboli by means of a generalized autoregressive conditional heteroskedasticity model[END_REF]), and neural networks ( [START_REF] Benoudjit | In vitro microemboli classification using neural network models and RF signals[END_REF]).

The starting point of this study was based on the concept of spectral correlation ( [START_REF] Girault | Spectral Correlation of the embolic blood Doppler signal[END_REF]) that can be viewed as a process of synchronization. Unlike the simulation studies previously presented by [START_REF] Girault | Spectral Correlation of the embolic blood Doppler signal[END_REF], this study evaluated the levels of performance of the synchronous detector from clinical recordings.

This new class of detector uses the fact that the nature of the microembolic Doppler signal is different from that of the blood Doppler signal. Here we contrast the cyclic nature of the blood Doppler signal to the unpredictable nature (i.e. uncyclic) of the Doppler signal of microemboli.

Our new class of detector was compared to the most frequently used commercial detectors, i.e. non-synchronized energy detectors, to demonstrate its relevance.

STANDARD METHOD

The energy of the Doppler signal seems to provide the best information from which the presence of microemboli can be detected. This can mainly be explained by the fact that the energy of the backscattered Doppler signal is proportional to the size of the microembolus to the power of 6 while the Rayleigh scattering holds, in other words larger the microembolus, the higher the intensity of the Doppler signal. Two kinds of energy detector can be used, those based on time estimators and those based on spectral estimators. As spectral estimators are presented in the next section, only the time estimator is presented in this section. The time estimator of the instantaneous energy can be obtained by:

E(t) = +∞ -∞ +∞ -∞ x(τ )g * (t -τ ))dτ 2 dt,
where g(t) is a Hanning sliding window and * stands for the conjugation operation.

As previously explained, in commercial devices the decision information (DI) on which detection is based is currently the instantaneous energy E(t). This DI is then compared to a constant threshold. Two strategies are then possible:

• the constant threshold is fixed empirically by the user for the entire examination. This arbitrary threshold is patient-, operator-and devicedependent. As this threshold level is not the same at each examination, comparison is difficult;

• the constant threshold is set automatically from the instantaneous energy. This threshold is often evaluated as a mean value derived from previous energy over a few seconds. This threshold is thus a statistical threshold.

Whatever the method, DI is then compared to a non-time-varying threshold λ. If the decision information DI = E(t) is greater than the threshold λ then an embolus is considered to be detected (hypothesis H 1 ), otherwise no embolus is detected (hypothesis H 0 ). This can be expressed as follows:

H 1 E(t) ≷ λ.
(1) H 0

In this study only statistical thresholds were taken into consideration. This means that a training period was required to extract statistics. After observing that the statistics of the instantaneous energy followed gaussian law, the following threshold λ was proposed:

λ = µ + ασ,
where α was a parameter to be adjusted, and µ and σ were the mean value and the standard deviation of the instantaneous energy, respectively.

From our experience, a training period of 5 seconds and α ranging from 3 to 5 seemed to be optimal values.

NEW METHOD

The new type of detector that we propose is based on the use of a detector synchronized with the cardiac rhythm. The solution that we proposed consisted of four parts (Fig. Note that for standard commercial detectors introduced in the previous section, there is no step 2, and step 3 may or may not be present.

Energy estimator

Several estimators can be used to assess the instantaneous energy of Doppler signals. Here we focus on the spectral estimator since it is already included in the spectrogram calculation of commercial TCD systems (function A in Fig. 2).

The instantaneous energy is obtained from the spectrogram by:

S x (t, f ) = +∞ -∞ x(τ )g * (t -τ ))e -j2πf τ dτ 2 ,
where g(t) is a Hanning sliding window and * stands for conjugation operation.

An illustration of the spectrogram of a recorded Doppler signal is presented in Fig. 1c. The instantaneous energy over the entire spectrum for a fixed time t, can be obtained by:

E(t) = +∞ -∞ S x (t, f )df.
An illustration of the instantaneous energy of a Doppler signal with microemboli is presented in Fig. 1b.

As expected, the instantaneous energy fluctuated considerably since it revealed the stochastic nature of billions of red blood cells randomly located in the artery. Furthermore, repetitive patterns for each cardiac cycle can be seen. The pattern shape describes each part of the cardiac cycle, such as the systolic and the diastolic phases.

Cardiac rhythm estimator

The instantaneous cardiac rhythm can be obtained using different modalities (ultrasound and EKG) and different estimators based on the instantaneous frequency or on the instantaneous energy of the Doppler signal, estimators being temporal or spectral.

Many temporal algorithms based on the correlation function of the Doppler signals have been proposed to evaluate the fetal heart rate (e.g. [START_REF] Voicu | Robust Estimation of Fetal Heart Rate from US Doppler Signals[END_REF]). However, as the instantaneous energy had already been evaluated in the previous steps, it seemed preferable to use the instantaneous energy to evaluate the cardiac rhythm (function B in Fig. 2).

To determine the cardiac rhythm from the instantaneous energy E(t), we proposed extracting the fundamental component E 0 (t) of the instantaneous energy. This fundamental component was extracted from the spectrogram S x (t, f ) by using a low pass filter :

S 0 (f, t) = S x (f, t)G(f, t),
where G(f, t) is the square modulus of the time-varying transfer function of the low pass filter. This latter transfer function can be simply a rectangular function centered around the null frequency:

G(f, t) = G(f ) = Rect Fc (f ),
where F c is the cut-off frequency of the low pass filter. By using the inverse Fourier transform and an appropriate value of F c , the filtered energy signal is written simply as:

E 0 (t) = A 0 cos(2πf 0 t),
where f 0 corresponds to the cardiac frequency. This kind of filter extracted the fundamental component f 0 of the Fourier series of the cardiac rhythm. By detecting the maxima of the fundamental component of the instantaneous energy E 0 (t), it was possible to index each cardiac cycle. Note that it would also be possible to index each cardiac rhythm by detecting the minima or the zero-crossings.

As an illustration, we report on Fig. 3 the histograms of normalized cardiac rhythms obtained from two different recordings. The first histogram with a gaussian shape had a relatively smaller variance compared to the second histogram. The difference was due to the large number of ventricular extra-systoles in the second recording. This could be problematic, since our synchronous detector was based on the regularity of the cardiac rhythm. To overcome this problem, all cardiac cycles that exceeded 20% of the mean value of the normalized cardiac rhythm were removed. Derived from the analysis of our database, this value seemed to be the best trade-off between i) keeping enough cardiac cycles to maintain the rate of non-detection of microemboli at its lowest and ii) removing sufficient outlier cardiac cycles so that the synchronization process was not too disturbed.

Statistical threshold

This step was probably the most crucial. It consisted of determining the cardiac statistical threshold by extracting each Doppler energy signal from previous time indices (function C in Fig. 2). However, as the cardiac rhythm can fluctuate slightly , each extracted energy signal was resampled in order to give them all the same length. For a gaussian statistic, the statistical threshold can be evaluated as follows:

λ(t) = µ(t) + ασ(t),
where α is a parameter to be adjusted, and µ(t) and σ(t) the mean value and the standard deviation of the energy signal, respectively.

From study of our data set it seemed best to set α ranging from 3 to 5. Note that to maintain a certain degree of adaptativity, accuracy and robustness in our approach, a sufficient duration to determine the statistical threshold was required. It was obvious that the longer the duration of the training phase, the more accurate the detection. On the other hand, the shorter the duration, the easier it is for the system to adapt to the presence of artefacts or any other changes. Our experience indicated that a duration of 5 seconds was a good trade-off to evaluate the statistical threshold. Note that the threshold was updated for each cardiac cycle. This involved an overlap of four cardiac cycles since the duration was of five cardiac cycles.

An illustration is given in Fig. 4 where the instantaneous energy was evaluated over 12 seconds. A high intensity transient signal was observed in the tenth cardiac cycle. After indexing and resampling, energy signals E i (t) of normalized duration were superimposed on each other. i represents the index of the cardiac cycle. The tenth cardiac cycle is superimposed in red in both Fig. 4a and Fig. 4b. The microembolus signature is highlighted in Fig. 4b. This type of example explains why our detector has the potential to detect the smallest microemboli even though their energy signatures were below the systolic energy.

Detection and enumeration

In addition to the other previously introduced strategies, a third strategy consisting of determining a time-varying threshold during the cardiac cycle can be proposed. Detection is said to be "synchronized" if comparison of the energy E with the threshold λ is performed at the same time position t 0 in the normalized cardiac cycle T , i.e. periodically for all t = t 0 +kT . For a time position t 0 in a normalized cardiac cycle and the normalized cardiac duration T = 1, synchronous detection consists of comparing an energy value E(t 0 ) at time t 0 , to a threshold λ(t 0 ). This statistical threshold, set for example at λ(t 0 ) = µ(t 0 ) + 5σ(t 0 ), is therefore obtained from a preliminary setting. µ(t 0 ) and σ(t 0 ) are the mean and the standard deviation taken at time position t 0 of the energy cycle, respectively. For example, µ(t 0 ) is obtained by adding the energy for each time value t = t 0 + kT as described below:

µ(t 0 ) = 1 N N -1 k=0 E(t 0 + kT ).
A setting phase (of a few seconds) is therefore necessary during the examination to evaluate statistics. Statistics are updated periodically thereafter during the examination. Note that this synchronized detection is equivalent to detection based on an adaptive threshold λ(t 0 ), this adaptive threshold itself being based on the statistics of the signal being analyzed:

H 1 E(t, T ) ≷ λ(t, T ).
(2) H 0 As the comparison is performed periodically with the normalized cardiac period T , E(t) and λ(t) become E(t, T ) and λ(t, T ). This latter point is very important, because now the threshold is no longer obtained from an empirical study but rather from a statistical study of a non-stationary process.

Parameters

Gold

Standard Synchronous standard detector detector N 4.2 3.1 5.5 P ct 100% 75% 130%

RESULTS AND DISCUSSION

Protocol

The new kind of detector proposed here was tested on 20 clinical recordings. These 30-minute recordings were carried out in the Ultrasound Department of Bretonneau Hospital in Tours, France . The waky-TCD T M of Atys Medical was used. A 2 MHz ultrasound probe was used to explore the middle cerebral artery. The pulse repetition frequency was mostly set at 6.4 kHz and the US power did not exceed 150 mW/cm 2 . Volunteer patients with carotid stenoses took part in the measurement protocol which was approved by the Clinical Investigation Center CIC-IT 806 CHRU of Tours.

The gold standard corresponded to visual and audible detection of microemboli by both non-hospital practitioners and hospital practitioners. The automatic thresholds were set in order to guarantee the detection of all microemboli obtained with the gold standard.

Results

The first interesting findings showed that the false alarm rate for all automatic detectors was less than 2%. The second important result, shown in Table1, demonstrated that about 25% of microemboli were not detected with the automatic standard detector compared to the gold standard. The third interesting result was that the synchronous detector counted about 30% of detections more than the gold standard and about 75% of detections more than the standard energy detector. Note that such results were obtained from 92% of the whole recording. Indeed, 8% of cardiac cycles were unusable according to our validity criterion for cardiac cycles. This means that 8% of cardiac cycles were outside the authorized range of 20% around the mean cardiac rhythm. Most of these unusable cardiac rhythms corresponded to ventricular extra-systoles. The percentage of unusable cardiac rhythms reached 37% for some patients and sometimes 1%. This means that the quasi-cyclostationarity hypothesis was validated on the whole, but prudence is required.

In order to be sure that the 30% of over-detections were true microemboli undetected by the gold standard procedure, recordings were listened to again at a playing speed reduced by a factor of two and visualized twice on the spectrogram. One spectacular case is reported in Fig. 1.

Discussion

The results summarized in Table 1 confirmed that the standard detector was not a good automatic detector since it missed 35% of microemboli detected by visualizing the spectrogram on the TCD screen and by listening to audible signatures of microemboli. However, these levels of performance were not too bad since detection was performed automatically, i.e. without operator-dependent pre-setting. The levels of performance of our new kind of detector were therefore very encouraging since it made it possible to detect inaudible microemboli. The results confirmed that the human auditory system is not perfect. However, although our new type of detector surpassed the standard detector, these good results must be confirmed on a larger dataset.

One short term prospect for this work would be to improve further the levels of performance of the automatic detectors tested. Several approaches might be explored, the first being proposing other values for the training and updating periods, and the second proposing other values of α. These two approaches seem to be relevant since their are patient-dependent.

Finally, note that no microemboli were detected with the gold standard or the standard energy detector in the unusable cardiac rhythms. However, there is no guarantee that microemboli might not sometimes appear in unusable cardiac cycles. There is currently no solution to this possibility and this will be the objective of further research.

CONCLUSION

Automatic detectors that were not operator-dependent were tested in this study. The levels of performance of the new proposed detector surpassed those of the gold standard and the standard detector. However, although the synchronous detector demonstrated the best levels of performance, it is clear that this detector based on pseudo-cyclo-stationary properties might sometimes fail for certain patients with high levels of variations in cardiac rhythm. Further quantitative study should be planned to disprove or confirmed this possibility.

Figure 1 :

 1 Figure 1: Doppler signal with two inaudible microemboli recorded in a patient with stenosis. (top) Time representation of the Doppler signal. (middle) Instantaneous energy of the Doppler signal. (bottom) Spectrogram of the Doppler signal.

Figure 2 : 1 .

 21 Figure 2: Function diagram of the new system detector without artefact rejection.

Figure 3 :

 3 Figure 3: Histograms of normalized cardiac rhythms obtained from two different recordings. a) Narrow gaussian histogram obtained for one patient. b) Wide gaussian histogram obtained for another patient.

Figure 4 :

 4 Figure 4: Microembolus signature superimposed on the Doppler energy. (a) Doppler energy signal of 12 seconds estimated from our data set. (b) After indexing and resampling of each energy signal, energy signals were superimposed on each other. Each energy signal had a normalized duration.

Table 1 :

 1 (First line) Mean number N of microemboli detected from clinical recordings of 30 minutes. (Second line) Percentage of detection P ct comparison to the Gold standard. Around 8% of cardiac cycles were unusable.


Nina Aguillon

Frédéric Lagoutière

Nicolas Seguin

Convergence of finite volumes schemes for the coupling between the inviscid Burgers equation and a particle

Keywords: Fluid-particle interaction, Burgers equation, Non-conservative coupling, moving interface, convergence of finite volume schemes, PDE-ODE coupling 2010 Mathematics Subject Classification: 35R37, 65M12, 35L72

schemes consistent with a maximal part of the germ 3.1 Convergence of the fluid's part . . . . . . . . . . .

Introduction

We study the numerical convergence of finite volume schemes for the Cauchy problem

     ∂tu + ∂x u 2 2 = -λ(u -h ′ (t))δ h(t) (x), mph ′′ (t) = λ(u(t, h(t)) -h ′ (t)), u |t=0 = u 0 , h(0) = h 0 , h ′ (0) = v 0 .
(1) It models the behavior of a pointwise particle of position h, velocity h ′ and acceleration h ′′ with mass mp, immersed into a "fluid," whose velocity at time t and point x is u(t, x). The velocity of the fluid is assumed to follow the inviscid Burgers equation. This system is fully coupled: the fluid exerts a drag force D = λ(u(t, h(t)) -h ′ (t)) on the particle, where λ is a positive friction parameter. By the action-reaction principle, the particle exerts the force -D on the particle. The interaction is local: it applies only at the point where the particle is. This friction force tends to bring the velocities of the fluid and the particle closer to each other: as λ is positive, the particle accelerates if u(t, h(t)) is larger than h ′ (t) and vice-versa. This toy model was introduced in [START_REF] Lagoutière | A simple 1D model of inviscid fluid-solid interaction[END_REF] (see also [START_REF] Borsche | On the interactions between a solid body and a compressible inviscid fluid[END_REF] and [START_REF] Aguillon | Riemann problem for a particle-fluid coupling[END_REF] for related problems). In contrast with the model studied in [START_REF] Luis | Large time behavior for a simplified 1D model of fluid-solid interaction[END_REF], [START_REF] Hillairet | Asymptotic collisions between solid particles in a Burgers-Hopf fluid[END_REF] and [START_REF] Luis | Lack of collision in a simplified 1D model for fluid-solid interaction[END_REF], the particle and the fluid do not share the same velocity and the fluid is inviscid. In particular the fluid velocity is typically discontinuous through the particle. It yields to issues to define correctly the product (u -h ′ )δ h and the ODE for the particle in system (1). To do so, the idea is to regularize the Dirac measure in (1), and to remark that the values of the fluid velocity on both sides of this thickened particle are independent of the regularization. It allows to reformulate System (1) as an interface problem, where the traces around the particle u-(t) = lim x→h(t) -u(t, x) and u+(t) = lim x→h(t) + u(t, x) must belong to a set G λ (h ′ (t)), which takes into account the interface conditions. This study was done in details in [START_REF] Lagoutière | A simple 1D model of inviscid fluid-solid interaction[END_REF]. The germ is defined as follow.

Definition 1.1. For any given speed v ∈ R, the germ at speed v, G λ (v), is the set of all (u-, u+) in R 2 such that (u-,

u+) ∈ G 1 λ ∪ G 2 λ (v) ∪ G 3 λ (v), where G 1 λ = (u-, u+) ∈ R 2 : u-= u+ + λ , G 2 λ (v) = (u-, u+) ∈ R 2 : v ≤ u-≤ v + λ, v -λ ≤ u+ ≤ v
and u--u+ < λ , and G 3 λ (v) = (u-, u+) ∈ R 2 : -λ ≤ u+ + u--2v ≤ λ and u--u+ > λ . The germ G λ (0) and its partition are depicted on Figure 1 on the left (note that the germ G λ (v) is the translation of G λ (0) by the vector (v, v)). Here, we choose a slightly different partition of the germ than in [START_REF] Andreianov | Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes[END_REF] and [START_REF] Andreianov | Well-posedness for a one-dimensional fluid-particle interaction model[END_REF], which is depicted on the right of Figure (1). The reason is that we are able to find a class of finite volume schemes which are consistent with G 1 λ ∪ G 2 λ (0) with this choice, but not with the original partition. However, the essential property that G 1 λ ∪ G 2 λ (0) is a maximal part of the germ still holds true with the partition of Definition 1.1 (more details are given in Definition 1.4 and Proposition 3.8). Once the germ has been defined, System (1) is defined as an interface problem. The

u - u - u + u + -λ -λ λ λ G 1 λ G 1 λ G 2 λ (0) G 2 λ (0) G 3 λ (0) G 3 λ (0)
Figure 1: The germ for a motionless particle and its partitions. Left: the partition used in this work. Right: the partition used in [START_REF] Andreianov | Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes[END_REF] and [START_REF] Andreianov | Well-posedness for a one-dimensional fluid-particle interaction model[END_REF].

equation on the particle is reformulated to keep the conservation of total momentum mph ′ (t) +

• the function u is an entropy weak solution of the Burgers equations on the sets {(t, x), x < h(t)} and {(t, x), x > h(t)},

• for almost every positive time t,

mph ′′ (t) = (u-(t) -u+(t)) u-(t) + u+(t) 2 -h ′ (t) (2)
and (u-(t), u+(t)) ∈ G λ (h ′ (t)).

This definition requires the existence of traces along the particle's trajectory h. It follows from the works of Panov [START_REF] Yu | Existence of strong traces for quasi-solutions of multidimensional conservation laws[END_REF] and Vasseur [START_REF] Vasseur | Strong traces for solutions of multidimensional scalar conservation laws[END_REF]. When the particle is motionless, well-posedness in the BV setting was proved in [START_REF] Andreianov | Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes[END_REF], while for the fully coupled system (1), it is proved in [START_REF] Andreianov | Small solids in an inviscid fluid[END_REF] and [START_REF] Andreianov | Well-posedness for a one-dimensional fluid-particle interaction model[END_REF].

Remark that Definition 1.2 is not suitable to prove convergence of finite volume schemes in a general framework. Indeed, a scheme can create a numerical boundary layer near the particle, of several cells width. It does not prevent the scheme from converging in, say, L ∞ loc in time and L 1 in space; but in that case we cannot expect the numerical traces to converge to their correct values. Nevertheless we will prove the convergence of some schemes that create such boundary layers. The key point is to use, instead of Definition 1.1, an equivalent definition which does not contain the traces of u. We begin with some properties useful to decide if a pair (c-, c+) belongs to the germ G λ (v). We adopted the vocabulary of the theory of conservation law with discontinuous flux function of [START_REF] Andreianov | On vanishing viscosity approximation of conservation laws with discontinuous flux[END_REF] and [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF].

In the sequel, we denote by Φv the so-called Kruzhkov entropy flux associated with fv(u) = u 2 2 -vu:

Φv : R 2 -→ R (a, b) -→ sgn(a -b) a 2 2 -va -b 2 2 -vb
and we define

Ξv : R 2 × R 2 -→ R ((a-, a+), (b-, b+)) -→ Φv(a-, b-) -Φv(a+, b+) Proposition 1.3. If both (a-, a+) and (b-, b+) belong to G λ (v), then Ξv((a-, a+), (b-, b+)) ≥ 0. Conversely, if (a-, a+) is such that ∀(b-, b+) ∈ G λ (v), Ξv((a-, a+), (b-, b+)) ≥ 0,
then (a-, a+) belongs to the germ.

Definition 1.4. A subset H λ (v) of G λ (v) is said to be maximal if any (a-, a+) that satisfies ∀(b-, b+) ∈ H λ (v), Ξv((a-, a+), (b-, b+)) ≥ 0 (3)
belongs to the germ G λ (v).

We will prove in Proposition 3.8 that G 1 λ ∪ G 2 λ (v) is maximal. In the sequel H λ (v) always denotes a maximal part of G λ (v). We now focus on alternative traceless characterizations of entropy solutions. For all (c-, c+) we denote by c the piecewise constant function

c(t, x) = c-1 x<h(t) + c+1 x≥h(t) ,
and by dist1(a, X) the L 1 -distance of a point a := (a-,

a+) of R 2 to a set X included in R 2 : dist1((a-, a+), X) = inf (x -,x + )∈X |a--x-| + |a+ -x+|.
Proposition 1.5. Let h be a function of W 2,∞ loc (R+) and let u be a function of L ∞ loc (R+ × R), which is an entropy solution of the Burgers equation on the sets {(t, x), x < h(t)} and {(t, x), x > h(t)}. The following assertions are equivalent.

• For almost every time t > 0, (u-(t), u+(t)) belongs to G λ (h ′ (t)).

• For almost every time t > 0, for all (c-, c+) ∈ R 2 , there exist δ ∈ (0, t) and a constant A depending only on ||u 0 ||∞, λ, (c-, c+) and ||h ′ ||∞ such that for every nonnegative function

ϕ in C ∞ 0 ((t -δ, t + δ) × R), R + R |u -c|(s, x)∂tϕ(s, x -h(s)) + Φ h ′ (t) (u, c)(s, x)∂xϕ(s, x -h(s))dx ds ≥ -A R + dist1((c-, c+), H λ (h ′ (s)))ϕ(s, 0) ds. (4)
Proof. For the sake of completeness we reproduce here the main ingredients of the proof that can be found in [START_REF] Andreianov | Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes[END_REF]. Let ϕ be in C ∞ 0 ((t -δ, t + δ) × R), where δ belongs to (0, t). For positive ε, we introduce the function

ζε(z) = 1 -min(1, |z|/ε),
whose support is (-ε, ε). The support of the function

ψε(t, x) = (1 -ζε)ϕ(t, x -h(t))
is included in {(t, x), t > 0, x = h(t)}. The function u is a entropy solution of the Burgers equation on the sets {(t, x), x < h(t)} and {(t, x), x > h(t)}, thus for all real κ,

R + ×R |u(s, x) -κ|∂sψε(s, x) + Φ0(u(s, x), κ)∂xψεdx ds ≥ 0. But ∂sψε(s, x) = ∂s((1 -ζε)ϕ)(s, x -h(s)) -h ′ (s)∂x((1 -ζε)ϕ)(s, x -h(s)
), and we using the fact that

Φv(a, b) = Φ0(a, b) -v|a -b|,
we obtain

R + ×R |u -c|(s, x)(∂s(1 -ζε)ϕ)(s, x -h(s)) + Φ0(u, c)(s, x)∂x((1 -ζε)ϕ)(s, x -h(s))dx ds ≥ 0.
Thus we have

R + ×R |u -c|(s, x)∂tϕ(s, x -h(s)) + Φ h ′ (t) (u, c)(s, x)∂xϕ(s, x -h(s))dx ds ≥ lim inf ε→0 R + ×R |u -c|(s, x)(∂t(ζεϕ))(s, x -h(s)) + Φ h ′ (t) (u, c)(s, x)(∂x(ζεϕ))(s, x -h(s))dx ds = R + Φ h ′ (s) (u-(s), c-) -Φ h ′ (s) (u+(s), c+)ϕ(s, 0)ds = R + Ξ h ′ (s) ((u-(s), u+(s)), (c-, c+))ϕ(s, 0)ds
For all s for which the pair (u-(s), u+(s)) exists and belongs to G λ (h ′ (s)), we denote by (c-(s), c+(s)) a L 1 -projection of (c-, c+) on H λ (h ′ (s)). We have

Ξ h ′ (s) ((u-(s), u+(s)), (c-(s), c+(s))) ≥ Ξ h ′ (s) ((u-(s), u+(s)), (c-(s), c+(s))) -|Ξ h ′ (s) ((u-(s), u+(s)), (c-(s), c+(s))) -Ξ h ′ (s) ((u-(s), u+(s)), (c-(s), c+(s)))|.
Since (c-(s), c+(s)) belongs to H λ (h ′ (s)), Proposition 1.3 yields

R + Ξ h ′ (s) ((u-(s), u+(s)), (c-(s), c+(s)))ϕ(s, 0)ds ≥ 0.
On the other hand

|Ξ h ′ (s) ((u-(s), u+(s)), (c-(s), c+(s))) -Ξ h ′ (s) ((u-(s), u+(s)), (c-(s), c+(s)))| ≤ |Φ h ′ (s) (u-(s), c-(s)) -Φ h ′ (s) (u-(s), c-(s))| + |Φ h ′ (s) (u+(s), c+(s)) -Φ h ′ (s) (u+(s), c+(s))|
which is smaller than a constant depending only on ||h ′ ||∞, ||u||∞, c and λ (since c → c depends on λ), multiplied by the L 1 -distance between (c-, c+) and (c-(s), c+(s)), and we obtain the result. Conversely, using a sequence of test functions ϕ concentrating at a time t for which u has traces in Proposition 4, we obtain that for all (c-,

c+) in H λ (h ′ (t)), Ξ h ′ (t) ((u-(t), u+(t)), (c-, c+)) ≥ 0,
and thus by Proposition 1.3, (u-(t), u+(t)) belongs to the germ G λ (h ′ (t)).

Proposition 1.6. Let u in L ∞ loc (R+×R) be a solution of the Burgers equation on the sets {(t, x), x < h(t)} and {(t, x), x > h(t)}. Consider a function h in W 2,∞ loc (R+) which verifies (2) with initial data h(0) = h 0 and h ′ (0) = v 0 almost everywhere if and only if for all ξ ∈ C ∞ 0 ([0, T )) and for all ψ ∈ C ∞ 0 (R) such that ψ(0) = 1

- T 0 mph ′ (t)ξ ′ (t)dt = mpv 0 ξ(0) + R T 0 u 2 2 (s, x)ξ(s)ψ ′ (x -h(s))ds dx + R T 0 u(s, x)[ξ ′ (s) -h ′ (s)ψ ′ (x -h(s))]ds dx + R u 0 (x)ψ(x -h(0))ξ(0)dx. (5)
Proof. This characterization were proved in [START_REF] Andreianov | Small solids in an inviscid fluid[END_REF]. It follows from the application of the Green-Gauss theorem and the fact that u is an entropy solution of the Burgers equation away from the particle:

T 0 R u 2 2 (s, x)ξ(s)ψ ′ (x -h(s)) + u(s, x)[ξ ′ (s) -h ′ (s)ψ ′ (x -h(s))]ds dx = T 0 R u 2 2 (s, x)∂x(ξψ(x -h(s))) + u(s, x)∂s(ξψ(x -h(s)))ds dx = - T 0 x =h(t) ∂x u 2 2 + ∂tu (ξψ)dx ds - R u 0 (x)ψ(x -h(0))ξ(0)dx + T 0 ξ(s) u 2 -(s) 2 -h ′ (s)u-(s) - u 2 + (s) 2 -h ′ (s)u+(s) ds = T 0 mph ′′ (s)ξ(s)ds - R u 0 (x)ψ(x -h(0))ξ(0)dx.
We now present the family of finite volume schemes for which we prove convergence. The proof follows the guidelines of the Lax-Wendroff theorem. In Section 2, we obtain a BV bound on the fluid velocity and a W 2,∞ bound on the particle's trajectory that allows to extract convergent subsequences in L 1 loc (R+ × R) and W 1,∞ loc (R+). The difficulties are to treat numerically the interface conditions enclosed in the germ and the coupling between an ODE and a PDE. More precisely:

• First, we have to take into account at the numerical level the interface condition of Definition 1.1.

We will use schemes that preserves a "sufficiently large" part of the germ.

• Second, to deal with a moving particle. It is crucial that the particle lies at an interface of the mesh at the beginning of the time step. To do so and avoid the problem of the replacement of the particle, we use a mesh that tracks the particle and we update the particle's velocity by conservation of total impulsion.

Let us fix a time step ∆t and a space step ∆x. In the sequel we suppose that the time step and the space step are proportional, and we denote by µ = ∆t ∆x their ratio. We propose to approximate the solution of (1) with a finite volume scheme. We use a mesh that follows the particle, which is placed between the cells numbered 0 and 1. The speed of the particle is approximated by a piecewise constant (v n ) n∈N . Given the solution a time n∆t: we consider that the particle has constant velocity v n on the whole time step (n∆t, (n + 1)∆t) to update the fluid velocity, then we update v n by conservation of the total impulsion. The interface 1/2 where the particle lies is special, and we have to use appropriate fluxes at this interface. Due to the source term, the equation is not conservative around the particle, thus we have two different fluxes f n,- 1/2 and f n,+ 1/2 on the left and on the right of the particle respectively. Away from the particle, Equation (1) writes as a scalar conservation law, and we can use any standard flux for the Burgers equation. The scheme is initialized with

∀j ∈ Z, u 0 j = 1 ∆x x 0 j+1/2 x 0 j-1/2 u 0 (x) dx.
From the integration of the first equation of (1) on the space time cell

C n j = {(n∆t + s, x n j-1/2 + y + sv n ), 0 ≤ s < ∆t, 0 ≤ y < ∆x},
we obtain the finite volume scheme

               u n+1 j = u n j -µ(f n j+1/2 (v n ) -f n j-1/2 (v n )) for j ∈ Z, j / ∈ {0, 1}, u n+1 0 = u n 0 -µ(f n 1/2,-(v n ) -f n -1/2 (v n )), u n+1 1 = u n 1 -µ(f n 3/2 (v n ) -f n 1/2,+ (v n )), v n+1 = v n + ∆t mp (f n 1/2,-(v n ) -f n 1/2,+ (v n )), x n+1 j = x n j + v n ∆t. (6)
Here we emphasized the dependency of the flux on the particle's velocity. In the sequel we denote by u∆t the constant by cell function

u∆t(t, x) = u n j if (t, x) ∈ C n j . (7)
and by v∆t and h∆t the constant and linear by cell functions:

v∆t(t) = v n if n∆t ≤ t < (n + 1)∆t, h∆t(t) = h 0 + ∆t n-1 m=0 v m + v n (t -n∆t) if n∆t ≤ t < (n + 1)∆t. (8)
Another way to proceed is to performed the change of variable

ũ(t, x) = u(t, x + h(t))
in (1). This function verifies the PDE

∂t ũ + ∂x ũ2 2 -h ′ (t)ũ = -λ(ũ -h ′ )δ0(x) (9)
The particle is now motionless but the flux depends on time. We denote by fv(u) = u 2 2 -vu. Integrating (9) on [n∆t, (n + 1)∆t] × [x 0 j-1/2 , x 0 j+1/2 ], and using special flux around the particle (still placed at interface 1/2), we obtain the finite volume scheme

           ũn+1 j = ũn j -µ(f v n ,n j+1/2 -f v n ,n j-1/2 ) for j ∈ Z \ {0, 1}, ũn+1 0 = ũn 0 -µ(f v n ,n- 1/2 -f v n ,n -1/2 ), ũn+1 1 = ũn 1 -µ(f v n ,n 3/2 -f v n ,n+ 1/2
),

v n+1 = v n + ∆t mp (f v n ,n- 1/2 -f v n ,n+ 1/2
).

(10)

The two points of view are illustrated on Figure 2.

∂ t u + ∂ x u 2 2 = -λ(u -h ′ (t))δ h(t) (x) ∂ t ũ + ∂ x ũ2 2 -h ′ (t)ũ = -λ(ũ -h ′ )δ 0 (x) t 1 t 2 v n x 0 1/2 x 0 1/2 x 1 1/2
Figure 2: To approximate the solution of (1), we can either use a mesh that follows the particle (on the left) or straighten the particle's trajectory and approximate the solution of (9). In both case, the particle's trajectory is the bold line.

The fluxes

f n j-1/2 (v n ) with j = 1/2 (or f n± 1/2 (v n ) if j = 1/2) are strongly related to the fluxes f v n ,n j-1/2 : in (6), f n j+1/2 (v n ) is an approximation of 1 ∆t (n+1)∆t n∆t (f 0 (u) -v n u)(t, x n j+1/2 + v n t)dt while in (10), f v n ,n -1/2 is an approximation of 1 ∆t (n+1)∆t n∆t f v n (ũ)(t, x 0 j+1/2 )dt
In the following we prove the convergence of Scheme (6) under a set of assumptions on the fluxes f n j+1/2 , f n 1/2,-and f n j+1/2,+ and a Courant-Friedrichs-Lewy condition. We restrict the study to two-points fluxes

f n j+1/2 = g(u n j , u n j+1 , v n ) and f n 1/2,± = g ± λ (u n j , u n j+1 , v n ).
The assumptions on the flux f n j+1/2 away from the particle are the classical ones: • consistency with the modified Burgers equation:

∀a ∈ R, ∀v ∈ R, g(a, a, v) = a 2 2 -va, (11)
• monotonicity with respect to the first two arguments:

∀(a, b) ∈ R 2 , ∀v ∈ R, ∂1g(a, b, v) ≥ 0 and ∂2g(a, b, v) ≤ 0. ( 12
)
• g is locally Lipschitz-continuous; (13) they ensure convergence of the scheme to an entropy solution of the Burgers equation away from the particle.

The assumptions on the fluxes around the particle are the following. We first have some consistency assumptions, which ensure that some particular solutions corresponding to a large enough part of the germ are exactly preserved by the numerical scheme. We do not ask the flux to preserve the whole germ though, but only, in Section 3 with a maximal part of the germ, and in Section 4, with G 1 λ . More precisely, the hypothesis on the fluxes g ± λ are: • consistency the part G 1 λ of the germ:

∀v ∈ R, ∀(a, b) ∈ G 1 λ , g - λ (a, b, v) = a 2 2 -va and g + λ (a, b, v) = b 2 2 -vb. (14)
In Section 4, we make the stronger assumption that g is consistent with a maximal subset

H λ of G λ (see Definition 1.4) ∀v ∈ R, ∀(a, b) ∈ H λ (v), g - λ (a, b, v) = a 2 2 -va and g + λ (a, b, v) = b 2 2 -vb. (15)
Hypothesis ( 14) will be used to prove BV estimates on the fluid part (u n j ) j∈Z,n∈N . We also assume that • if the particle has the same velocity than the fluid, its velocity does not change:

∀v ∈ R, g - λ (v, v, v) = g + λ (v, v, v). ( 16
)
This hypothesis will be used to prove a L ∞ bound on the particle velocity (v n ) n∈N . We add two classical conditions of regularity and monotonicity, also used to prove the BV bound on (u n j ) j∈Z,n∈N . We assume that:

• both g - λ and g + λ are locally Lipschitz-continuous; (17) • g - λ and g + λ are nondecreasing with respect to their first arguments, and nonincreasing with respect to their second arguments. (18) Just like in [START_REF] Andreianov | Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes[END_REF], we need a dissipativity property to prove discrete entropy inequalities. Moreover, it will also be a key assumption to prove the bounds on the particle's velocity.

• The function g - λ -g + λ is nondecreasing with respect to its first two arguments. (19) For this family of finite volume schemes, we are able to prove the following convergence theorem.

Theorem 1.7. Consider a finite volume scheme of the form (6) that satisfies the set of hypothesis (11-14) and (16-19), and (15) in Section 4. Suppose that u 0 belongs to BV (R) ∩ L 1 (R). Let us denote by L the largest Lipschitz constant of g, g + and g -on the set

[m, M ] 2 × [v, v], where          m = min{ess inf R -u 0 -λ, ess inf R + u 0 }, M = max{ess sup R -u 0 , ess sup R + u 0 + λ}, v = min(m, v 0 ), v = max(M, v 0 ).

Then, under the Courant-Friedrichs-Lewy condition

Lµ ≤ 1 2 , ( 20
)
the sequence (u∆t) converges in L 1 loc (R+ × R) toward u and the sequence (h∆t) converges in W 1,∞ loc (R+) toward h when ∆t tends to 0, where (h, u) is the solution of (1).

The next three Sections are devoted to the proof. In Section 2, we prove bounds on the total variation of the fluid and on the acceleration of the particle, which permit us to extract converging subsequences. Then in Section 3, we prove Theorem 1.7 under Hypothesis (15), which is sufficient to obtain a discrete version of (4). In Section 4, we drop hypothesis (15) and prove the convergence of the family of schemes such that

g - λ (a, b, v) = g(a, b + λ, v), g + λ (a, b, v) = g(a -λ, b, v)
, where g satisfies assuptions (11-13). This type of schemes was introduced in [START_REF] Andreianov | Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes[END_REF]. They only preserve the part G 1 λ of the germ, in the sense that if

(a, b) belongs to G 1 λ , then g - λ (a, b, v) = f v (a) and g - λ (a, b, v) = f v (b). We recall that G 1 λ is not a maximal subset of G λ (v).
Under the set of assumptions specified above (except (15)) we extend the proof of convergence of [START_REF] Andreianov | Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes[END_REF] to the fully coupled case (1).

A priori bounds

In the sequel we suppose that u 0 belongs to L 1 (R) ∩ BV (R), that Hypothesis (11), ( 12) and (13) on the flux g are fulfilled, and that the monotonicity and regularity assumptions ( 18) and (17) on g ± are verified. We will specify the consistency hypothesis on g ± along the way. We first consider the uncoupled problem where (v n ) n∈N is fixed.

Proposition 2.1. Let u 0 be in BV (R) ∩ L 1 (R). Let (v n ) n∈N be given and v and v in R such that ∀n ∈ N, v ≤ v n ≤ v.
Consider the finite volume scheme

     u n+1 j = u n j -µ(g(u n j , u n j+1 , v n ) -g(u n j-1 , u n j , v n )) for j ∈ Z \ {0, 1}, u n+1 0 = u n 0 -µ(g - λ (u n 0 , u n 1 , v n ) -g(u n -1 , u n 0 , v n )), u n+1 1 = u n 1 -µ(g(u n 1 , u n 2 , v n ) -g + λ (u n 0 , u n 1 , v n )).
Suppose that the fluxes g ± verify (14) and that the CFL condition (20) holds. Then we have the following L ∞ and BV estimates in space on u∆t, with m and M the constants of Theorem 1.7:

∀n ≥ 0, ∀j ∈ Z, m ≤ u n+1 j ≤ M (21) and ∀n ∈ N, j∈Z |u n j -u n j-1 | ≤ j∈Z |u 0 j -u 0 j-1 | + 2λ. ( 22
)
Proof. Due to the presence of the particle, the maximum and the total variation of the exact solution u of (1) can increase through time. For example if u 0 is constant equals to 0 and if [START_REF] Lagoutière | A simple 1D model of inviscid fluid-solid interaction[END_REF], Lemma 5.7). This prevents us for applying the LeRoux and Harten lemma (see [START_REF] Harten | On a class of high resolution total-variation-stable finite-difference schemes[END_REF] and [START_REF] Leroux | A numerical conception of entropy for quasi-linear equations[END_REF]) directly to (u n j ) j∈Z, n∈N . Yet it can be applied to the sequence (w n j ) j∈Z, n∈N defined by

v 0 > λ, then ||u(0 + , •)|| L ∞ (R) = ||u 0 || L ∞ (R) + λ and ||u(0 + , •)|| BV (R) = ||u 0 || BV (R) + 2λ (see
w n j = u n j -λ 2 if j ≤ 0, u n j + λ 2 if j ≥ 1.
Let us prove that there exists two families of real (C n j+1/2 ) j∈Z,n∈N and (D n j+1/2 ) j∈Z,n∈N such that for all j in Z, for all n in N,

w n+1 j = w n j + C n j+1/2 (w n j+1 -w n j ) -D n j-1/2 (w n j -w n j-1 ), (23)
and

0 ≤ 1 -C n j+1/2 -D n j+1/2 ≤ 1, 0 ≤ C n j+1/2 ≤ 1 and 0 ≤ D n j+1/2 ≤ 1.
In other words, w n+1 j writes as a convex combination of w n j-1 , w n j and w n j+1 and therefore,

∀n ≥ 0, min k w n k ≤ w n+1 j ≤ max k w n k .
As a consequence, for all n ∈ N and for j ≤ 0,

min k w 0 k + λ/2 ≤ u n j ≤ max k w 0 k + λ/2 which rewrites min{• • • , u 0 0 , u 0 1 + λ, • • • } ≤ u n j ≤ max{• • • , u 0 0 , u 0 1 + λ, • • • }. Similarly, for all n ∈ N and for all j ≥ 1, min{• • • , u 0 0 -λ, u 0 1 , • • • } ≤ u n j ≤ max{• • • , u 0 0 -λ, u 0 1 , • • • },
hence the L ∞ bound (21) is proven. Moreover, the LeRoux and Harten lemma yields

∀n ∈ N, j∈Z |w n+1 j -w n+1 j-1 | ≤ j∈Z |w n j -w n j-1 |,
and thus (22).

Let us go back to the existence of C n j+1/2 and D n j-1/2 . In the sequel we denote by |a, b| the interval [min(a, b), max(a, b)]. Suppose first that (23) holds for some n ∈ N. Then for every j ≤ -1, there exists

wn j-1/2 ∈ |w n j-1 , w n j | and wn j+1/2 ∈ |w n j , w n j+1 | w n+1 j = w n j -µ g(u n j , u n j+1 , v n ) -g(u n j-1 , u n j , v n ) = w n j -µ g λ w n j + λ 2 , w n j+1 + λ 2 , v n -g λ w n j-1 + λ 2 , w n j + λ 2 , v n = w n j -µ ∂1g λ wn j-1/2 + λ 2 , w n j+1 + λ 2 , v n (w n j -w n j-1 ) +∂2g λ w n j-1 + λ 2 , wn j+1/2 + λ 2 , v n (wj+1 -wj ) Both triplets wn j-1/2 + λ 2 , w n j+1 + λ 2 , v n and w n j-1 + λ 2 , wn j+1/2 + λ 2 , v n belong to [m, M ] 2 × [v, v].
The CFL condition (20), and the fact that ∂1g ≥ 0 and ∂2g ≤ 0, yield (23) with

D n j-1/2 = µ∂1g λ wn j-1/2 + λ 2 , w n j+1 + λ 2 , v n , C n j+1/2 = -µ∂2g λ w n j-1 + λ 2 , wn j+1/2 + λ 2 , v n .
The case j ≥ 2 can be treated in the exact same way. We now turn to the trickier case j = 0. The facts that g - λ is consistent with G 1 λ and that g is consistent (Hypothesis ( 14) and ( 11)) imply that

g - λ w n 0 + λ 2 , w n 0 - λ 2 , v n = g λ w n 0 + λ 2 , w n 0 + λ 2 , v n ,
which allows us to write

w n+1 0 = w n 0 -µ g - λ (u n 0 , u n 1 , v n ) -g(u n -1 , u n 0 , v n ) = w n 0 -µ g - λ w n 0 + λ 2 , w n 1 - λ 2 , v n -g λ w n -1 + λ 2 , w n 0 + λ 2 , v n = w n 0 -µ g - λ w n 0 + λ 2 , w n 1 - λ 2 , v n -g - λ w n 0 + λ 2 , w n 0 - λ 2 , v n +g λ w n 0 + λ 2 , w n 0 + λ 2 , v n -g λ w n -1 + λ 2 , w n 0 + λ 2 , v n
Thus, there exists wn

-1/2 ∈ |w n -1 , w n 0 | and wn 1/2 ∈ |w n 0 , w n 1 | such that w n+1 0 = w n 0 -µ ∂2g - λ w n 0 + λ 2 , wn 1/2 - λ 2 , v n (w n 1 -w n 0 ) +∂1g λ wn -1/2 + λ 2 , w n 0 + λ 2 , v n (w n 0 -w n -1 )
Once again, both triplets

w n 0 + λ 2 , wn 1/2 -λ 2 , v n and w n 0 + λ 2 , wn -1/2 + λ 2 , v n belong to [m, M ] 2 × [v, v].
The monotonicity on g and g - λ allow to conclude with

D n -1/2 = µ∂1g λ wn -1/2 + λ 2 , w n 0 + λ 2 , v n , C n 1/2 = -µ∂2g - λ w n 0 + λ 2 , wn 1/2 -λ 2 , v n .
The case j = 1 can be treated in the exact same way, using the consistency assumption

g + λ w n 1 + λ 2 , w n 1 - λ 2 , v n = g λ w n 1 - λ 2 , w n 1 - λ 2 , v n .
We now turn to the case where the particle's velocity is updated from time to time, and focus on the estimates on the velocity and acceleration of the particle.

Proposition 2.2. Suppose that the fluxes g ± verify (16), ( 19) and (20) that the time step verifies

4L mp ∆t ≤ 1, (24)
Then, the sequence (u n j ) j∈Z,n∈N (defined by (6)) verifies Estimates (21) and (22), while (v n ) n∈N verifies the following estimates:

∀n ∈ N, v ≤ v n ≤ v, (25)
and

∀n ∈ N, v n+1 -v n ∆t ≤ 2L mp (||u 0 ||∞ + λ + ||v||∞). ( 26
)
The constants v and v are defined in Theorem 1.7.

Proof. We proceed by induction. Let us first remark that if the estimate (25) on v n is fulfilled at time t n , the proof of Proposition 2.1 yields the L ∞ and BV estimates on (u n+1 j ) j∈Z . Therefore, we focus on the estimate on v n+1 . Using Hypothesis (16), we introduce the null quantity g

- λ (v n , v n , v n ) -g + λ (v n , v n , v n ) and write v n+1 = v n + ∆t mp (g - λ (u n 0 , u n 1 , v n ) -g + λ (u n 0 , u n 1 , v n )) = v n + ∆t mp 1 0 ∂s(g - λ (v n + s(u n 0 -v n ), v n + s(u n 1 -v n ), v n ))ds - 1 0 ∂s(g + λ (v n + s(u n 0 -v n ), v n + s(u n 1 -v n ), v n ))ds ,
and we obtain

v n+1 = v n + ∆t mp 1 0 (u n 0 -v n )∂1(g --g + )(v n + s(u n 0 -v n ), v n + s(u n 1 -v n ), v n )ds + 1 0 (u n 1 -v n )∂2(g --g + )(v n + s(u n 0 -v n ), v n + s(u n 1 -v n ), v n )ds (27)
Suppose now that v n ≤ min(u n 0 , u n 1 ). Then both (u n 1 -v n ) and (u n 0 -v n ) are nonnegative. Moreover, the dissipativity assumption (19) implies that ∂1(g --g + ) and ∂2(g --g + ) are also nonnegative. Hence we have v n+1 ≥ v n and Hypothesis (24) yields

v n+1 ≤ v n + 2L ∆t mp (u n 0 -v n + u n 1 -v n ) ≤ 1 - 4L∆t mp v n + 4L∆t mp max(u n 0 , u n 1 ) ≤ v.
We now treat the case

u n 0 ≤ v n ≤ u n 1 . The only difference is that u n 0 -v n is now negative. The integral form (27) of v n+1 and Hypothesis (25) yield v ≤ v n -2L ∆t mp (v n -u n 0 ) ≤ v n+1 ≤ v n + 2L ∆t mp (u n 1 -v n ) ≤ v.
Once the L ∞ bounds on (u n j ) j∈Z,n∈N and (v n ) n∈N are proven, the bound of the particle's acceleration (26) is an easy consequence of the integral form of v n+1 .

Remark 2.3. Condition (24) is fulfilled for small enough ∆t. Thus it is not a restriction to prove the convergence of the scheme. However from the numerical point of view, one has to check Condition (24) in addition to the CFL condition (20). This restriction is severe if the particle is very light. It is possible, at the cost of solving a nonlinear system, to use an implicit version of Scheme (6) for the particle's velocity, i.e.

               u n+1 j = u n j -µ(f n j+1/2 (v n+1 ) -f n j-1/2 (v n+1 )) for j ∈ Z, j / ∈ {0, 1}, u n+1 0 = u n 0 -µ(f n 1/2,-(v n+1 ) -f n -1/2 (v n+1 )), u n+1 1 = u n 1 -µ(f n 3/2 (v n+1 ) -f n 1/2,+ (v n+1 )), v n+1 = v n + ∆t mp (f n 1/2,-(v n+1 ) -f n 1/2,+ (v n+1 )), x n+1 j = x n j + v n ∆t.
In that case, we obtain Bounds (25) and (26) without Constraint (24) on the time step. The proof is exactly the same than the one of Proposition 2.2. For example in the case where v n+1 ≤ min(u n 0 , u n 1 ), we obtain

v n+1 ≤ v n + 2L ∆t mp (u n 0 -v n+1 + u n 1 -v n+1 ),
and thus, without any constraint on ∆t other than (20),

v n+1 ≤ v n + 2L ∆t mp (u n 0 + u n 1 ) 1 + 4L ∆t mp ≤ v. △
We are now in position to extract converging subsequences of (u∆) and (h∆) (defined in (7) and ( 8)). In Section 3, we will prove that their limits are solutions of the Cauchy problem (1) for the fully coupled problem.

Proposition 2.4. Assume that u 0 belongs to BV (R) ∩ L 1 (R), and that Hypothesis (11-14) and (16-19) are verified. Moreover, suppose that the CFL condition (20) holds. Then there exists u in BV loc (R+ × R) and h in W 2,∞ loc (R+) such that, up to a subsequence, the sequence (u∆t) converges in L 1 loc (R+ × R) toward u and the sequence (h∆t) converges in W 1,∞ loc (R) toward h as ∆t tends to 0.

Proof. Let us first fix a time T > 0 and a constant A > 0 and prove the convergence in L 1 ([0, T ] × [-A, A]) and W 1, ∞ ([0, T ]). By Proposition 2.1, we can use Helly's theorem to prove the convergence in

L 1 ([0, T ] × [-A, A]) of (u∆t), toward a function u in BV (([0, T ] × [-A, A])).
Similarly Proposition 2.2 allows us to apply Arzelà-Ascoli's theorem to prove convergence in W 1,∞ ([0, T ]) of (h∆t) to a function h belonging to W 2,∞ ([0, T ]). The result is extended to the whole time-space R+ × R thanks to the Cantor diagonal extraction argument.

Remark 2.5. Up to the same subsequence, (v∆t) converges toward h ′ in L 1 loc . Moreover the sequence of functions (c∆t) defined by

c∆t(t, x) = c-if t < h∆x(t), c+ if t > h∆x(t), converges in L 1 loc toward c(t, x) = c-if t < h(t), c+ if t > h(t).

Indeed, we have

A -A T 0 |c∆t(t, x) -c(t, x)|dtdx ≤ |c+ -c-| T 0 |h∆t(t) -h(t)|dt ≤ 2LT ∆t.
△ 3 Convergence of schemes consistent with a maximal part of the germ

For now on, we suppose that all the hypotheses of Proposition 2.4 are fulfilled, and that both Conditions (20) and ( 24) are verified. The aim of this section is to prove Theorem 1.7. To that purpose, we prove that under Condition (15), which states that the fluxes g ± λ around the particle are consistent with a maximal subset H λ of the germ (see Definition 1.4), the limit (u, h) of the scheme is the solution of (1).

The fact that the Cauchy problem (1) is well posed in BV (R) is proven in [START_REF] Andreianov | Well-posedness for a one-dimensional fluid-particle interaction model[END_REF]. Once we know that Scheme (6) converges toward a solution of (1), the uniqueness of the solution yields that the whole sequence (u∆t, h∆t) converges. Theorem 1.7 gives a different way to prove the existence of a solution (but not the uniqueness).

Convergence of the fluid's part

The aim of this subsection is to prove that the limit u of (u∆t) verifies (4). We prove in Proposition 3.2 that (u n j ) j∈Z,n∈N verifies a discrete version of (4). In the sequel, for all reals number a and b we denote by a⊤b = max(a, b) and by a⊥b = min(a, b).

In the following proposition, we establish a discrete entropy inequality.

Proposition 3.1. Assume that Hypothesis (11-19) hold (included (15)) and that the CFL condition (20) is fulfilled. Then for all (c-, c+) in R 2 , there exists a constant A, depending only on λ, ||u 0 ||∞, ||v||∞ and (c-, c+), such that for all j ∈ Z, for all n ∈ N, the following inequality holds:

|u n+1 j -cj | -|u n j -cj | ∆t + G n j+1/2,--G n j-1/2,+ ∆x ≤ εj A ∆x dist1((c-, c+), H λ (v n )), (28)
where ∀j = 0, G n j+1/2,-= G n j+1/2,+ = G n j+1/2 , with G n j+1/2 = g(u n j ⊤cj, u n j+1 ⊤cj+1, v n ) -g(u n j ⊥cj, u n j+1 ⊥cj+1, v n ), G n 1/2,± = g ± λ (u n 0 ⊤c0, u n 1 ⊤c1, v n ) -g ± λ (u n 0 ⊥c0, u n 1 ⊥c1, v n ), cj = c-if j ≤ 0, c+ if j ≥ 1, and εj = 1 if j ∈ {0, 1}, 0 otherwise.
Proof. We follow the guidelines of proofs of classical entropy inequalities. They rely on the identity

|u n+1 j -cj | = u n+1 j ⊤cj -u n+1 j ⊥cj.
For j ∈ Z \ {0, 1}, we use the condensed notation u n+1 j = H(u n j-1 , u n j , u n j+1 , v n ). Hypothesis (12) on the monotonicity of the fluxes and the CFL condition (20) ensure that for every v, H is increasing with respect to its first three arguments. Moreover if j ∈ Z \ {0, 1}, cj-1 = cj = cj+1 and we use the consistency of the flux away from the particle (11) to write cj = H(cj-1, cj , cj+1, v n ). It follows that

u n+1 j ⊤cj = H(u n j-1 , u n j , u n j+1 , v n )⊤H(cj-1, cj , cj+1, v n ) ≤ H(u n j-1 ⊤cj-1, u n j ⊤cj , u n j+1 ⊤cj+1, v n ) u n+1 j ⊥cj = H(u n j-1 , u n j , u n j+1 , v n )⊥H(cj-1, cj , cj+1, v n ) ≥ H(u n j-1 ⊥cj-1, u n j ⊥cj , u n j+1 ⊥cj+1, v n )
and that

|u n+1 j -cj| ≤ H(u n j-1 ⊤cj-1, u n j ⊤cj , u n j+1 ⊤cj+1, v n ) -H(u n j-1 ⊥cj-1, u n j ⊥cj, u n j+1 ⊥cj+1, v n ) ≤ u n j ⊤cj -u n j ⊥cj -µ(G n j+1/2 -G n j-1/2 ) ≤ |u n j -cj | -µ(G n j+1/2 -G n j-1/2 )
Let us now focus on the more complicated case j = 0 (the case j = 1 can be treated in the exact same way). We denote by (c n 0 , cn 1 ) a projection of (c-, c+) = (c n 0 , c n 1 ) on H λ (v n ) for the L 1 -norm, and by (c n j ) j∈Z,n∈N and ( Gn j+1/2 ) j∈Z,n∈N the analogues of (cj ) j∈Z and (G n j+1/2 ) j∈Z,n∈N constructed with c:

∀j = 0, Gn j+1/2,-= Gn j+1/2,+ = Gn j+1/2 = g(u n j ⊤cj , u n j+1 ⊤cj+1, v n ) -g(u n j ⊥cj , u n j+1 ⊥cj+1, v n ), Gn 1/2,± = g ± λ (u n 0 ⊤c0, u n 1 ⊤c1, v n ) -g ± λ (u n 0 ⊥c0, u n 1 ⊥c1, v n ). Let us first remark that |u n+1 0 -c0| -|u n 0 -c0| ≤ |u n+1 0 -cn 0 | + |c n 0 -c0| -|u n 0 -cn -| -|c n 0 -c0| ≤ |u n+1 0 -cn 0 | -|u n 0 -cn -| + 2|c n 0 -c0|.
Thus we have

|u n+1 0 -c0| -|u n 0 -c0| ∆t + G n 1/2,--G n -1/2 ∆x ≤ |u n+1 0 -cn 0 | -|u n 0 -cn 0 | ∆t + G n 1/2,--G n -1/2 ∆x + 2 ∆t dist1((c-, c+), H λ (v n )) ≤ G n 1/2,--G n -1/2 ∆x - Gn 1/2,--Gn -1/2 ∆x + 2 ∆t dist1((c-, c+), H λ (v n )).
Indeed, as (c n 0 , cn 1 ) belongs to H λ (v n ), Hypothesis (15) yields that cn 0 = H λ (c n -1 , cn 0 , cn 1 , v n ), and we obtain as before

|u n+1 0 -cj | ≤ |u n j -cj | -µ( Gn j+1/2 -Gn j-1/2
). We now attempt to bound

G n 1/2,--Gn 1/2,-= g - λ (u n 0 ⊤c0, u n 1 ⊤c1, v n ) -g - λ (u n 0 ⊥c0, u n 1 ⊥c1, v n ) -g - λ (u n 0 ⊤c n 0 , u n 1 ⊤c n 1 , v n ) + g - λ (u n 0 ⊥c n 0 , u n 1 ⊥c n 1 , v n ).
As (v n ) n∈Z is bounded (Proposition 2.2), the maximum and minimum over n of cn ± is a bounded function of (c-, c+) and ||v||∞. Thus the set

[min(m, c-, c+, cn -, cn + ), max(M, c-, c+, cn -, cn + )] 2 × [v, v].
is compact. Therefore, with Lc the Lipschitz constant of g - λ over this set, we have

|g - λ (u n 0 ⊤c0, u n 1 ⊤c1, v n ) -g - λ (u n 0 ⊤c n 0 , u n 1 ⊤c n 1 , v n )| ≤ |g - λ (u n 0 ⊤c0, u n 1 ⊤c1, v n ) -g - λ (u n 0 ⊤c n 0 , u n 1 ⊤c1, v n )| + |g - λ (u n 0 ⊤c n 0 , u n 1 ⊤c1, v n ) -g - λ (u n 0 ⊤c n 0 , u n 1 ⊤c n 1 , v n )| ≤ Lc dist1((c-, c+), H λ (v n )),
and similarly

|g - λ (u n 0 ⊥c0, u n 1 ⊥c1, v n ) -g - λ (u n 0 ⊤c n 0 , u n 1 ⊤c n 1 , v n )| ≤ Lc dist1((c-, c+), H λ (v n )),
which concludes the proof with A = 2Lc + 2∆x ∆t .

We are now in position to obtain a discrete version of (4).

Proposition 3.2. Let (ϕ n j ) j∈Z,n∈N be a compactly supported sequence of nonnegative reals. If (28) holds for all n in N and j in Z, then

∆t∆x j∈Z,n∈N |u n+1 j -cj | ϕ n+1 j -ϕ n j ∆t + ∆x i∈Z |u 0 j -cj |ϕ 0 j + ∆t∆x j∈Z * ,n∈N G n j+1/2 ϕ n j+1 -ϕ n j ∆x + ∆t∆x n∈N G n j+1/2,+ ϕ n 1 -ϕ n 0 ∆x ≥ -A∆t n∈N dist1(c, H λ (v n ))(ϕ n 0 + ϕ n 1 ). (29)
Proof. Classically, the starting point is to multiply Equation ( 28) by ϕ n j and to sum over j ∈ Z and n ∈ N. Then the different terms are rearranged to bring out discrete time and space derivatives of ϕ. However, this is not straightforward around the particle, because two different fluxes are used on its left and on its right. The first term of (28) yields j∈Z,n∈N

|u n+1 j -cj | -|u n j -cj | ∆t ϕ n j = j∈Z,n∈N |u n+1 j -cj| ϕ n j -ϕ n+1 j ∆t - 1 ∆t j∈Z |u 0 j -cj |ϕ 0 j ,
and the second term yields j∈Z,n∈N

G n j+1/2,--G n j-1/2,+ ∆x ϕ n j = j∈Z * ,n∈N G n j+1/2 ϕ n j -ϕ n j+1 ∆x + n∈N ϕ n 0 ∆x G n 1/2,-- ϕ n 1 ∆x G n 1/2,+ = j∈Z * ,n∈N G n j+1/2 ϕ n j -ϕ n j+1 ∆x + n∈N ϕ n 0 ∆x (G n 1/2,--G n 1/2,+ ) + n∈N ϕ n 0 -ϕ n 1 ∆x G n 1/2,+ .
We almost have a discrete version of (4). The following lemma ensures that the corrective term

n∈N ϕ n 0 ∆x (G n 1/2,--G n 1/2,+ )
has the correct sign.

Lemma 3.3. If g - λ -g + λ is nondecreasing with respect to its first two arguments then we have the dissipativity property

G n 1/2,--G n 1/2,+ ≥ 0.
g - λ (a, b, v n ) -g - λ (ã, b, v n )) ≥ g + λ (a, b, v n ) -g + λ (ã, b, v n ),
which is a straightforward consequence of the monotonicity of g - λ -g + λ with respect to its two first variables.

Let us go back to the proof of Lemma 3.2. Hypothesis (19) exactly says that g - λ -g + λ is nondecreasing with respect to its two first arguments. Thus we can apply Lemma 3.3 to obtain j∈Z,n∈N

G n j+1/2,--G n j-1/2,+ ∆x ϕ n j ≥ j∈Z * ,n∈N G n j+1/2 ϕ n j -ϕ n j+1 ∆x + n∈N ϕ n 0 -ϕ n 1 ∆x G n 1/2,+ .
Eventually, we have

j∈Z,n∈N εj A ∆x dist1((c-, c+), H λ (v n ))ϕ n j = A ∆x n∈N dist1((c-, c+), H λ (v n ))(ϕ n 0 + ϕ n 1 )
and ( 29) is obtained by regrouping all the terms and changing their signs, and multiplying by ∆t∆x.

Passing to the limit ∆t → 0 in Equation (29), we obtain the following proposition.

Proposition 3.4. If u 0 belongs to BV (R) ∩ L 1 (R), if the CFL condition (20) holds and if Hypothesis (11-19), ( included (15)), are fulfilled, then the limit u of (u∆t) verifies Inequality (4) for any nonnegative function ϕ in C ∞ 0 (R+ × R).

Proof. For small enough ∆t, Condition (24) is verified. Let us fix (c-, c+) in R 2 , and prove that for every nonnegative ϕ in C ∞ 0 , the discrete inequality (29) converges to the continuous entropy inequality (4), where the sequence (ϕ n j ) j∈Z,n∈N is defined by ϕ n j = ϕ(n∆t, x n j -h n ). We recall that C n j is the space-time cell C n j = {(n∆t + s, x n j-1/2 + y + sv n ), s ∈ [0, ∆t), y ∈ [0, ∆x)}, that h n is the discrete position of the particle's trajectory deduced from its velocity:

h n+1 = h n + v n ∆t
and that the mesh is moving with the particle: x n+1 j = x n j + v n ∆t. We first treat the first term of (29). The sequence of piecewise constant functions (ζ∆t) defined by

ζ∆t(t, x) = ϕ n+1 j -ϕ n j ∆t if (t, x) ∈ C n+1 j
converges uniformly to the function (t, x) → (∂tϕ)(t, x -h(t)). Indeed, for every (t, x) ∈ C n+1 j , there exists t ∈ [n∆t, (n + 1)∆t] such that

|ζ∆x(t, x) -(∂tϕ)(t, x -h(t))| = ϕ((n + 1)∆t, x n+1 j -h n+1 ) -ϕ(n∆t, x n j -h n ) ∆t -(∂tϕ)(t, x -h(t)) = |(∂tϕ)( t, x n j -h n ) -(∂tϕ)(t, x -h(t))| ≤ C(| t -t| + |x -x n j | + |h n -h(t)|) ≤ C(∆t + ∆x + ||h∆t -h||∞)
We used the fact that x n+1 j -h n+1 = x n j -h n . We conclude thanks to Remark 2.5 :

∆t∆x j∈Z,n∈N |u n+1 j -cj | ϕ n+1 j -ϕ n j ∆t = j∈Z,n∈N C n+1 j |u∆t -c∆t|ζ∆tdt dx = R R + 1 t≥∆t |u∆t -c∆t|ζ∆tdt dx -→ R R + |u -c|(∂tϕ)(t, x -h(t))dt dx
On the other hand,

∆t∆x j<0,n∈N G n j+1/2 ϕ n j+1 -ϕ n j ∆x = x<-∆x 2 R + G∆tξ∆tdt dx
where for every

(t, x) in C n j+1/2 = {(n∆t + s, xj + y + v n s), 0 ≤ s < ∆t, 0 ≤ y < ∆x}, G∆t(t, x) = G j+1/2 =g λ u∆t t, x - ∆x 2 ⊤c-, u∆t t, x + ∆x 2 ⊤c-, v∆t(t) -g λ u∆t t, x - ∆x 2 ⊥c-, u∆t t, x + ∆x 2 ⊥c-, v∆t(t)
and for every

(t, x) in C n j+1/2 , ξ∆t(t, x) = ϕ n j+1 -ϕ n j ∆x .
The sequence (ξ∆t) converges uniformly to (t, x) → ∂xϕ(t, x -h(t)). By continuity of translations in L 1 , the sequences (u∆t(t, • + ∆x 2 ))∆t and (u∆t(t, • -∆x 2 ))∆t converge in L 1 loc , and therefore up to extraction almost everywhere, toward u. On the other hand, (v∆t) converges almost everywhere toward h ′ . The consistency of the germ implies that G∆t converges almost everywhere to

g(u⊤c-, u⊤c-, h ′ ) -g(u⊥c-, u⊥c-, h ′ ) = sgn(u -c-) u 2 2 -h ′ u - c 2 - 2 -h ′ c- .
As (u∆t) and (v∆t) are uniformly bounded in L ∞ , the dominated convergence theorem yields

∆t∆x j<0,n∈N G n j+1/2 ϕ n j+1 -ϕ n j ∆x -→ R -R + Φ h ′ (t) (u(t, x), c-)∂xϕ(t, x -h(t))dt dx.
The second and fourth terms of (29) are easily treated:

∆x i∈Z |u 0 j -cj |ϕ 0 j -→ R |u 0 -c|ϕ(0, x)dx and ∆t∆x n∈N G n j+1/2,+ ϕ n 1 -ϕ n 0 ∆x -→ 0.
Eventually, we study the convergence of

∆t n∈N dist1(c, H λ (v n ))(ϕ n 0 + ϕ n 1 ) = 2 R + dist1(c, H λ (v∆t)) ϕ∆t(t, -∆x 2 ) + ϕ∆t(t, ∆x 2 ) 2 dt.
Clearly,

ϕ ∆x (t,-∆x 2 )+ϕ ∆x (t, ∆x 2 ) 2 converges uniformly to ϕ(•, 0). Moreover, | dist1(c, H λ (v∆t)) -dist1(c, H λ (h ′ ))| = | dist1(c, (v∆t -h ′ , v∆t -h ′ ) + H λ (h ′ )) -dist1(c, H λ (h ′ ))| = | dist1(c -(v∆t -h ′ , v∆t -h ′ ), H λ (h ′ )) -dist1(c, H λ (h ′ ))| ≤ |v∆t -h ′ | and ∆t n∈N dist1(c, H λ (v n ))(ϕ n 0 + ϕ n 1 ) -→ 2 R + dist1(c, H λ (h ′ ))ϕ(t, 0)dt,
which concludes the proof.

Remark 3.5. In [START_REF] Cancès | Error estimate for Godunov approximation of locally constrained conservation laws[END_REF], the authors are able to derive error estimates for the Godunov scheme adapted to a conservation law with a discontinuous flux (with respect to the space variable). The jump in such a flux can be related to the presence of the particle in our case, and a treatment partially consistent with the interface is also proposed in this paper. A careful investigation of the interface enables the authors to prove adapted BV bounds, which are one of the main difficulties for obtaining error estimates. Due to the particular fluxes we use around the particle, we can also prove here BV bounds, see Proposition 2.1, and one may expect to adapt the proof of [START_REF] Cancès | Error estimate for Godunov approximation of locally constrained conservation laws[END_REF] and thus obtain error estimates for our numerical methods. △

Convergence of the particle's part

We now prove that the limit h of (h∆t) verifies (5). To begin with, we prove that a discrete version of (5) holds.

Proposition 3.6. Let (u n j ) n∈N,j∈Z and (v n ) n∈N be given by Scheme (6). Then, for every compactly supported sequences (ξ n ) n∈N and (ψ n j ) n∈N,j∈Z such that ψ n 0 = ψ n 1 = 1 for all integrer n, -m∆t

n∈N * v n ξ n -ξ n-1 ∆t = mv 0 ξ 0 + ∆x∆t n∈N * ,j∈Z u n j ψ n j ξ n -ψ n-1 j ξ n-1 ∆t + ∆x j∈Z u 0 j ξ 0 ψj + ∆t∆x n∈N,j =0 f n j+1/2 ξ n ψ n j+1 -ψ n j ∆x . (30)
Proof. We write

m n∈N v n+1 ξ n = m n∈N v n ξ n + ∆t n∈N (f n 1/2,--f n 1/2,+ )ξ n + ∆x n∈N j / ∈{0,1} (u n j -u n+1 j ) -µ(f n j+1/2 -f n j-1/2 ) ξ n ψ n j + ∆x n∈N (u n 0 -u n+1 0 ) -µ(f n 1/2,--f n -1/2 ) ξ n + ∆x n∈N (u n 1 -u n+1 1 ) -µ(f n 3/2 -f n 1/2,+ ) ξ n .
This comes from the fact that the sum of the last three lines is zero. We now rearrange the different terms. On the one hand we have:

n∈N,j≤-1 (f n j+1/2 -f n j-1/2 )ξ n ψ n j = n∈N,j≤-1 f n j+1/2 ξ n (ψ n j -ψ n j+1 ) + n∈N ξ n f n -1/2 ,
and on the other hand we have:

n∈N,j≥2 (f n j+1/2 -f n j-1/2 )ξ n ψ n j = n∈N,j≥1 f n j+1/2 ξ n (ψ n j -ψ n j+1 ) - n∈N ξ n f n 3/2 .
It follows that

m n∈N v n+1 ξ n = m n∈N v n ξ n + ∆x n∈N,j∈Z (u n j -u n+1 j )ξ n ψ n j -∆t n∈N,j =0 f n j+1/2 ξ n (ψ n j -ψ n j+1 ).
To conclude, we just have to rearrange the sum over n. Being careful with n = 0 we obtain

n∈N (v n+1 -v n )ξ n = n∈N * v n (ξ n-1 -ξ n ) -v 0 ξ 0 and n∈N,j∈Z (u n j -u n+1 j )ξ n ψ n j = n∈N * ,j∈Z u n j (ψ n j ξ n -ψ n-1 j ξ n-1 ) + j∈Z u 0 j ξ 0 ψ 0 j ,
and the result follows by regrouping all the terms.

We can now pass to the limit ∆t → 0 in Proposition 3.6 to prove that h verifies (5).

Proposition 3.7. Suppose that Hypothesis (11-19) hold, and that the CFL condition (20) is fulfilled. For all test functions ξ and ψ such that ψ(0) = 1, the limit h of (h∆t) verifies Inequality (5).

Proof. Define ψ n j = ψ(x n j -h n ) and ξ n = ξ(n∆t). Proposition 3.6 applies if ψ n 0 = ψ n 1 = 1. Here, we only have

∀j ∈ {0, 1}, ψ n j -1 ≤ C∆x.
The equality (30) holds up to the following corrections appearing in the left hand side:

∆x∆t n∈N * ,j∈{0,1} u n j (1 -ψ n j )ξ n -(1 -ψ n-1 j )ξ n-1 ∆t + ∆x j∈{0,1} u 0 j ξ 0 (1 -ψ 0 j ) + ∆x∆t n∈N f n -1/2 ξ n (1 -ψ n 0 ) ∆x -f n 1/2 ξ n (1 -ψ n 1 ) ∆x ,
which all tends to zero since ψ n 0 -1 = O(∆x) and ψ n 1 -1 = O(∆x). The sequence

ζ∆t(t, x) = ψ n j ξ n -ψ n-1 j ξ n-1 ∆t if (t, x) ∈ C n j
converges uniformly to the function (t, x) → ψξ ′ . Indeed, by definition of the moving mesh, x n j -h n = x n-1 j -h n-1 . Therefore, ψ n j = ψ n-1 j and

ψ n j ξ n -ψ n-1 j ξ n-1 ∆t = ψ n j ξ n -ξ n-1 ∆t
which converges uniformly toward the expected function. Now, define F∆t by

F∆t(t, x) = g λ u∆t t, x - ∆x 2 , u∆t t, x + ∆x 2 , v∆t(t)
in such a way that for all (t, x) in C n j+1/2 ,

F∆t(t, x) = f n j+1/2 .
By continuity of translations in L 1 , the sequences (u∆t(t, •+ ∆x 2 ))∆t and (u∆t(t, •-∆x 2 ))∆t converge in L 1 loc , and therefore up to extraction, almost everywhere, toward u. On the other hand, (v∆t) converges almost everywhere toward h ′ . The consistency of the flux (11) implies that F∆t converges almost everywhere to

g(u, u, h ′ ) = u 2 2 -h ′ u.
• If λ ≤ u-and u+ ≤ -λ, the result

u 2 --v 2 - 2 + u 2 + -v 2 + 2 ≥ 0 is a straightforward consequence of u 2 -+ u 2 + 2 ≥ λ 2 ≥ v 2 -+ v 2 + 2 .
• Eventually, if λ ≤ u-and -λ ≤ u+ ≤ 0, let us first suppose that u+ ≤ v+ and prove

u 2 --v 2 - 2 + u 2 + -v 2 + 2 ≥ 0. It follows from v 2 + + v 2 - 2 ≤ u 2 + + v 2 - 2 ≤ u 2 + + u 2 - 2 .
Suppose now that u+ > v+ and u+ ≥ -u-+ λ. The result

u 2 --v 2 - 2 - u 2 + -v 2 + 2 ≥ 0 comes from u 2 --u 2 + 2 ≥ -2λu+ + λ 2 2 ≥ λ 2 2 ≥ v 2 --v 2 + 2 .
It is possible to find fluxes that verifies (15) with H λ = G 1 λ ∪ G 2 λ and (18). Proposition 3.9. The family of finite volume schemes defined by

g - λ (u-, u+, v) = g(u-, min(u+ + λ, max(u-, v)), v) g + λ (u-, u+, v) = g(max(u--λ, min(u+, v)), u+, v) (32)
is consistent with G 1 λ ∪ G 2 λ (v) and verifies the monotonicity assumptions ∂1g ± λ ≥ 0 and ∂2g ± λ ≤ 0.

Proof. The proof consists in a simple verification. We first check that for all u-and u+ in R, g - λ (u-, u--λ, v) = g(u-, min(u-, max(u-, v)), v) = g(u-, u-, v) and g + λ (u+ + λ, u+, v) = g(max(u+, min(u+, v)), u+, v) = g(u+, u+, v). Then, we verify that for all u+ in

[v -λ, v], g - λ (v, u+, v) = g(u-, min(u+ + λ, max(v, v)), v) = g(v, v, v) and g + λ (v, u+, v) = g(max(v -λ, min(u+, v)), u+, v) = g(u+, u+, v) while for every u-in [v, v + λ], g - λ (u-, v, v) = g(u-, min(v + λ, max(u-, v)), v) = g(u-, u-, v) and g + λ (u-, v, v) = g(max(u--λ, min(v, v)), v, v) = g(v, v, v).
Eventually, the monotonicity properties are implied by those on g as soon as soon as the first component is not u+ and the second is not u-. But if the first component is u+, then u+ < v and ∂2g

+ λ = u+ -v ≤ 0, while if the second component is u-, then u-> v and ∂2g - λ = u--v ≥ 0.
It remains to prove that Assumption (19) holds. This is not the case for every choice of flux g (a counterexample can be found in [START_REF] Andreianov | Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes[END_REF]), but we can check it for three classical fluxes.

Proposition 3.10. The family of finite volume schemes (32) verifies that g - λ -g + λ is nondecreasing with respect to its two first variables if g is the Godunov, the Rusanov or the Engquist-Osher numerical flux. Proof. Let us divide the phase space (u-, u+) in six zones, depending on which values are taken by g - and g + :

g - λ (u-, u+, v) =    g(u-, u-, v) if v ≤ u-≤ u+ + λ zone I, g(u-, v, v) if u-≤ v ≤ u+ + λ zone II, g(u-, u+ + λ, v) if u+ + λ ≤ max(u-, v) zone III, while g + λ (u-, u+, v) =    g(u+, u+, v) if u--λ ≤ u+ ≤ v zone 1, g(v, u+, v) if u--λ ≤ v ≤ u+ zone 2, g(u--λ, u+, v) if min(u+, v) ≤ u--λ zone 3.
These zones are depicted on Figure 3. If u+ belongs to zones 1 or 2, g + does not depends on u-and g - λ -g + λ is nondecreasing with respect to its first argument. Similarly, if u-belongs to zones I or II, g - λ -g + λ is nondecreasing towards its second argument. We focus on the case where u-belongs to zone III or u+ belongs to zone 3. Let us first remark that the case where u-belongs to zone III and u+ is in zone 3 reduces to the choice of flux studied in [START_REF] Andreianov | Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes[END_REF], where the monotonicity property has been proven for the Godunov, Rusanov and Engquist-Osher scheme. Suppose that case u-is in zone I and u+ is in zone 3. Then we have

(g - λ -g + λ )(u-, u+, v) = g(u-, u-, v) -g(u--λ, u+, v).
For the sake of simplicity we assume that v = 0.

• If g is the Godunov flux, as u+ + λ ≥ u-≥ λ, the Riemann problem between u--λ and u+ is a shock traveling faster than v. It follows that

(g - λ -g + λ )(u-, u+, 0) = (u-) 2 2 - (u--λ) 2 2 = λu-- λ 2 2 is nondecreasing toward its first two arguments. • If g is the Rusanov flux, (g - λ -g + λ )(u-, u+, 0) = (u-) 2 2 - (u--λ) 2 + u 2 + 4 -(u--λ) u+ -(u--λ) 2
and we have

∂1(g - λ -g + λ )(u-, u+, 0) = u-- u--λ 2 - u+ -(u--λ) 2 + u--λ 2 = -u-+ 3λ + u+ 2 .
As u-belongs to zone I, u+ + λ ≥ u-, and the last quantity is larger than λ. On the other hand,

∂2(g - λ -g + λ )(u-, u+, 0) = - u+ -(u--λ) 2
and this last quantity is nonnegative because u+ belongs to zone 3.

• Eventually, if g is the Engquist-Osher scheme, the fact that 0 ≤ u--λ ≤ u+ implies that

(g - λ -g + λ )(u-, u+, 0) = (u-) 2 2 - (u--λ) 2 2 = λu-- λ 2 2
is once again nondecreasing with respect to its first two arguments. The case where u-is in zone III while u+ is in zone 1 can be treated in a symmetrical way.

4 Convergence of schemes only consistent with G 1 λ

In this section, we no longer require Hypothesis (15) to be fulfilled, and prove convergence of a family of finite volume schemes that verifies only (14). The difficulty is that G 1 λ is not a maximal part of the germ, and we cannot prove a discrete version of (4) directly. The key point is to study the convergence of the solution of Scheme (6) for initial data in the maximal subset of the germ G 1 λ ∪ G 2 λ . We then extend the comparison argument of [START_REF] Andreianov | Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes[END_REF] to prove convergence for arbitrary initial data.

Proof of convergence

Let us now focus on fluxes that do not preserve a maximal part of the germ (in the sense of Hypothesis (15)), but only the straight line G 1 λ , i.e. that verifies (14) but not (15). Our aim is to prove the following theorem.

Theorem 4.1. If the numerical fluxes around the particle are given by

f n 1/2,-(u n 0 , u n 1 , v n ) = g(u n 0 , u n 1 + λ, v n ), f n 1/2,+ (u n 0 , u n 1 , v n ) = g(u n 0 -λ, u n 1 , v n ), (33)
where g is a numerical flux verifying (11-14) and (16-19), and if the CFL condition (20) holds, Scheme (6) converges toward the solution of (1).

Proof. Let us first remark that Proposition 2.1 and Proposition 2.2 did not use Hypothesis (15), thus we can extract converging subsequences as we did in the previous Section. Now, consider a test function ϕ supported in {x < 0} or {x > 0}, we have ϕ n 0 = ϕ n 1 = 0 for small enough ∆x. We easily obtain, as in Proposition 3.1, that for all c in R, for all j ≤ -1, and we straightforwardly obtain that the limit u of the scheme is an entropy solution of the Burgers equation on the sets {x < h} (and similarly on {x > h}). It remains to prove that the traces around the particle belong to the germ for almost every time. Let us fix a time t0 such that h ′ and the traces u-(t0) and u+(t0) exist. Fix (c-, c+) in H λ (h ′ (t0)). Our aim is to prove a discrete version of (4). Let us first suppose that (c-, c+) belongs to the straight line G 1 λ but not to the closed square G 2 λ (h ′ (t0)). By continuity of h ′ , there exists δ > 0 such that, ∀t ∈ (t0 -δ, t0 + δ), dist1((c-, c+), G 1 λ ) = dist1((c-, c+), H 1 λ (h ′ (t)))

(see Figure 1). Up to taking a smaller δ, this equality is also true at the numerical level for small enough ∆t, since from Proposition 2.4, (vn) n∈N converges. Therefore, passing to the limit in (29) with ϕ supported in time in (t0 -δ, t0 + δ), we directly obtain (4). We now treat the case where (c-, c+) belongs to the interior of G 2 λ (h ′ (t0)). The principle of the proof is to compare the numerical solution with another one, for which the initial data is much simpler as it corresponds to an element of G 2 λ (h ′ (t0)). Since h ′ is continuous, there exists δ such that ∀t ∈ (t0 -δ, t0 + δ), (c-, c+) ∈ G 2 λ (h ′ (t))

Thus the limit c of the scheme (35) with initial data (36) at time t0 -δ is such that c is larger than h ′ on x < h and smaller on x > h. It allows to prove that c is, on {(t, x) : x < h(t)}, the solution of      ∂tu + ∂x u 2 2 = 0 ∀t ∈ (t0 -δ, t0 + δ), ∀x < h(t), u(t0 -δ, x) = c-∀x < h(0) u(t, h(t)) = h ′ (t) ∀t ∈ (t0 -δ, t0 + δ).

(37) As c-is larger than h ′ on the whole time interval, the boundary condition is inactive and the solution is u = c-. Let us recall the definition given by Bardos, LeRoux and Nedelec in [START_REF] Bardos | First order quasilinear equations with boundary conditions[END_REF] of this conservation law on a bounded domain. A function u in L ∞ is a solution of      ∂tu + ∂xf (u) = 0 ∀t > 0, ∀x < h(t), u(t = 0, x) = u 0 (x) ∀x < h(0), u(t, h(t)) = u b (t) ∀t > 0, if for all real κ and for all nonnegative function ϕ ∈ C ∞ 0 (R+ × R), the following inequality holds:

t>0 x<h(t)
|u(t, x) -κ|∂tϕ(t, x -h(t)) + Φ h ′ (t) (u(t, x), κ)∂xϕ(t, x -h(t))dx dt

+ x<h(0)
|u 0 (x) -κ|ϕ(0, x)dx + t>0 sgn(κ -u b (t)){f (u(t, h(t) -)) -f (κ)}ϕ(t, 0) ≥ 0.

(

) 38
The convergence of finite volume schemes for scalar conservation laws in a bounded domain has been proven in [START_REF] Vovelle | Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains[END_REF] for instance. We are here in a favorable case: we can obtain a discrete version of (38) by summing (28) multiplied by ∆t ∆x ϕ n j over n ≥ 0 and j ≤ -1. We obtain |c--κ|ϕ(0, x)dx + t>0 sgn(κ -c(t, h(t))){f (c(t, h(t) -)) -f (κ)}ϕ(t, 0) ≥ 0.

To conclude we check that sgn(κ -h ′ (t)){f (c(t, h(t) -)) -f (κ)} ≥ -sgn(c(t, h(t) -κ)){f (c(t, h(t) -)) -f (κ)}.

This relies strongly on the fact that c remains larger than h ′ .

• If h ′ ≤ κ ≤ c, the inequality reduces to {f (c(t, h(t) -)) -f (κ)} ≥ -{f (c(t, h(t) -)) -f (κ)} which holds because f is increasing on (0, +∞). with (c-, c+) ∈ G 2 λ (v 0 ). In Appendix A, we prove the convergence for this specific initial data directly, without using the local in time comparison with the one-way scheme (35) in which the velocity of the particle is fixed. △

Proof of Lemma 3. 3 .

 3 Let us denote by a = u n 0 ⊤c0, ã = u n 0 ⊥c0, b = u n 1 ⊤c1 and b = u n 1 ⊥c1, such that a ≥ ã and b ≥ b. The dissipativity property holds if and only if

Figure 3 :

 3 Figure 3: Choice of the fluxes in the family of finite volume schemes (32).

  Multiplying by ∆t∆xϕ n j and summing over n ∈ N and j ≤ -1, we obtain as in Proposition 3

  Passing to the limit yieldst>0 x<h(t) |c(t, x) -κ|∂tϕ(t, x -h(t)) + Φ h ′ (t) (c(t, x), κ)∂xϕ(t, x -h(t))dx dt + x<h(0)

•

  If h ′ ≤ c ≤ κ or κ ≤ h ′ ≤ c the inequality reduces to {f (c(t, h(t) -)) -f (κ)} ≥ {f (c(t, h(t) -)) -f (κ)} or -{f (c(t, h(t) -)) -f (κ)} ≥ -{f (c(t, h(t) -)) -f (κ)},which are both trivial.Remark 4.3. Of course, Theorem 4.1 applies when the initial data isu 0 (x) = c-1x<0 + c+1 x≥0 ,

A family of scheme consistent with a maximal part of the germ

In this section we exhibit a family of schemes that verifies the set of Assumptions (11)(12)(13)(14)(15)(16)(17)(18)(19). Let us clarify which maximal subset of G λ is used.

Proposition 3.8. The part H λ (v) = G 1 λ ∪ G 2 λ (v) is a maximal subset of the germ.

Proof. Following [START_REF] Andreianov | Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes[END_REF] (see Equations ( 13) and ( 14) in this reference), it suffices to show that if Ξv((u-, u+), (v-, v+)) ≥ 0 for any (v-, v+) ∈ G 2 λ (v),

then the stronger following property holds Ξv((u-, u+), (v-, v+)) ≥ 0 for any (v-, v+) ∈ G 1 λ ∪ G 2 λ (v).

In the sequel we suppose that v = 0. The general case follows by translation. The two main arguments are first, that Proposition 1.4 implies that this is automatically verified if (u-, u+) belongs to the germ, and second, that for all (v-,

In the sequel, (v-, v+) always denotes an element of G 2 λ . We proceed by a tedious, but not difficult, disjunction of cases.

• If u-≥ λ and u+ ≥ 0, then we want to prove that

If we apply Equation (31) to (λ, 0), we obtain that

and the result follows.

• If 0 ≤ u-≤ λ and u+ ≥ 0, then (u-, u+) belongs to the germ. Indeed, Equation (31) applied to

2 ≥ 0 and therefore, u+ = 0. • If u-≤ 0 and u+ ≥ 0, then (u-, u+) belongs to the germ. Indeed, Equation (31) applied to (0, 0) yields

and therefore, u-= u+ = 0.

• If u-≤ 0 and -λ ≤ u+ ≤ 0, then (u-, u+) belongs to the germ. Indeed, Equation (31) applied to (0, u+) yieldsu 2 -2 ≥ 0 and therefore, u-= 0. • If u-≤ 0 and ≤ u+ ≤ -λ, then we want to prove that

If we apply Equation (31) to (0, -λ), we obtain

and the result follows.

• If 0 ≤ u-≤ λ and u+ ≤ -λ, let us first suppose that u-≥ v-. We have to prove that

But 0 ≤ v-≤ u-and 0 ≥ v+ ≥ u+, and we have the result:

We now suppose that u-≤ v-. We want to prove that

Moreover, (u-, u+) does not belong to the germ G λ , and therefore u+ ≤ -u--λ and

and on the time interval (t0 -δ, t0 + δ), (4) becomes

Up to reducing δ and for small enough ∆t, this is also true at the numerical level. Now, for (u n j ) j∈Z,n∈N and (v n ) n∈N given by the fully coupled scheme (6), consider (c n j ) j∈Z,n∈N * the sequence given by the scheme

with initial data

We recall that (c-, c+) belongs to G 2 λ (h ′ (t0)). Simple modifications of Propositions 3.1 and 3.2 yield

where ϕ is a test function supported in (t0 -δ, t0 + δ), we obtain (34) by passing to the limit. We now study this convergence.

Lemma 4.2. Suppose that at iteration n, the sequence (c n j ) j∈Z given by the scheme (35) is nondecreasing on j ≤ 0 and on j ≥ 1, and such that

Proof. The monotonicity of (c n+1 j ) j≤0 follows from the monotonicity of H λ under the CFL condition (20). For j ≤ -2, we have

Moreover, for j ≤ -1, both c n j-1 , c n j and c n j+1 are between c-and c-+ λ, thus the same holds at iteration n + 1. For j = 0, as c+ ≤ c-(because (c-, c+) belongs to G 2 λ (h ′ (t0))), we conclude by remarking that

The results for positive integers j are obtained in a similar way. Let us now prove that u n+1

, there exists a positive δ such that h ′ (t) stays in the interval (c+, c-) on the time interval (t0 -δ, t0 + δ). For small enough ∆t, it is also true at the numerical level. Up to reducing slightly δ, (c-, c+) belongs to G 2 λ (v n ) for small enough ∆t and for all iteration in time such that t n belongs to (t0 -δ, t0 + δ), and in particular c+ ≥ v n ≥ c-.

Appendices

A Detailed analysis when the initial data belongs to G 2 λ (v 0 )

Our aim in this section is to prove directly that if

with (u-, u+) in G 2 λ (v 0 ), Scheme (6) converges toward the exact solution, which in that case is given by

In this section only and for technical reasons, we consider a finite volume scheme on a bounded space domain [-a, a], subdivided with 2Mc cells and with periodic boundary conditions. The scheme under consideration writes

We recall that the ratio of the time step ∆t and the cell size ∆x is equals to µ. We fixed the final time T . At each time step, four new cells (one of both part of the particle and one of each extremities of the interval because of the periodic boundary conditions) are influenced by Scheme (40), in the sense that their values were constant equals to u-or u+ before. We take a large enough so that the influence of the particle does not interact with the influence of the boundary condition, and stays in the interval [-a/3, a/3] during the time interval [0, T ] (see Figure 4 below). This is achieved by taking a larger than problem (1).

Proposition A.1. Suppose that the numerical flux g verifies (11-14) and (16-19), that

and that ∂3g is decreasing with respect to its first two arguments. Under Condition (20) and for the initial data (39), Scheme (40) converges toward the solution of (1) on {(t, x) : t < T and -a/3 + h(t) < x < a/3 + h(t)}.

Proof. We prove, as we did in Section 4, that (u n j ) -Mc/3≤j≤0 converges toward the solution of (37), with a Neumann boundary condition on the left of the particle. The key point is to prove that v n remains smaller than c-on the whole time interval [0, T ], in which case the boundary condition is inactive and we obtain the result. Similarly on the right of the particle, the boundary condition is inactive if v n remains larger than c+.

To prove that c+ ≤ v n ≤ c-, we apply the Crandall-Tartar lemma [START_REF] Michael | Some relations between nonexpansive and order preserving mappings[END_REF] to the application

Lemma A.2 (Crandall-Tartar). Let (Ω, µ) be a measured space, and let S be a subset of L 1 (Ω) stable by sup:

Consider a function T : S → S such that

In our case, Ω = R 2Mc × R and

It is straightforward to verify that Scheme (40) preserves the norm || • || L 1 . The fact that T takes its values in S is proven exactly as in the proof of Lemma 4.2. We prove in Lemma A.4 that T is order preserving. Applying the Crandall-Tartar lemma to ((u 0 j ), v 0 ) and (ū 0 j , v) = ((u 0 j ),

), we obtain

The result follows since vn+1 =

(see Lemma A.3 below).

Lemma A.3. If g verifies (41) and if the initial data is

, then Scheme (40) verifies v n = v 0 for all integer n.

Proof. We prove by induction the following stronger result:

The symmetry of the initial data ensures that this is verified for n = 0. Suppose that this is verified for some n ≥ 0. Hypothesis (41) on the flux and the induction hypothesis yield

Hence, the velocity remains constant. A similar reasoning can be applied to the fluid velocity. Let us give some details for j ≤ -1:

and for j = 0:

Lemma A.4. Suppose that ∂3g is decreasing with respect to its first two arguments, and that (12), ( 20

Proof. The case where v n is equal to vn is a straightforward. On the one hand the monotonicity assumption (12) on g and CFL condition (20) yield as usual

On the other hand,

is nonnegative by Hypothesis (19). We now focus on the case where (u n j ) j∈Z is equal to (ū n j ) j∈Z and v n ≤ vn . For j ≤ -1 and j ≥ 2, a straightforward computation gives that there exists a n j+1/2 ∈ [u n j , u n j+1 ] and b n j-1/2 ∈ [u n j-1 , u n j ]

Moreover, u n j-1 ≤ u n j ≤ u n j+1 because we are considering elements of S, thus if ∂3g is decreasing with respect to its first two variables, u n+1 j ≤ ūn+1 j . The same reasoning extends to j ∈ {0, 1} because u n 0 -u n 1 ≤ λ. Eventually, vn+1 -v n+1 = vn -v n + ∆t mp (g(u n 0 , u n 1 + λ, vn ) -g(u n 0 , u n 1 + λ, v n ))

which is nonnegative if (42) holds.