Louis Lecrosnier

Rémi Boutteau

Pascal Vasseur

Xavier Savatier

Friedrich Fraundorfer

Vision based vehicle relocalization in 3D line-feature map using Perspective-n-Line with a known vertical direction

Common approaches for vehicle localization propose to match LiDAR data or 2D features from cameras to a prior 3D LiDAR map. Yet, these methods require both heavy computational power often provided by GPU, and a first rough localization estimate via GNSS to be performed online. Moreover, storing and accessing 3D dense LiDAR maps can be challenging in case of city-wide coverage.

In this paper, we address the problem of camera global relocalization in a prior 3D line-feature map from a single image, in a GNSS denied context and with no prior pose estimation. We propose a dual contribution.

(1) We introduce a novel pose estimation method from lines, (i.e. Perspective-n-Line or PnL), with a known vertical direction. Our method benefits a Gauss-Newton optimization scheme to compensate the sensor-induced vertical direction errors, and refine the overall pose. Our algorithm requires at least 3 lines to output a pose (P3L) and requires no reformulation to operate with a higher number of lines.

(2) We propose a RANSAC (RANdom SAmple Consensus) 2D-3D line matching and outliers removal algorithm requiring solely one 2D-3D line pair to operate, i.e. RANSAC1. Our method reduces the number of iteration required to match features and can be easily modified to exhaustively test all feature combinations.

We evaluate the robustness of our algorithms with a synthetic data, and on a challenging sub-sequence of the KITTI dataset.

I. INTRODUCTION

Recently, a significant effort has been put into autonomous driving technology to meet safety and efficiency standards. One of the key aspects in autonomous driving is to accurately localize a vehicle on a local scale. Knowing this exact location and orientation, also called pose, is the first step to path planning and navigation. This step is often handled by matching sensor data against a prior map of the environment.

3D maps are commonly acquired with LiDAR (Light Detection and Ranging) sensors. Depending on the model, they output either a sparse or dense depth map of the environment in the form of a point cloud.

A common approach for vehicle localization is to equip the vehicle with a LiDAR sensor, and match the sensor data with the embed environment map. A high-end LiDAR sensor dedicated to autonomous navigation, such as the Velodyne *This work is part of the COPTER project, and is co-funded by the European Union and the Région Normandie. Europe is involved in Normandy through the European Regional Development Fund (ERDF) 1 are with Normandie Univ, UNIROUEN, ESIGELEC, IRSEEM, 76000 Rouen, France 2 is with Normandie Univ, UNIROUEN, UNIHAVRE, INSA Rouen, LITIS, 76000 Rouen, France 3 is with Institute for Computer Graphics and Vision, Graz University of Technology, 8010 Graz, Austria and German Aerospace Center (DLR), Remote Sensing Technology Institute, Germany Fig. 1: Image from the vehicle (top), with manually extracted 2D features (green), reprojection of 3D features with the estimated pose (blue) and reprojected 3D features with the ground truth pose (yellow). Reconstructed 3D point cloud of the KITTI sequence used (bottom) HDL-64e, can span up to a million measurement points per second, and an acquisition of only a few minutes can lead to an uncompressed point cloud of several gigabytes. Because of the size of the point cloud map, matching directly sensor data is a challenging process. Moreover, these kind of sensors often cost several times the value of the vehicle carrying them.

While GNSS (Global Satellite Navigation System) provides reliable information about the global vehicle location, it can suffers inaccuracies of several meters on a local scale when confronted to multi-path effects in urban canyons or with low satellite coverage, making GNSS localization insufficient on its own. To increase localization accuracy, GNSS can be coupled with an inertial measurement unit (or IMU) with vehicle kinematic-aware filters, and to a ground GNSS antenna (e.g. RTK-GPS). While requiring costly hardware, this method can produce a centimetric localization, but drifts over time in case of degraded or lost GNSS signal.

Our work focuses on estimating the pose of a monocular camera in a prior 3D-lines map. When replacing a dense 3D LiDAR map with a line-feature map, we drastically reduce the storage space needed while allowing an increased overall map size. Moreover, by using geometric features such as lines, we rely on time-invariant information. While methods focusing on photometric appearance might struggle over night and day illumination changes, building and street shape are less likely to change over a long period of time.

Because we rely solely on a monocular camera and an affordable IMU sensor for pose estimation, we avoid the substantial cost of a LiDAR sensor. Over the sequence we selected in the KITTI dataset, we show that even a minimalist 3D feature map is sufficient to estimate the vehicle pose online, without the use of a GNSS for a rough pose estimate.

In this paper, our first contribution is a perspective-n-Line algorithm relying on a prior 3D lines map and known vertical direction to accurately estimate the pose a vehicle from a single image. Our algorithm requires only 3 2D-3D line correspondence to operate, and formulates the PnL problem as a set of linear equations that are solved using a linear least squares method, followed by a Gauss-Newton-based pose refinement.

Our second contribution is a RANSAC algorithm requiring a single line correspondence to match 2D and 3D line features, i.e. RANSAC1. We combine these two approaches, and we rely on a prior 3D feature map to leverage the vehicle relocalization in a GNSS-denied environment. We evaluate our algorithms on synthetic data and on a challenging sequence of the KITTI odometry dataset.

II. RELATED WORK

Most of the work regarding visual localization is targeted towards matching photometric features of the environment. This matching process can be either performed by comparing image descriptors, such as SURF [START_REF] Bay | Surf: Speeded up robust features[END_REF] or SIFT [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF], or by comparing image intensity value directly. However, this approach rapidly become ineffective when the environment appearance varies, e.g. in case of seasonal change, or with night and day lighting conditions. To solve this issue, [START_REF] Churchill | Experience-based navigation for longterm localisation[END_REF] propose to store multiple images of the same place over time. [START_REF] Naseer | Robust visual slam across seasons[END_REF] rely on a similarity matrix to match trajectories.

Aside from photometric feature comparison, other localization methods rely on geometric feature matching from LiDAR map. [START_REF] Wolcott | Visual localization within lidar maps for automated urban driving[END_REF] localize a vehicle in an urban environment by comparing a camera image to a set of synthetic images computed from the intensity values of a LiDAR map. Final result is obtained by maximizing the mutual information, and provides a 3 DoF (degree of Freedom) pose. [START_REF] Pascoe | Direct visual localisation and calibration for road vehicles in changing city environments[END_REF] present a 6 DoF camera pose by combining photometric and geometric data into their appearance prior map, which they use to render and match a view to a live camera image through normalized information distance minimization.

The topic of camera pose estimation from geometric features is itself a well studied subject in computer vision. However, feature matching is often assumed, which is unpractical for outdoor vehicle localization.

When related to point features, pose estimation is referred as Perspective-n-Points (or PnP) [START_REF] Dhome | Determination of the attitude of 3d objects from a single perspective view[END_REF] [8] [9] [START_REF] Ferraz | Very fast solution to the pnp problem with algebraic outlier rejection[END_REF]. Pose estimation from lines, or PnL, is a less studied subject. PnL methods formulate the pose estimation problem as a system of linear or polynomial equations, that is solved by minimizing an algebraic or geometric error (e.g. reprojection error).

PnL is introduced by [START_REF] Dhome | Determination of the attitude of 3d objects from a single perspective view[END_REF] and [START_REF] Chen | Pose determination from line-to-plane correspondences: existence condition and closed-form solutions[END_REF] in 1989. [START_REF] Chen | Pose determination from line-to-plane correspondences: existence condition and closed-form solutions[END_REF] state that the problem can only be solved with a minimum of three 2D/3D line correspondences, i.e. P3L, spanning a least 8 solutions, while [START_REF] Dhome | Determination of the attitude of 3d objects from a single perspective view[END_REF] propose the first closed-form solution to PnL with a polynomial approach.

In a more recent work, [START_REF] Ansar | Linear pose estimation from points or lines[END_REF] propose a PnL formulation relying on a lifting approach to linearize a polynomial system formed with the elements of the rotation matrix. Using 3 line-correspondences, [START_REF] Mirzaei | Globally optimal pose estimation from line correspondences[END_REF] present a PnL method spanning 23 solutions, but more robust to noise, and operating with a minimum of 3 lines. [START_REF] Zhang | Robust camera pose estimation from unknown or known line correspondences[END_REF] introduce a P4L algorithm operating on 3-line subsets. They recover solutions from the derivatives of a 16 th order cost function.

[15] extend the work from [START_REF] Zhang | Robust camera pose estimation from unknown or known line correspondences[END_REF] with ASPnL (Accurate Subset based PnL), which is outlier-sensitive, but is more accurate with a small dataset. ASPnL includes a Gauss-Newton pose refinement.

[16] [START_REF]Absolute pose estimation from line correspondences using direct linear transformation[END_REF] present a DLT-based (Direct Linear Transform) PnL algorithm, inspired by the work of [START_REF] Hartley | Multiple view geometry in computer vision[END_REF]. They parameterize lines using the Plücker line representation. Because their method relies on singular value decomposition to estimate the pose, is very efficient and accurate for a dataset with a very high number of lines (hundreds to thousands), but is inaccurate for a small dataset. Some of the most recent work assume that the camera setup is coupled with an Inertial Measurement Unit (IMU), that provides a vertical direction vector, or up-vector. [START_REF] Horanyi | Multiview absolute pose using 3d-2d perspective line correspondences and vertical direction[END_REF] introduced NPnLUpL and NPnLUpC (N-camera PnL Upvector Linear and Cubic), and rely on modified Plücker coordinates of lines to formulate the PnL problem as a set of linear or cubic equations. Their algorithms are designed to operate in a multi-camera setup with known correspondences between 2D lines across all cameras, but can be used in a monocular setup.

While outliers rejection schemes exist within the PnL literature, feature matching is often assumed. [START_REF] Zhang | Robust camera pose estimation from unknown or known line correspondences[END_REF] handles unknown feature correspondence using weak pose prior. [START_REF] Ferraz | Very fast solution to the pnp problem with algebraic outlier rejection[END_REF] propose an outliers rejection algorithm for PnP based on algebraic error minimization. This method inspired the work of [START_REF] Xu | Pose estimation from line correspondences: A complete analysis and a series of solutions[END_REF] and [START_REF]Absolute pose estimation from line correspondences using direct linear transformation[END_REF], and handles efficiently large dataset but is less accurate with small and noisy datasets.

Aside from algebraic error minimization, other outliers rejection methods rely on a statistical approach. [START_REF] Fischler | Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography[END_REF] proposed the RANSAC (RAndom SAmple Consensus) algorithm in 1991. The key idea is to select a random sample of the dataset to estimate the parameters of a model and validate the parameters on the entire dataset. This process is iterated until a minimal error criterion is met. [START_REF] Xu | Pose estimation from line correspondences: A complete analysis and a series of solutions[END_REF] present RANSAC3 and RANSAC4 algorithms, but the runtime of the two methods for a high number of lines is problematic for online applications.

In this paper, we handle both the line matching and outliers rejection process by introducing a RANSAC1 algorithm relying on a known vertical direction. By reducing the number of lines to be considered by the RANSAC algorithm from lines, we leverage the runtime issue of the latter RANSAC3 and RANSAC4 methods. We extend our work to a pose estimation from lines algorithm, with a known vertical direction. We parameterize lines with Plücker coordinates, in the same manner as [START_REF] Přibyl | Camera pose estimation from lines using plucker coordinates[END_REF], but propose a linear formulation, that we solve with a least squares solver. We also propose a subsequent Gauss-Newton based pose refinement scheme.

III. THE PERSPECTIVE-N-LINE PROBLEM

In this section, we use the pinhole camera model, and we assume that an IMU calibrated with our camera provides the vertical direction (i.e. two of the three rotations are known).

A. 3D Line parameterization

We parameterize the lines in Plücker coordinates, as shown in Fig. 2.

The Plücker representation of a 3D line L can be expressed from two 3D points in homogeneous coordinates

A = (a 1 , a 2 , a 3 , a 4 ) T and B = (b 1 , b 2 , b 3 , b 4 ) T . L is a homo- geneous 6-vector L = (U T , V T ) T such as: L = (L 1 , L 2 , L 3 , L 4 , L 5 , L 6 ) T (1)
where

U = (a 1 , a 2 , a 3 ) × (b 1 , b 2 , b 3 ) V = a 4 .(b 1 , b 2 , b 3 ) -b 4 .(a 1 , a 2 , a 3 ), (2)
and '×' represents the cross-product. U is the normal vector to the projection plane Π. In Plücker coordinates, L must satisfy the bilinear constraint U T .V = 0. L W in the world frame {W } can be expressed in the camera frame {C} with the line motion matrix M 6×6 as:

L C = M.L W (3) with M = R -R.T [×] 0 3×3 R . (4)
'[×]' is the skew symmetric matrix of a 3-vector. T = (t x ,t y ,t z ) T represents a translation vector and R is a 3 × 3 rotation matrix.

B. Line projection

The 3D line l W is projected onto the image plane Π in the camera frame {C} using the projection matrix P such as :

P = R -R.T [×] . (5)
We obtain the projected 2D line l C such as

l C ∼ P.L W , (6)
where l C = (l 1 , l 2 , l 3 ) T . We use here the same parameterization as [START_REF] Přibyl | Camera pose estimation from lines using plucker coordinates[END_REF], derived from [START_REF] Bartoli | The 3d line motion matrix and alignment of line reconstructions[END_REF]. This framework provides a linear projection, that can easily be used as input for a linear least squares pose solver.

C. Pose estimation with a known vertical direction

Let l i = (l i1 , l i2 , l i3 ) T be the 2D projection of the 3D line

L i = (L i1 , L i2 , L i3 , L i4 , L i5 , L i6
) T , l i and L i being the i th element of a n pairs of lines dataset. We have

L i = (U T i , V T i ) T .
Pose estimation through P3L requires solving a 6DoF problem, i.e estimation three rotations ρ, θ , ψ and three translations t x , t y , and t z . With a prior knowledge of the upvector, two of the three rotation are known, reducing the problem to a 4 DoF problem.

By definition, L C i lies on the projection plane of the 2D line l i , leading to the constraint

(R CW .V W i ) T .l C i = 0, (7)
where R CW is the rotation matrix composed with the remaining rotation to be determined, such as

R CW = R z .R y .R x =   c z -s z 0 s z c z 0 0 0 1   .R y .R x (8)
where c z = cos(ψ) and s z = sin(ψ). With a known up-vector, only the two unknowns c z and s z remain. This means that two line correspondences are required to obtain a unique solution for the complete rotation. In a noiseless case, we directly obtain R CW . However, real data can lead to a badly scaled rotation matrix R, which does not satisfy the trigonometric constraint. We enforce this constraint with a singular value decomposition of the rotation matrix R, such as

U.Σ.V * = R R CW = U.V * T (9)
In order to estimate the translation vector T CW , we reformulate [START_REF] Pascoe | Direct visual localisation and calibration for road vehicles in changing city environments[END_REF]

into l C [×] .P.L W = 0 3×1 . ( 10
)
For a single 2D/3D line correspondence, we obtain from (10) a rank deficient system of three equations [START_REF] Hartley | Multiple view geometry in computer vision[END_REF]. We need to stack the equations from at least three 2D/3D line correspondences to obtain a rank 3 linear system and recover the three remaining unknowns t x , t y and t z . We rearrange this linear system such as

M.T CW = N, (11)
each column of M being respectively expressed in term of t x , t y and t z . T CW is the translation matrix such as T CW = (t x ,t y ,t z ) T , and N contains all terms independent from t x , t y and t z . We finally solve (11) using a linear least squares method, i.e.

T CW = (M T .M) -1 M T .N. ( 12
)
We refer to this method as VPnL LS (Vertical Perspectiven-Line with Least Squares solver).

D. Gauss-Newton optimization for the rotation and the translation

In our PnL formulation, the rotation and translation are sequentially estimated. We propose rotation and translation refinement methods that can also be sequentially applied to the estimated pose.

Our PnL method relies on the known up-vector to estimate the complete pose. However, the up-vector is prone to sensor error. For this reason, our rotation optimization scheme refines the PnL-estimated rotation angle ψ, as well as the up vector.

Let A W and B W be two 3D points in the world frame defining the 3D line L W , and l C be its projection in the camera frame. V W and V C are respectively the 3D line direction in the world and camera frames. For a set of n lines, we express the constraint [START_REF] Dhome | Determination of the attitude of 3d objects from a single perspective view[END_REF] as a set of n functions f = ( f 1 , ..., f n ) depending on the three rotations ρ, θ and ψ defined as in [START_REF] Lepetit | Epnp: An accurate o (n) solution to the pnp problem[END_REF], so that

f i (R CW ) = (R CW .V W i ) T .l i = 0. ( 13
)
We can then derive this function for each rotation angle ρ, θ and ψ and form a jacobian matrix. We then proceed with a Gauss-Newton optimization scheme.

Any point included in the 3D line L W is included in the projection plane of l C . We express this constraint similarly to [START_REF] Mirzaei | Globally optimal pose estimation from line correspondences[END_REF] in a set of n functions g = (g 1 , ..., g n ) to refine the translation estimation in

g i (T CW ) = (R CW .A W i + T CW ) T .l i = 0. ( 14
)
We rely on this geometric constraint to compose a second jacobian matrix, which we use to optimize the translation. When coupled with our PnL algorithm, these two optimizations are referred as VPnL GN (GN stands for Gauss-Newton solver). We observed that both our optimizations converge for a very small number of iterations. For this reason, we empirically choose to stop the algorithms after 20 iterations.

E. Line pairing and outliers rejection with RANSAC1

Algorithm 1 presents the line pairing process with outliers rejection. First, a random 2D-3D pair of lines is set as quadratic system of equations such as

l i T .R CW .V j c 2 z + s 2 z = 1 . ( 15
)
Algorithm 1 RANSAC1 Pairing

l i = normalized 2D line, i = 1..n 4:
V j = normalized direction of the 3D line in Plücker coord., j = 1..m From all m.n 2D-3D pair, select one at index k 17:

Estimate R CW1 and R CW2 with a quadratic equation solver 18:

From R CW1 and R CW2 , estimate ε ε ε 1 and ε ε ε 2 ∀ possible pairs return line pairs for ε ε ε 2 < criterion Solving this quadratic system provides two rotation matrix R CW1 and R CW2 expressing rotation between the world frame to the camera frame. For each orientation proposition and all possible pairs of lines, we compute an error vector ε ε ε of size m.n.

We split ε ε ε 1 and ε ε ε 2 into (m.n)/6 quantiles, m.n being the number of unique 2D/3D line combinations, and observe the value of the first quantile. If this value is smaller than the given threshold, we assume that the pairs returning the smallest error are highly probable inliers, and stop the algorithm. If not, we iterate until this condition is met, or all line pairs have been examined. In the latter case, we still return the pairs that generated the smallest error over all iterations. Note here that the number of quantiles (m.n)/6 is found empirically.

IV. RESULTS

A. Synthetic data

In this section, we evaluate quantitatively the performance and robustness of the different steps of our method.

a) Synthetic dataset setup: For this synthetic dataset, we simulate a 640×480 pixel camera with 655 pixel of focal length and no radial or tangential distortion.

We generate a set of random 2D pairs of points on the image plane, each pair defining a 2D line being at least 70 pixel long. Each line is expressed in metric coordinates using the camera intrinsic parameters, and is then given a depth between 2 and 20 meters. We finally express the previously obtained 3D lines in the world frame {W } by applying the transform Tr CW = [R CW (ρ, θ , ψ)|T CW ] with ρ, θ , ψ being random rotations between 0 rad and 2π rad, and T CW being a random translation between 2 to 20m around the camera optical center.

In order to evaluate the robustness to noise, a normally distributed noise is added to the endpoints of 2D and 3D lines, displacing the points by respectively σ pixels and σ 3D mm. This modifies both the position and direction of the lines.

IMU sensors are prone to inaccuracies when measuring the vertical direction, with errors between 0.02°for high-end IMUs, to 0.5°for low-cost sensors. For a fair comparison with methods estimating the complete pose instead of relying on the vertical direction, we add a constant 0.5°noise to the upvector used as input for the PnL algorithms.

As for metrics, we measure the translation error, ie. the euclidean distance between the estimated location and the ground truth location. We also refer to rotation error as defined in [START_REF] Huang | Sensor pose estimation from multi-center cylindrical panoramas[END_REF], ie. R error = arccos((tr(R RT ) -1)/2), with R the ground truth rotation and R the estimated rotation. We consider the recall rate, ie. the number of correct pairs returned over the number of correct pairs possible for a given image, and the precision rate, ie. the number of correct pairs returned over the number of pairs returned.

We compare our PnL algorithm with optimization, VPnL GN, and without, VPnL LS, with the ASPnL algorithm from [START_REF] Xu | Pose estimation from line correspondences: A complete analysis and a series of solutions[END_REF], NPnLUpL and NPnLUpC from [START_REF] Horanyi | Multiview absolute pose using 3d-2d perspective line correspondences and vertical direction[END_REF]. While the two latter are designed for multiview pose estimation, they accept a single view configuration as input. ASPnL estimates the 6 DoF pose, and both our algorithms and NPnLUpL consider a 4 DoF problem.

NPnPUpL, NPnLUpC and ASPnL are known to be very sensitive to outliers, and do not include an outliers rejection scheme. For this reason, we compare our RANSAC1 algorithm to RANSAC3 and RLPnL Enull from [START_REF] Xu | Pose estimation from line correspondences: A complete analysis and a series of solutions[END_REF]. Because the two latter algorithms do not include feature pairing, we can only evaluate the performance against outliers. To do so, we introduce outliers into the synthetic dataset, from none to 60 %, with a fixed 2D noise of 5 pixels and an up-vector noise or 0.5 °. b) PnL algorithm evaluation: Fig. 3 show a noise robustness evaluation of our methods in comparison with ASPnL, LPnLUpL and LPnLUpC.

With our test setup, ASPnL performs better in the noiseless case, because it does not suffer the up-vector error. Otherwise, regarding translation error, our two PnL algorithm outperform all other considered algorithms. In term of rotation error, when confronted to 2D noise up to 20 pixels, our algorithms perform better at first, then similarly to NPnLUpL. In case of 3D noise, our two algorithms lead to similar results for noise up to 200 mm. With a higher Fig. 3: Rotation and translation errors in case of 2D and 3D noises. VPnL LS is our PnL formulation with a known vertical direction and a linear least squares solver only. VPnL GN is the same method with Gauss-Newton optimization on the complete pose. ASPnL refers to [START_REF] Xu | Pose estimation from line correspondences: A complete analysis and a series of solutions[END_REF] method, and NPnLUpL is the PnL linear formulation of [START_REF] Horanyi | Multiview absolute pose using 3d-2d perspective line correspondences and vertical direction[END_REF] operating on a singlecamera setup. Left column evaluates robustness against 2D noise, right column evaluates robustness against 3D noise.

3D noise, VPnL LS is outperformed by both LPnLUpL and LPnLUpC, but VPnL GN provides similar results than NPnLUpL.

In Fig. 4, we show how the number of lines impacts the pose estimation in terms of rotation and translation error. For a medium 2D or 3D noise and a number of line pairs between 4 to 40, both our algorithms outperform the others. Performance increase up to 20 lines, where accuracy increases slower.

Overall, we suggest a hybrid use of VPnL GN and VPnL LS.

When confronted to high 3D noise (more then 500mm), rotation refinement provides a more accurate rotation estimate, while our linear VPnL LS provides a more accurate translation.

In case of moderate 2D or 3D noise, the complete pose refinement VPnL GN performs better.

Finally, with low 2D or 3D noise, we suggest the use of VPnL LS. c) RANSAC1 as outliers removal module: Fig. 5 shows the robustness of our RANSAC1 algorithm to outliers in presence of moderate 2D or 3D noise and high noise on the up-vector. When combined with our two PnL methods, we observe a rotation and translation error similar to Fig. 3, when RANSAC3 and RLPnL Enull show resulst unsuitable for vehicle application. Moreover, as seen in I, our RANSAC1

B. Experiment on real data

In this section, we evaluate simultaneously our PnL method as well as our pairing/outliers rejection algorithms with real data using the KITTI dataset [START_REF] Geiger | Are we ready for autonomous driving? the kitti vision benchmark suite[END_REF]. The KITTI dataset provides images from high resolution color and grayscale stereo camera. The image acquisition is synchronized with a Velodyne HDL-64-E, providing a 3D point cloud. Additionally, the camera pose in the world frame is provided by an OXTS RT3003 inertial and GPS navigation system.

Among the various sequences in the odometry section of KITTI, we selected in the sequence 00 a challenging subsequence composed 50 images with two consecutive sharp turns, taking place in an urban environment, starting from image 1223 to image 1276. We recomposed a unique 3D point cloud for the entire sub-sequence using the GPS/IMU pose. The KITTI dataset is known to be prone to GPS/IMU errors for small sub-sequences, especially in challenging consecutive turns.

For this reason, we used the 3DTK toolkit [24], that took as input the GPS/IMU poses and the multiple point clouds of our sequence, and produced a concatenated point cloud with the estimated camera poses over the sequence.

To illustrate the KITTI ground truth inaccuracies on short sequences, we compared the last pose of our sequence in the KITTI dataset to the 3DTK corresponding pose. As seen in 6, we observed a 54cm translation error and a 5.40°error.

We extracted a set of 17 3D lines on this global point cloud, creating this way a map of 3D lines which our vehicle has to localize itself in. We used our RANSAC1 algorithm to match this 3D feature set to 2D lines manually extracted from all images of our sequence. We extracted between 4 to 7 lines per image. Each extracted 2D line has a correspondence in the 3D lines map. We finally localize the camera in our 3D lines map using our pairing and pose estimation algorithms.

Because of the vehicle turns in the sequence, all 3D lines do not have 2D correspondence on each image. This means that for each pose estimation, our pairing algorithm has to deal with 58% up to 76% outliers.

To illustrate the GPS/IMU pose errors, Fig. 6 compares the KITTI poses ground truth and the 3DTK poses, as well as the results of our PnL algorithm. We observe that the consecutive PnL-estimated poses are consistent with the 3DTK poses. It is to be noted that all the PnL-estimated poses are independently estimated, ie. no filter was applied over the trajectory. Results on table II show that our PnL method coupled with our pairing algorithm achieves a mean translation error of 16.1cm, with a mean rotation error of 0.51°, for a 100% precision rate and 100% recall rate. Over the entire sequence, we measure a mean runtime for the combined line matching, pose estimation and pose refinement of 13.5 ms.

V. CONCLUSION

In this paper, we presented and evaluated a line-based pose estimation algorithm and a pairing/outliers removal algorithm relying on a known vertical direction. Using these two combined algorithm, we are able to localize a car in 3D lines map of an urban environment with only a very small number of 2D and 3D feature correspondences, and from a single image. Over a challenging sequence of the KITTI dataset, where the GPS/IMU provided ground truth poses with inaccuracies up to 54cm and 5°, we located successfully our vehicle within 16cm and with 0.5°error.

With a prior acquisition of a 3D feature map, our algorithms can be used to rapidly relocate a vehicle where a GNSS/IMU localization operates in degraded mode because of signal loss.

Fig. 2 :

 2 Fig. 2: 3D line and its projection onto the image plane. Points A and B form the 3D line L in the world frame {W}. 2D line l in the coordinate system {C} lies at the intersection of the projection plane and the image plane Π and is defined by the normal u of the projection plane.

ε 1 = first quantile of ε ε ε 1 12 :ε 2 = first quantile of ε ε ε 2

 11222 1 = error vector such as l i T .R CW1 .V j , length m.n 10: ε ε ε 2 = second error vector, using R CW2 11: for k = 1..m.n do 16:

19 :

 19 Split each error vectorε ε ε 1 and ε ε ε 2 into mif (min(ε 1 , ε 2 ) < criterion then 22: if min(ε 1 , ε 2 ) = ε 1 then 23:return line pairs for ε ε ε 1 < criterion 24:else 25:

Fig. 6 :

 6 Fig. 6: Comparison between PnL-estimated trajectory and ground truth

TABLE I :

 I Runtime results on a simulation dataset

	Method	Runtime (ms)
	VPnL LS	1.9
	VPnL GN	3.6
	RLPnL Enull	2.1
	RANSAC3	45.8


Joannes Falade
email: joannes.falade@ingroupe.com

Sandra Cremer
email: sandra.cremer@ingroupe.com

Christophe Rosenberger
email: christophe.rosenberger@ensicaen.fr

Keywords: Biometrics, matching, indexing, fingerprint, minutiae, quality metrics, NFIQ2, Locality-sensitive Hashing

Identifying individuals from average quality fingerprint reference templates, when the best do not provide the best results !

I. INTRODUCTION

Biometric systems use physical and behavioral characteristics (fingerprint, face, iris, hand shape, signature, gait) for people recognition. Fingerprint biometric systems are most used due to their properties as uniqueness, permanency and its ease of acquisition for government or private applications. For example, Aadhaar is the most large biometric database project (1 billion subjects) built for indian people recognition. For people administrative document as passport, ID card and residence permit, fingerprint is used as the biometric modality to identify the owner. Also, for migration issues and security against terrorist attack, many gorvernments use fingerprint databases to enhance the security of border control. For people recognition and in order to deliver authentic administrative papers, these large databases need to be searched on time. In general, biometric fingerprint systems have three functions: enrollment, authentication (verification) and identification.

The enrollment is the basic record function for all biometric systems. It consists in representing an individual uniquely by its fingerprint reference template that is associated with a corresponding identifier (name, pin, id) in the database. It is the first step of feature extraction and registration in database. The reference template must contain all the most reliable and discriminant information about each enrolled individual. That is why the quality of the reference template needs to be controled before the next two critical recognition tasks of authentication and indentification.

Authentication systems use an input fingerprint (sample or query) to verify or confirm if a person is what he/she claims to be. Thus, the verification task has to achieve comparison between an input fingerprint and some reference templates in the database; that's called 1-1 matching. Matching algorithms are evaluated through two performance metrics FAR and FRR. The FAR means False Acceptance Rate and FRR denotes False Reject Rate. The ultimate aim in the best case is to have both F AR → 0 and F RR → 0; that is called the compromise of verification systems because when one of these metrics tends to 0 the second pulls away from 0.

Identification systems use an input fingerprint and have as objective to check if the person has already been enrolled in the database before under another id. In this case, the system attempts to give result after searching against all entries of database. This identification task can be viewed as many successives verfications; That's why identification is called one-to-many or 1-N matching. In this case, identification requires a long computation time to carry out the verifications against all database entries. It has also been shown that doing all comparisons induces a high error rate in identification system performance [START_REF] Asker | Indexing Fingerprint Databases Based on Multiple Features[END_REF]. Figure 1 describes all steps for different fingerprint systems functions. So, classification, clustering and indexing strategies are important to achieve such an identification task in limited time. The first objective is to split and filter a large database in order to reduce the list of candidates to verify and the global identification time. This is measured by the Penetration Rate that we denote P R. P R = #candidates * 100 database Size . The second objective is to avoid decreasing identification precision because of the candidate preselection step. This is measured by the Hit Rate we denote HR. HR = #W ell Identif ied Candidates * 100 #T otal Identif ied Candidates T est . The ultimate aim in the best case is to have P R → 1 and HR → 100 . That's called the compromise of identification systems because when P R → 1, HR declines quickly. The major challenge for fingerprint identification comes from the difficulty of achieving both high speed and accuracy in recognition systems.

Many algorithms are proposed in the state of the art in the last decade to solve such an identification problem. These algorithms have different recognition and computation time performances; they are also treated in different conditions on different databases. We also think that identification algorithms can be more efficient with the constraint of choosing the ideal fingerprint reference for the step of enrollment. Notice that for the same finger, the image capture is different for each sample acquired several times. This is the real problem of enrollment because there is not unique features model for a fingerprint. In this paper, we provide a comparative study of different enrollment strategies with indexing algorithms for identification from the literature. We evaluate the impact of choosing the quality of the reference template during the enrollment process on performance for the identification step. We build a benchmark to compare all methods on the same databases and with the same conditions. To our knowledge, such a comparative study of fingerprint quality enrollment has never been proposed in the literature, this is our main contribution to an important topic in biometrics.

The rest of this paper is organized as follow. First, in section 2, we present various state of the art fingerprint indexing strategies for identification. Next, in section 3 we describe the proposed experimental protocol. We use the NFIQ 2.0 quality metric to change the quality of the reference template during the enrollment and finally provide comparisons between them. We conclude in section 4 and give an overview of perspectives.

II. RELATED WORKS

For any biometric task, fingerprint features are extracted according two major strategies: one uses global features and the other one local features. For global features, fingerprint representation uses the core point, ridges shapes, ridges number and singular points such as delta and loop [START_REF] Maltoni | Handbook of Fingerprint Recognition[END_REF]. On the other hand, local features use fingerprint minutiae points and pores in order to build different solid geometrics or neighborhood features (Triplets, quadruplets, circle or cylinder arround minutiae) [START_REF] Asker | Indexing Fingerprint Databases Based on Multiple Features[END_REF].

The problem we discuss in this work is about the impact of the quality of the reference template in order to get the best accuracy performance for identification task on large databases. In general, databases can be searched in three ways: classification, clustering and indexing. Classification is a supervised learning approach while clustering is known as unsupervised classification. Indexing techniques combine classification rules and clustering to realize a quick search in a database by assigning an index number to each database entry. In the special case of fingerprint database searching, classification and indexing are the most used methods to quickly search for a candidates list from a query fingerprint.

The basic method for the identification task is also known as the naive method in the state of the art. It consists in considering identification as multiple and successive verifications. For an input fingerprint I in the database, we try all possible verifications against all M fingerprints and return the best scores as candidates list. This method is experimental method and deterministic. The performance of identification is dependent of the matching algorithm that is used. The Cascade method is a combination of successive naive methods. This method relies on using the advantages of many matchers to search the database. In general, the cascade is formed by the fastest matching algorithms at the front and the most accurate algorithm at the end. Indeed, generally the faster is a matching algorithm, the less precise it is. So, the cascade is built so as to quickly do a first preselection thank to the fastest algorithm and finish with the most precise algorithm for the last selection step.

The first classification method has been proposed by Henry [START_REF] Henry | Classification and uses of finger prints[END_REF] in 1900 based on global fingerprint feature extraction. He proposed to organize large fingerprint databases in five classes: right loop, left loop, whorl, arch and tented arch. In [START_REF] Jain | A multichannel approach to fingerprint classification[END_REF], Jain and al. also proposed a five-class based classification like Henry and tried to improve P R and HR. To achieve this performance, authors used the core point and built 48 areas of interest arround this point. They added a KNN algorithm to build the five classes so as to maximise the accuracy of the method. In [START_REF] Zhang | Fingerprint classification based on extraction and analysis of singularities and pseudoridges[END_REF], [START_REF] Najim | Classification of Fingerprint Images Using Neural Networks Technique[END_REF], [START_REF] Karu | Fingerprint Classification[END_REF], other authors worked on fingerprint database classification on five or at most nine classes. However, exclusive classification into five fingerprint classes is limited by the short number of classes and the fact that fingerprints are not uniformly distributed under the five classes defined by Henry. So, P R and HR can be improved and that is why fingerprint indexing methods are also most used in recent works.

Although, these new methods use an indexing strategy, they focus more on the representation of fingerprint as a discriminating factor. In [START_REF] Parmar | Fingerprint Indexing Approches for Biometric Database: A Review[END_REF], Parmar and Degadwala present a good, and useful review of fingerprint indexing methods. In [START_REF] Iloanusi | Indexing fingerprints using minutiae quadruplets[END_REF] Iloanusi and al. present an original indexing fingerprint method based on minutiae quadruplets. Javad and Ali Khodadoust [START_REF] Khodadoust | Fingerprint indexing based on minutiae pairs and convex core point[END_REF], [START_REF] Khodadoust | Fingerprint indexing based on expanded Delaunay triangulation[END_REF] use an expanded Delaunay triangulation (triplets) and minutiae pairs to build an interesting fingerprint indexing method with good performances. Wei Zhou and al. [START_REF] Zhou | Fingerprint indexing based on combination of novel minutiae triplet features[END_REF] present a new triplets scheme for fingerprint while in [START_REF] Cappelli | Candidate List Reduction Based on the analysis of Fingerprint Indexing Scores[END_REF] authors propose a global method to reduce possible fingerprint index candidates lists. Notice that indexing methods are in general based on local features due to the fact that they are more discriminant. In [START_REF] Gyaourova | A Novel Coding Scheme for Indexing Fingerprint Patterns[END_REF], Arun Ross introduced an indexing method that we called the matcher method for fingerprint indexing. This method proposes to build a new input fingerprint index by using other specific fingerprints as references. Fingerprints which produce a high matching score variance against other fingers are chosen as references (discriminative property). These n reference fingerprints are very few (at most 5%of M ). Then, Minutiae Cylinder Code (MCC) is a binary representation of a fingerprint proposed by Capelli et al. in [START_REF] Cappelli | BMinutia Cylinder-Code: A New Representation and Matching Technique for Fingerprint Recognition[END_REF]. Authors describe each minutiae of a fingerprint as a discretized cylinder which represents its neighborhood. In [START_REF] Cappelli | Short papers: Fingerprint Indexing Based on Minutia Cylinder-Code[END_REF], an usefull algorithm is proposed for indexing MCC fingerprint during identification task.

III. EXPERIMENTAL PROTOCOL

We detail the protocol we used in this comparative study.

A. NFIQ2 Metric and enrollment

In this part, we present the fingerprint quality metric that we use for our experimental study about fingerprint quality during enrollment. The metric we use is NFIQ2 which all details is reported in [START_REF]NFIQ 2.0 NIST Fingerprint Image Quality[END_REF]. NFIQ2 is the second version of NFIQ. NFIQ2 is a software developed by the NIST for fingerprint quality measurement. For each fingerprint, NFIQ2 returns a value between 0 and 100 related to the fingerprint quality. So, in order to compare two samples of fingerprint, notice that the best is the sample whith the high value of NFIQ2. NFIQ2 sometimes returns value 255 of out range when the quality cannot be computed for a fingerprint.

During our study, we simulate three scenarios during enrollment which are the Max, Mean and Min quality enrollment. As we explain in the section below, enrollment is the step of the first registration of an individual fingerprint in the database. We suppose that for this step, we get more than one sample of fingerprint of an individu and we have to choose one of these samples as reference. We compute the quality for all samples and save the best quality of NFIQ2 as enrollment reference in Max case. In Min case, we choose the worst quality of NFIQ2 as reference. About the Mean case, the image that we choose as reference is the NFIQ2 value of fingerprint which is the closest to the mean of all samples of fingerprint from an individual. In general, let's note that Max is the best case value of NFIQ2, the Mean is the Mean or normal case and Min is the worst case.

B. Fingerprint Databases

For experimental databases, we consider nine fingerprint databases from FVC competition. FVC 2000: DB1, DB2, DB3, DB4; 2002: DB1, DB4 and 2004: DB2, DB3, DB4. These fingerprint databases are the most used for testing, and particularly the 2004 databases which are considered as the most difficult ones. Each database contains 100 individuals and 8 samples per individual. We use one sample (using NFIQ2 metric) for enrollment and the other seven as query fingerprints.

C. Identification algorithms and parameters

Here, we present the algorithms we use for fingerprint identification in order to test our enrollment experiment protocol. We also present the set up of important parameters for each identification method as described in [START_REF] Falade | Comparative study of fingerprint database indexing methods[END_REF] in a previous comparative study of identification algorithms. To evaluate the impact of reference choice in identification performance, we consider four identification strategies from the state of the art that we have implemented. The identification methods we use are, the basic method, the cascade method, the matcher method and the MCC method. For each of these methods, we vary the three strategies of enrollment of individuals and we compute the CMC curve corresponding of each strategy to evaluate the differences. For the basic, cascade and matcher identification methods, it is important to note that they are dependent on the matching algorithms. In table I, we show different matching algorithms we use to perform experimentations. We use four algorithms and notice that the higher the algorithm number is, the more accurate it is and the slower it is to perform a fingerprint comparison. Algo4 has better accuracy than algo3 but is slower. In order to get a high bound limit for accuracy, we use algo4 for the basic method. Figure 2 shows that we use algo1, algo2, algo4 for cascade chain in our test. We use algo3 for matcher method and the rest of parameters to be fixed are written in table II. As MCC method does not depend on any matching algorithm for preselection, we set up MCC indexing method parameters in table III.

D. Experimental results

The principle of the proposed method is to evaluate the impact of enrollment on the identification task. Although we did our experience on all FVC database, we decide to show and explain our result against FVC2000DB3 which generalize our observation and can be see as difficult database also. We also used the same minutiae extractor to build all fingerprint templates. We ensure this hypothesis because, in a previous study, we observed that the minutiae detection step is very important and discriminant for matching algorithms performance. For all databases, we compute CMC (Cumulative Matches Curve), a curve that combines P R and HR as the main performance tool for indexing methods. For implementation, we use C++ with no particular programming optimization. Our computers run with windows 7 on 2.60 GHz Intel core i7. Figures 3,[START_REF] Cappelli | Short papers: Fingerprint Indexing Based on Minutia Cylinder-Code[END_REF][START_REF] Falade | Comparative study of fingerprint database indexing methods[END_REF][START_REF] Gyaourova | A Novel Coding Scheme for Indexing Fingerprint Patterns[END_REF] show the performance of different identification methods against the same database FVC2000DB3 and the variations of the curves corresponds to each enrollment strategy. By observing the figures, we notice globally that the basic method and cascade have the best performance. These methods as we explain in [START_REF] Falade | Comparative study of fingerprint database indexing methods[END_REF] are theoretical, consume long time and unsusable for large databases. Mcc indexing is known as better than Matcher method which is considered as the worst approach. Figures 3,[START_REF] Cappelli | Short papers: Fingerprint Indexing Based on Minutia Cylinder-Code[END_REF] show that the better is the fingerprint reference, the better is the identification by using cascade method or basic because the green curve is the higher; the next is the orange and last is red which respectively represent the Max, Mean and Min reference enrollment. The reason of this is also that we compute at first all possibles (1-1) matching to make identification decision and our matching algorithms have a good recognition performance. These methods are totally deterministic. When we use a non deterministic method for identification like matcher and Mcc methods, the idea is to be more flexible, tolerate errors slightly on the enrollment step in order to be general during the identification task (general and probabilistic idea of the machine learning concept). Based on this concept, enrollment of the best reference as possible can be seen as overfitting. This situation can cause a trouble during identification step and especially if the identification method depends on global 1-1 matching result like the matcher method. The enrollment of the worst reference quality can cause the inverse of overfitting which is known as underfitting. So, to avoid this situation the mean enrollment strategy seems to be a good alternative as it can be seen on Figure 5 for the matcher method. As, the Mcc method is a very discriminative fingerprint representation, the CMC curve of identification shows a good robustness. But, our theory remains valid, because beyond a penetration rate of P R > 30%, the CMC curve of Max, Min and Mean enrollment strategies are the same. Otherwise, using the strategy of Mean for reference enrollment is the best way in fact, the orange curve is the best when P R < 30% (see Figure 6). This paper provides a study on the impact of the quality of reference templates on four fingerprint identification methods. We use a well known fingerprint quality assessment metric NFIQ2 to realize a real, simple and reproductible experiments on the effect of fingerprint quality enrollment during the identification step. In practice, for industrial purposes, the best sample reference is chosen but our study reveals that it is usefull sometimes to get the Mean quality for reference enrollment; that is to say not try to choose the best quality of fingerprint as reference and avoid also a very worst quality as reference too. Sometimes, the identification method which uses local representation of the fingerprint like Mcc seems to be more robust against enrollment variation. This study is done on FVC databases and our tests consist in identifying an input fingerprint in population of 100 people.

Perspectives of this study are to work on more large databases. We will realize tests on databases of ten million fingerprints generated by the Sfinge synthetic fingerprint. We are also continuing to implement other fingerprint identification methods from state of the art in order to provide a complete effect of enrollment strategies on identification methods. This will be helpful for all to get a real idea of state of the art and also to fix and correct the general idea of getting the best fingerprint quality as reference for identification.

Figure 1 .

 1 Figure 1. Three functions of enrollment, authentication and identification of fingerprint systems.

Figure 2 .

 2 Figure 2. Cascade chain for fingerprint identification

Figure 3 .

 3 Figure 3. CMC identification performance on database FVC2000DB3 using basic method

Figure 4 .

 4 Figure 4. CMC identification performance on database FVC2000DB3 using Cascade method

Figure 6 .

 6 Figure 6. CMC identification performance on database FVC2000DB3 using MCC method

  Table I PERFORMANCE EVALUATION OF USED MATCHING ALGORITHMS IN TERMS OF FAR, FRR AND COMPARISON TIME. NOTE THAT FAR VALUE IS EQUAL TO FRR IN OUR CASE IS EXPRESSED BECAUSE WE USE THE EQUAL ERROR RATE(EER) FOR OUR ALGORITHM. t i IN MILLISECOND. NOTE THAT algo3 IS MCC 1-1 MATCHING ALGORITHM. THE OTHER ALGORITHMS ARE COMMERCIAL ONES.

	Matcher algo1	algo2	algo3	algo4
	FAR	0.095	0.078	0.04	10 -2
	FRR	0.095	0.078	0.04	10 -2
	t i	10 -1	1.1 * 10 -1	2 * 10 -1	6.7 * 10 -1
		Matcher method Values
			b	2
			n	5
			d	3
			Table II
	MATCHER INDEXING METHOD PARAMETERS WE USE FOR
	IMPLEMENTATION. THESE VALUES ARE CHOSEN BY READING [6] AND
	SET UP THE BEST PARAMETERS FOR OUR MCC MATCHING ALGORITHM
	WHICH IS THE BEST SUITED FOR THIS INDEXING STRATEGY.
		MCC parameters Values
			n	1532
			h	32
			l	25
			min	5
			Table III
	MCC INDEXING PARAMETERS WE USE FOR IMPLEMENTATION. THE
	min VALUE IS USED IN THE MCC INDEXING ALGORITHM TO CREATE
	REJECT AND AVOID CONSIDERING MINUTIAE WITH TOO MUCH ZERO BY
			HASHING FUNCTION.


Angela A Felisia

Jan Chiarello

Paola Friedrich

Simone Goatin

Oliver Göttlich

Kolb

F A Chiarello

J Friedrich

P Goatin

S Göttlich

O Kolb

A non-local traffic flow

Keywords: non-local scalar conservation laws, upwind scheme, macroscopic traffic flow models on networks AMS subject classifications: 35L65, 65M12, 90B20

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

Introduction

In recent years, conservation laws with non-local flux gained growing attention for a wide field of applications. Indeed, they turned out to be suitable to describe several phenomena: flux functions depending on space-integrals of the unknown appear for example in models for sedimentation [START_REF] Betancourt | On nonlocal conservation laws modelling sedimentation[END_REF], supply chains [START_REF] Gröschel | Regularity theory and adjoint-based optimality conditions for a nonlinear transport equation with nonlocal velocity[END_REF], conveyor belts [START_REF] Göttlich | Modeling, simulation and validation of material flow on conveyor belts[END_REF], crowd motions [START_REF] Colombo | A class of nonlocal models for pedestrian traffic[END_REF] and traffic flows [START_REF] Chiarello | Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel[END_REF][START_REF] Chiarello | Non-local multi-class traffic flow models[END_REF][START_REF] Friedrich | A Godunov type scheme for a class of LWR traffic flow models with non-local flux[END_REF]. For this type of equations, general existence and uniqueness results have been established in [START_REF] Amorim | On the numerical integration of scalar nonlocal conservation laws[END_REF][START_REF] Chiarello | Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel[END_REF][START_REF] Friedrich | A Godunov type scheme for a class of LWR traffic flow models with non-local flux[END_REF] for specific classes of scalar equations in one space-dimension, and in [START_REF] Aggarwal | Nonlocal systems of conservation laws in several space dimensions[END_REF][START_REF] Chiarello | Non-local multi-class traffic flow models[END_REF] for systems of equations coupled through the non-local term.

In this paper, we propose a one-dimensional scalar model, arising in traffic flow modeling. The main difference with respect to the above mentioned literature is that the flux function may involve different velocity functions on different parts of the road. The model focuses on a non-local mean downstream velocity and can therefore describe the behavior of drivers on two stretches of a road with different velocities and capacities, without violating the maximal density constraint on each road segment. Hence, we are modelling a 1-to-1 junction and this model can be seen as a first step towards a network formulation for traffic flow models with non-local flux. Note that the work [START_REF] Shen | Traveling Waves for Conservation Laws with Nonlocal Flux for Traffic Flow on Rough Roads[END_REF] is dealing with a similar setting, where the author studies traveling waves profiles of conservation laws with non-local flux functions, describing traffic flow on a road with just different maximum velocities. We approximate the solution using an adapted Godunov or rather upwind type scheme. Deriving several properties of the scheme and relying on a Kružkov type entropy condition, we are able to prove the well-posedness of the model.

Since it is still an open question whether the solution of the non-local model tends to the solution of the corresponding local equation when the support of the kernel function tends to zero, see for example [START_REF] Colombo | Recent results on the singular local limit for nonlocal conservation laws[END_REF] for an overview, we investigate this issue only from the numerical point of view.

The paper is organized as follows: In Section 2, we present our model and the main result of this work. In Section 3, we prove the Lipschitz continuous dependence of weak entropy solutions with respect to the initial data, which implies their uniqueness. In Section 4, we introduce an adapted upwind type scheme and derive important properties: the maximum principle, uniform total variation (BV) estimates and a discrete entropy inequality. Afterwards, we prove the convergence of the scheme and the main theorem in Section 5. In the last section, we present some numerical simulations of our non-local junction model and we numerically investigate the behaviour of the corresponding solutions as the support of the kernel function tends to zero.

Modeling

Based on the model presented in [START_REF] Friedrich | A Godunov type scheme for a class of LWR traffic flow models with non-local flux[END_REF] we consider the following conservation law

∂ t ρ(t, x) + ∂ x f (t, x, ρ) = 0, x ∈ R, t > 0, (2.1)
where f (t, x, ρ) := ρ(t, x)V 1 (t, x) + g(ρ(t, x))V 2 (t, x), (2.2) with g(ρ) := min{ρ, ρ 2 max }, (2.3)

V 1 (t, x) := min{x+η,0} min{x,0}
v 1 (ρ(t, y))ω η (y -x)dy, (2.4)

V 2 (t, x) := max{x+η,0} max{x,0} v 2 (ρ(t, y))ω η (y -x)dy, (2.5)
for any η > 0. We couple the equation (2.1) with the initial datum

ρ(0, x) = ρ 0 (x) ∈ BV(R), s.t. ρ 0 (x) ∈ [0, ρ 1 max ] for x < 0 and ρ 0 (x) ∈ [0, ρ 2 max ] for x > 0.
(2.6)

The model assumes that drivers adapt their speed based on a weighted mean of downstream velocities. In the considered setting, changes in road characteristics at x = 0 may translate in different velocity functions, v 1 and v 2 , and in different road capacities, ρ 1 max and ρ 2 max , for x < 0 and x > 0 respectively. An example of such a situation on a road is illustrated in Figure 1. Here, different maximum capacities and different velocity functions can be "seen" by the

ρ 1 max ρ 2 max η > 0 ρ x v 1 v 2
Figure 1: Illustration of cars on a road with different parameters on each segment. The grey area represents the non-local traffic downstream information of the car in light grey. In this model, this car slows down in advance with respect to the density of cars in front of it. The illustration describes a microscopic evolution but can also be used for our macroscopic model.

car in light grey. In (2.2), the flux also accounts for the maximum capacity of the second road segment.

The special structure of the flux function (2.2) does not fit into the framework proposed in e.g. [START_REF] Chiarello | Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel[END_REF][START_REF] Friedrich | A Godunov type scheme for a class of LWR traffic flow models with non-local flux[END_REF][START_REF] Keimer | Existence, uniqueness and regularity results on nonlocal balance laws[END_REF]. Only for v 1 ≡ v 2 and therefore ρ 1 max = ρ 2 max the model coincides with the one presented in [START_REF] Friedrich | A Godunov type scheme for a class of LWR traffic flow models with non-local flux[END_REF]. Therefore, we have to investigate its well-posedness in the general case.

We impose the following reasonable hypotheses on v i , i ∈ {1, 2} and ω η :

v i ∈ C 2 ([0, ρ i max ]; R + ) : v i ≤ 0, v i (ρ i max ) = 0, ω η ∈ C 1 ([0, η]; R + ) : ω η ≤ 0, η 0 ω η (x)dx = 1 ∀η > 0, (2.7)
where η represents the look-ahead distance of the drivers. Since the flux function (2.2) is continuous in x, entropy weak solutions of (2.1), (2.6) are intended in the following way: Definition 1 (Entropy weak solution (see [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF])). A measurable function

ρ ∈ C([0, +∞); L 1 (R)) : [0, +∞) × R → [0, max{ρ 1 max , ρ 2 max }]
is an entropy weak solution of the initial value problem (2.1)-(2.6) if for any test function

ϕ ∈ C 1 c ([0, +∞) × R; R + ) and for any constant c ∈ R, ∞ 0 R |ρ -c|∂ t ϕ + sgn(ρ -c)(f (t, x, ρ) -f (t, x, c))∂ x ϕ -sgn(ρ -c)∂ x f (t, x, c)ϕ dxdt + ∞ -∞ |ρ 0 (x) -c|ϕ(0, x) dx ≥ 0.
(2.8)

Remark 1. We note that the entropy condition is essential to obtain the uniqueness of solutions in the framework of our proof 's technique. Under suitable assumptions, an alternative could be to follow the approach considered in [START_REF] Keimer | Existence, uniqueness and regularity results on nonlocal balance laws[END_REF], where the uniqueness is obtained as a consequence of the Banach fixed point theorem and therefore no entropy condition is needed.

The main result of this paper is the following theorem:

Theorem 1. Let ρ 0 ∈ BV(R; [0, max{ρ 1 max , ρ 2 max }]
) such that ρ 0 (x) ≤ ρ 1 max for x < 0 and ρ 0 (x) ≤ ρ 2 max for x ≥ 0, and hypotheses (2.7) hold. Then the Cauchy problem

∂ t ρ(t, x) + ∂ x f (t, x, ρ) = 0, x ∈ R, t > 0, ρ(0, x) = ρ 0 (x), x ∈ R,
admits a unique entropy weak solution in the sense of Definition 1 and

0 ≤ ρ(t, x) ≤ ρ 1 max for a.e. x < 0, t > 0, 0 ≤ ρ(t, x) ≤ ρ 2 max
for a.e. x ≥ 0, t > 0.

(2.9)

Theorem 1 is proved at the end of Section 5.

Uniqueness

Let us start to prove the Lipschitz continuous dependence of weak entropy solutions with respect to the initial data, which ensures the uniqueness of entropy solutions of the model (2.1)-(2.6). We follow [START_REF] Blandin | Well-posedness of a conservation law with non-local flux arising in traffic flow modeling[END_REF][START_REF] Chiarello | Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel[END_REF][START_REF] Friedrich | A Godunov type scheme for a class of LWR traffic flow models with non-local flux[END_REF], using Kružkov's doubling of variables technique [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF].

Theorem 2. Under hypotheses (2.7), let ρ and ρ be two entropy solutions of (2.1) with initial datum ρ 0 and ρ0 , respectively. Then, for any T > 0, there holds

ρ(t, •) -ρ(t, •) L 1 ≤ exp(KT ) ρ 0 -ρ0 L 1 ∀t ∈ [0, T ], (3.1)
with K given by (3.7).

Proof. The functions ρ and ρ are weak entropy solutions of

∂ t ρ(t, x) + ∂ x ρ(t, x)V 1 (t, x) + g(ρ)V 2 (t, x) = 0, ρ(0, x) = ρ 0 (x), ∂ t ρ(t, x) + ∂ x ρ(t, x) Ṽ1 (t, x) + g(ρ) Ṽ2 (t, x) = 0, ρ(0, x) = ρ0 (x),
respectively. V i , Ṽi for i = 1, 2 are defined as in (2.4) and (2.5), where the convolution is computed over the velocity of ρ and ρ, respectively. They are bounded measurable functions and Lipschitz continuous w.r.t.

x since ρ, ρ ∈ (L ∞ ∩ BV) (R + × R; R).
Using the classical doubling of variables technique, see [START_REF] Karlsen | On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients[END_REF][START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF], we get the following inequality:

ρ(t, •) -ρ(t, •) L 1 ≤ ρ 0 -ρ0 L 1 + T 0 R ∂ x ρ(t, x) V 1 (t, x) -Ṽ1 (t, x) dxdt + T 0 R ∂ x ρ(t, x) V 2 (t, x) -Ṽ2 (t, x) dxdt (3.2) + T 0 R |ρ| ∂ x V 1 -∂ x Ṽ1 dxdt + T 0 R g(ρ) ∂ x V 2 -∂ x Ṽ2 dxdt,
where ∂ x ρ must be understood in the sense of measures. Applying the mean value theorem and using the properties of the kernel function, we deduce

V i (t, x) -Ṽi (t, x) ≤ ω η (0) v i ∞ ρ(t, •) -ρ(t, •) L 1 , for i = 1, 2. (3.3)
Using the Leibniz integral rule and again the mean value theorem, we can also obtain for a.e.

x ∈ R

∂ x V 1 (t, x) -∂ x Ṽ1 (t, x) =                          0, if x > 0, 0 x (v 1 (ρ(t, y)) -v 1 (ρ(t, y)))ω η (y -x)dy + v 1 (ρ(t, x)) -v 1 (ρ(t, x)) ω η (0) , if -η < x < 0, x+η x (v 1 (ρ(t, y)) -v 1 (ρ(t, y)))ω η (y -x)dy -v 1 (ρ(t, x + η)) -v 1 (ρ(t, x + η)) ω η (η) + v 1 (ρ(t, x)) -v 1 (ρ(t, x)) ω η (0) , if x < -η ≤ ω η ∞ v 1 ∞ ρ(t, •) -ρ(t, •) L 1 (3.4) + ω η (0) v 1 ∞ |ρ -ρ|(t, x + η) + |ρ -ρ|(t, x) .
Similarly, we obtain

∂ x V 2 (t, x) -∂ x Ṽ2 (t, x) ≤ ω η ∞ v 2 ∞ ρ(t, •) -ρ(t, •) L 1 (3.5) + ω η (0) v 2 ∞ |ρ -ρ|(t, x + η) + |ρ -ρ|(t, x) . (3.6)
Plugging (3.3), (3.4), (3.5) into (3.2), we obtain

ρ(t, •) -ρ(t, •) L 1 ≤ ρ 0 -ρ0 L 1 + max i=1,2 v i ∞ T 0 ρ(t, •) -ρ(t, •) L 1 dt 2ω η (0) sup t∈[0,T ] ρ(t, •) BV(R) + ω η ∞ sup t∈[0,T ] ρ(t, •) L 1 + sup t∈[0,T ] g(ρ(t, •)) L 1   + max i=1,2 v i ∞ ω η (0) sup t∈[0,T ] ρ(t, •) ∞ + sup t∈[0,T ] g(ρ(t, •)) ∞ T 0 R |ρ -ρ|(t, x + η) + |ρ -ρ|(t, x) dxdt ≤ ρ 0 -ρ0 L 1 + K T 0 ρ(t, •) -ρ(t, •) L 1 dt, with K := max i=1,2 v i ∞ 2ω η (0) sup t∈[0,T ] ρ(t, •) BV(R) + ω η ∞ sup t∈[0,T ] ρ(t, •) L 1 + sup t∈[0,T ] g(ρ(t, •)) L 1 +2ω η (0) sup t∈[0,T ] ρ(t, •) ∞ + sup t∈[0,T ] g(ρ(t, •)) ∞   . (3.7)
By Gronwall's lemma we get the thesis and for ρ 0 = ρ0 the uniqueness of entropy solutions.

Remark 2. Note that we cannot directly apply previous results in the literature [START_REF] Chiarello | Stability estimates for non-local scalar conservation laws[END_REF][START_REF] Colombo | Stability and total variation estimates on general scalar balance laws[END_REF][START_REF] Karlsen | On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients[END_REF] to the present model, because it does not fit precisely the assumptions therein. Moreover, direct computations allow to recover sharper estimates on the coefficients.

Numerical scheme

In order to prove the well-posedness of model (2.1)-(2.6), we prove the existence of solutions via a numerical scheme which is based on the scheme from [START_REF] Friedrich | A Godunov type scheme for a class of LWR traffic flow models with non-local flux[END_REF]. Even though this scheme has been introduced in [START_REF] Friedrich | A Godunov type scheme for a class of LWR traffic flow models with non-local flux[END_REF] as a Godunov type scheme, it reduces to an upwind type scheme.

For j ∈ Z and n ∈ N, let x j-1/2 = j∆x be the cell interfaces, x j = (j + 1/2)∆x the cells centers, corresponding to a space step ∆x such that η = N η ∆x for some N η ∈ N, and let t n = n∆t be the time mesh. In particular, x = x -1/2 = 0 is a cell interface. We aim at constructing a finite volume approximate solution

ρ ∆x such that ρ ∆x (t, x) = ρ n j for (t, x) ∈ [t n , t n+1 [ ×[x j-1/2 , x j+1/2 [.
To this end, we approximate the initial datum ρ 0 with the cell averages

ρ 0 j = 1 ∆x x j+1/2 x j-1/2 ρ 0 (x)dx, j ∈ Z.
Following [START_REF] Friedrich | A Godunov type scheme for a class of LWR traffic flow models with non-local flux[END_REF], we consider the numerical flux function

F n j+1/2 (ρ n j ) := ρ n j V 1,n j + g(ρ n j )V 2,n j (4.1) with V 1,n j = min{-j-2,Nη-1} k=0 γ k v 1 (ρ n j+k+1 ), V 2,n j = Nη-1 k=max{-j-1,0} γ k v 2 (ρ n j+k+1 ), (4.2)
γ k = (k+1)∆x k∆x ω η (x)dx, k = 0, . . . , N η -1, (4.3)
where we set, with some abuse of notation b k=a = 0 whenever b < a. In this way we can define the following finite volume numerical scheme

ρ n+1 j = ρ n j -λ F n j+1/2 (ρ n j ) -F n j-1/2 (ρ n j-1 ) with λ := ∆t ∆x . (4.4)
Note that, due to the accurate calculation of the integral in (4.3) and the definition of the convoluted velocities in (4.2), there holds

0 ≤ V 1,n j ≤ v 1 max , 0 ≤ V 2,n j ≤ v 2 max , 0 ≤ V 1,n j + V 2,n j ≤ max{v 1 max , v 2 max }, ∀ j ∈ Z, n ∈ N.
We set

v := max{ v 1 ∞ , v 2 ∞ }, v := max{ v 1 ∞ , v 2 ∞ }, ρ := max{ρ 1 max , ρ 2 max }
and consider the following CFL condition:

λ ≤ 1 γ 0 v ρ + v . (4.5)
We will show that, under this CFL condition, the numerical scheme (4.1)-(4.4) satisfies a maximum principle, uniform BV estimates and a discrete entropy inequality. Equipped with these properties, we will show in Section 5 that the sequence of approximate solutions ρ ∆x converges towards the entropy solution of (2.1)-(2.6). Note that, for v 1 ≡ v 2 , the scheme (4.1)-(4.4) coincides with the scheme in [START_REF] Friedrich | A Godunov type scheme for a class of LWR traffic flow models with non-local flux[END_REF].

In the following proofs, we will omit the dependence on n of the flux function and the velocity whenever possible, in order to simplify the notation.

Maximum principle

The solutions generated by the numerical scheme (4.4) stay always positive and they are bounded by the maximum road capacity of each road segment as stated by the following lemma.

Lemma 1. Under hypothesis (2.6) and the CFL condition (4.5), the sequence generated by the numerical scheme (4.1)-(4.4) satisfies the following maximum principle:

0 ≤ ρ n j ≤ ρ 1 max for j ≤ -1 and 0 ≤ ρ n j ≤ ρ 2 max for j ≥ 0, ∀n ∈ N.
Proof. We start by showing the positivity. We directly obtain

ρ n+1 j = ρ n j -λ F n j+ 1 2 (ρ n j ) -F n j-1 2 (ρ n j-1 ) ≥ ρ n j -λF n j+ 1 2 (ρ n j ) ≥ ρ n j -λ v ρ n j ≥ 0.
Here we used the CFL condition (4.5) and g(ρ n j ) ≤ ρ n j . The rest of the proof follows closely the proof of [START_REF] Friedrich | A Godunov type scheme for a class of LWR traffic flow models with non-local flux[END_REF]Theorem 3.1]. Therefore, we compute the differences of the velocities and obtain

V 1,n j-1 -V 1,n j =              Nη-1 k=1 (γ k -γ k-1 )v 1 (ρ n j+k ) -γ Nη-1 v 1 (ρ n j+Nη ) + γ 0 v 1 (ρ n j ), j ≤ -N η -1, -j-1 k=1 (γ k -γ k-1 )v 1 (ρ n j+k ) + γ 0 v 1 (ρ n j ), -N η ≤ j ≤ -2, γ 0 v 1 (ρ n -1 ), j = -1, 0, j ≥ 0, (4.6)
and

V 2,n j-1 -V 2,n j =              0, j ≤ -N η -1, -γ Nη-1 v 2 (ρ n 0 ), j = -N η , Nη-1 k=-j (γ k -γ k-1 )v 2 (ρ n j+k ) -γ Nη-1 v 2 (ρ n j+Nη ), -N η + 1 ≤ j ≤ -1, Nη-1 k=1 (γ k -γ k-1 )v 2 (ρ n j+k ) -γ Nη-1 v 2 (ρ n j+Nη ) + γ 0 v 2 (ρ n j ), j ≥ 0. (4.
7) It is easy to see that the following estimates hold:

V 1,n j-1 -V 1,n j ≤ γ 0 v 1 (ρ n j ) j ≤ -1, 0, j ≥ 0, and V 2,n j-1 -V 2,n j ≤ 0 j ≤ -1, γ 0 v 2 (ρ n j ), j ≥ 0.
Using v 1 (ρ 1 max ) = v 2 (ρ 2 max ) = 0 and the mean value theorem we get

V 1,n j-1 -V 1,n j ≤ γ 0 v (ρ 1 max -ρ n j ) j ≤ -1, 0, j ≥ 0, and V 2,n j-1 -V 2,n j ≤ 0 j ≤ -1, γ 0 v (ρ 2 max -ρ n j ), j ≥ 0.
Now we consider the case j ≤ -1 and multiply the first inequality by ρ 1 max , subtract V 1,n j ρ n j and we get

V 1,n j-1 ρ 1 max -V 1,n j ρ n j ≤ γ 0 v ρ + V 1,n j (ρ 1 max -ρ n j ).
Similarly, we get

V 2,n j-1 g(ρ 1 max ) -V 2,n j g(ρ n j ) ≤ V 2,n j g(ρ 1 max ) -g(ρ n j ) ≤ V 2,n j (ρ 1 max -ρ n j ).
Adding the last two inequalities we obtain,

V 1,n j-1 ρ 1 max -V 1,n j ρ n j + V 2,n j-1 g(ρ 1 max ) -V 2,n j g(ρ n j ) ≤ γ 0 v ρ + v (ρ 1 max -ρ n j ).
Due to the CFL condition (4.5), we have for j ≤ -1

ρ n+1 j ≤ ρ n j + λ V 1,n j-1 ρ 1 max -V 1,n j ρ n j + V 2,n j-1 g(ρ 1 max ) -V 2,n j g(ρ n j ) ≤ ρ 1 max .
For j ≥ 0 the bound

V 2,n j-1 ρ 2 max -V 2,n j ρ n j ≤ γ 0 v ρ + v (ρ 2 max -ρ n j )
follows analogously to above. Note that V 1,n j = 0 for j ≥ -1. Since g(ρ n j-1 ) ≤ ρ 2 max holds even for j = 0 and g(ρ n j ) = ρ n j for j ≥ 0, we obtain

ρ n+1 j ≤ ρ n j + λ V 2,n j-1 ρ 2 max -V 2,n j ρ n j ≤ ρ 2 max .
This concludes the proof.

Remark 3. The role of the limiter g given by (2.3) in the flux function (2.2) is essential for the discrete maximum principle above. Indeed, let us consider the model without this limiter.

In order to deal with meaningful velocities, we set v 2 (ρ) = 0 if ρ > ρ 2 max , such that we have

V 2 (t, x) = max{x+η,0} max{x,0} max{0, v 2 (ρ(t, y))}ω η (y -x)dy.
For this model, it is possible to prove a maximum principle on [0, max{ρ 1 max , ρ 2 max }] as above and similar BV estimates as below, so that the convergence to a solution is ensured. Nevertheless, this solution has an interesting behavior for η → ∞ and an initial datum of compact support; in this case, V 1 (t, x) + V 2 (t, x) converges pointwise to v 2 (0) and it is possible to prove that the solution will converge to the solution of a linear transport with velocity v 2 (0) (see [START_REF] Chiarello | Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel[END_REF]Corollary 1.3] for a similar result). Therefore, it is obvious that, if ρ 1 max > ρ 2 max , for any initial datum ρ 0 of compact support, such that ρ 2 max < ρ 0 (x) ≤ ρ 1 max for x < 0, the corresponding solution does not satisfy (2.9), i.e. ρ(x, t) > ρ 2 max for x ∈ ]0, a[, a > 0, and t and η large enough.

BV estimate

In addition to the L ∞ bound, we also need a uniform estimate on the total variation of the sequence of approximate solutions. The crucial part here lies in the presence of the limiter g at x = 0. Lemma 2. Let ρ ∆x be constructed by (4.1)-(4.4) and let the CFL condition (4.5) hold, then for every T > 0 the following discrete space BV estimate is satisfied:

T V (ρ ∆x (T, •)) ≤ exp T w η (0) 2 v + v ρ T V (ρ 0 ) + T 2ω η (0) v ρ =: K(T ). (4.8)
Proof. For all j ∈ Z, we set ∆ n j := ρ n j+1 -ρ n j . In the following we consider a regularization of the function g defined in (2.3), namely

g ε (ρ) = 1 2 ρ + ρ 2 max -(ρ -ρ 2 max ) 2 + ε , ε > 0. (4.9)
The function g ε is differentiable for every ε > 0 with g ε ≤ 1 for all ε > 0. This will allow us to use the mean value theorem in the following computations. In particular, we will denote by ξ n j a value between ρ n j and ρ n j+1 such that g ε (ξ n j )∆ n j = g ε (ρ n j+1 ) -g ε (ρ n j ) holds. We obtain:

∆ n+1 j =∆ n j -λ F n j+ 3 2 (ρ n j+1 ) -2F n j+ 1 2 (ρ n j ) + F n j-1 2 (ρ n j-1 ) =∆ n j -λ V 1,n j+1 + g ε (ξ n j )V 2,n j+1 ∆ n j -V 1,n j-1 + g ε (ξ n j-1 )V 2,n j-1 ∆ n j-1 +ρ n j V 1,n j+1 -2V 1,n j + V 1,n j-1 + g ε (ρ n j ) V 2,n j+1 -2V 2,n j + V 2,n j-1
.

Let us now consider the differences of the velocities. With the differences already computed in (4.6) and (4.7) and the help of the mean value theorem, where ζ n j is a value between ρ n j and ρ n j+1 for which

v i (ζ n j )∆ n j = v i (ρ n j+1 ) -v i (ρ n j ) for i ∈ {1, 2} holds, we derive V 1,n j+1 -2V 1,n j + V 1,n j-1 =                      Nη-1 k=1 (γ k-1 -γ k )v 1 (ζ n j+k )∆ n j+k + γ Nη-1 v 1 (ζ n j+Nη )∆ n j+Nη -γ 0 v 1 (ζ n j )∆ n j , j ≤ -N η -2, Nη-1 k=1 (γ k-1 -γ k )v 1 (ζ n j+k )∆ n j+k -γ Nη-1 v 1 (ρ n -1 ) -γ 0 v 1 (ζ n j )∆ n j , j = -N η -1, -j-2 k=1 (γ k-1 -γ k )v 1 (ζ n j+k )∆ n j+k + (γ -j-1 -γ -j-2 )v 1 (ρ n -1 ) -γ 0 v 1 (ζ n j )∆ n j , -N η ≤ j ≤ -3, (γ 1 -γ 0 )v 1 (ρ n -1 ) -γ 0 v 1 (ζ n j )∆ n j , j = -2, γ 0 v 1 (ρ n -1 ), j = -1, 0, j ≥ 0, and
V 2,n j+1 -2V 2,n j + V 2,n j-1 =                        0, j ≤ -N η -2, γ Nη-1 v 2 (ρ n 0 ), j = -N η -1, γ Nη-1 v 2 (ζ n j+Nη )∆ n j+Nη + (γ Nη-1 -γ Nη )v 2 (ρ n 0 ), j = -N η , Nη-1 k=-j (γ k-1 -γ k )v 2 (ζ n j+k )∆ n j+k + γ Nη-1 v 2 (ζ n j+Nη )∆ n j+Nη + (γ -j-2 -γ -j-1 )v 2 (ρ n 0 ), -N η + 1 ≤ j ≤ -2, Nη-1 k=1 (γ k-1 -γ k )v 2 (ζ n j+k )∆ n j+k + γ Nη-1 v 2 (ζ n j+Nη )∆ n j+Nη -γ 0 v 2 (ρ n 0 ), j = -1, Nη-1 k=1 (γ k-1 -γ k )v 2 (ζ n j+k )∆ n j+k + γ Nη-1 v 2 (ζ n j+Nη )∆ n j+Nη -γ 0 v 2 (ζ n j )∆ n j , j ≥ 0.
Putting everything together we have

∆ n+1 j = 1 -λ V 1,n j+1 + g ε (ξ n j )V 2,n j+1 -γ 0 a n j ∆ n j + λ V 1,n j-1 + g ε (ξ n j-1 )V 2,n j-1 ∆ n j-1 + λ Nη-1 k=1 (γ k-1 -γ k )b n j+k ∆ n j+k + λγ Nη-1 c n j+Nη ∆ n j+Nη + λd n j ρ j v 1 (ρ n -1 ) -g ε (ρ n j )v 2 (ρ n 0 ) , (4.10)
where

a n j =        v 1 (ζ n j )ρ n j , j ≤ -2, 0, j = -1, v 2 (ζ n j )ρ n j , j ≥ 0, b n j+k =        -v 1 (ζ n j+k )ρ n j , j + k ≤ -2, 0, j + k = -1, -v 2 (ζ n j+k )g ε (ρ n j ), j + k ≥ 0, c n j+Nη =        -v 1 (ζ n j+Nη )ρ n j , j ≤ -N η -2, 0, j = -N η -1, -v 2 (ζ n j+Nη )g ε (ρ n j ), j ≥ -N η , d n j =                  0, j ≤ -N η -2, γ Nη-1 , j = -N η -1, γ -j-2 -γ -j-1 , -N η ≤ j ≤ -2, -γ 0 , j = -1, 0, j ≥ 0.
Since the coefficients in (4.10) are positive due to the CFL condition (4.5), we take absolute values, sum over j and rearrange the indices, which gives us

j |∆ n+1 j | ≤ j 1 -λ V 1,n j+1 + g ε (ξ n j )V 2,n j+1 -γ 0 a n j |∆ n j | + λ V 1,n j-1 + g ε (ξ n j-1 )V 2,n j-1 |∆ n j-1 | + λ Nη-1 k=1 (γ k-1 -γ k )b n j+k |∆ n j+k | + λγ Nη-1 c n j+Nη |∆ n j+Nη | +λ|d n j | ρ j v 1 (ρ n -1 ) -g ε (ρ n j )v 2 (ρ n 0 ) = j 1 -λ V 1,n j+1 + g ε (ξ n j )V 2,n j+1 -V 1,n j -g ε (ξ n j )V 2,n j + λ γ 0 a n j + Nη-1 k=1 (γ k-1 -γ k )b n j + γ Nη-1 c n j |∆ n j | + j λ|d n j | ρ j v 1 (ρ n -1 ) -g ε (ρ n j )v 2 (ρ n 0 )
. Now we use that V i,n j -V i,n j+1 ≤ γ 0 v and g ε ≤ 1 for the first term and for the second term we have a n j ≤ 0 and b n j , c n j ≤ v ρ , which gives us

j |∆ n+1 j | ≤ 1 + λγ 0 2 v + v ρ j |∆ n j | + j λ|d n j | ρ j v 1 (ρ n -1 ) -g ε (ρ n j )v 2 (ρ n 0 ) .
Since j |d n j | = 2γ 0 holds, using also λγ 0 ≤ ∆tω η (0) we finally obtain

j |∆ n+1 j | ≤ 1 + ∆tω η (0) 2 v + v ρ j |∆ n j | + ∆t2ω η (0) v ( ρ + √ ε 2 ).
This estimate holds for any ε > 0 and for ε → 0 we obtain the following estimate for the total variation

T V (ρ(T, •)) ≤ 1 + ∆tω η (0) 2 v + v ρ T /∆t T V (ρ 0 ) + T 2ω η (0) v ρ ≤ exp ω η (0) 2 v + v ρ T T V (ρ 0 ) + T 2ω η (0) v ρ .
To finally apply Helly's Theorem we also need an estimate for the discrete total variation in space and time, which we are now able to provide. Lemma 3. Let ρ ∆x be constructed by (4.1)-(4.4) and let the CFL condition (4.5) hold, then for every T > 0 the following discrete space and time total variation estimate is satisfied:

T V (ρ ∆x ; R × [0, T ]) ≤ T K(T )(1 + v ρ + v )
with K(T ) defined as in (4.8).

Using the regularization of g given by (4.9), the proof is entirely analogous to the one of [START_REF] Friedrich | A Godunov type scheme for a class of LWR traffic flow models with non-local flux[END_REF]Theorem 3.3].

Discrete Entropy Inequality

In the following, we use the notation a ∧ b = max{a, b}, a ∨ b = min{a, b} and follow [START_REF] Amorim | On the numerical integration of scalar nonlocal conservation laws[END_REF][START_REF] Chiarello | Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel[END_REF][START_REF] Friedrich | A Godunov type scheme for a class of LWR traffic flow models with non-local flux[END_REF].

ρ n+1 j -c ≤ ρ n j -c -λ H n j+1/2 (ρ n j ) -H n j-1/2 (ρ n j-1 ) (4.11) -λ sgn(ρ n+1 j -c) F n j+1/2 (c) -F n j-1/2 (c) ,
where

H n j+1/2 (u) = F n j+1/2 (u ∧ c) -F n j+1/2 (u ∨ c). Proof. Let G n j (u, w) = w -λ(F n j+1/2 (w) -F n j-1/2 (u)
). Under the CFL condition (4.5) and using the regularization (4.9) of g, G j is monotone in both its arguments, since we obtain

∂G n j ∂w = 1 -λ(V 1,n j+ 1 2 + g ε (w)V 2,n j+ 1 2 ) ≥ 0, ∂G n j ∂u = λ(V 1,n j-1 2 + g ε (u)V 2,n j-1 2 ) ≥ 0.
The monotonicity implies that

G n j (ρ n j-1 ∧ c, ρ n j ∧ c) ≥ G n j (ρ n j-1 , ρ n j ) ∧ G n j (c, c) (4.12) G n j (ρ n j-1 ∨ c, ρ n j ∨ c) ≤ G n j (ρ n j-1 , ρ n j ) ∨ G n j (c, c). (4.13)
Subtracting (4.13) from (4.12), we obtain

G n j (ρ n j-1 , ρ n j ) -G n j (c, c) ≤ ρ n j -c -λ H n j+1/2 (ρ n j ) -H n j-1/2 (ρ n j-1 ) . (4.14)
The left side of (4.14

) is ρ n+1 j -c + λ(F n j+1/2 (c) -F j-1/2 (c))
, and we get

ρ n+1 j -c + λ(F n j+1/2 (c) -F n j-1/2 (c)) ≥ sgn(ρ n+1 j -c) ρ n+1 j -c + λ(F n j+1/2 (c) -F n j-1/2 (c)) = ρ n+1 j -c + λ sgn(ρ n+1 j -c) F n j+1/2 (c) -F n j-1/2 (c) . (4.15)
The proof is completed by combining (4.14) and (4.15).

Convergence

Lemma 5. Let ρ = ρ(t, x) ∈ L ∞ ∩ BV([0, +∞) × R; [0, max{ρ 1 max , ρ 2 max }) be the L 1 loc -limit of approximations ρ ∆x generated by the upwind scheme (4.4) and let c ∈ R, ϕ ∈ C 1 c ([0, +∞)×R). Then ρ satisfies the entropy inequality given by (2.8).

Proof. Let ϕ ∈ C 1 c ([0, +∞) × R) and set ϕ n j = ϕ(t n , x j ). We multiply the discrete entropy inequality (4.11) by ϕ n j ∆x, and then apply summation by parts to get

∆x∆t n≥0 j∈Z ρ n+1 j -c (ϕ n+1 j -ϕ n j )/∆t + ∆x j ρ 0 j -c ϕ 0 j (5.1) + ∆x∆t n≥0 j∈Z H n j-1/2 (ϕ n j -ϕ n j-1 )/∆x (5.2) -∆x∆t n≥0 j∈Z sgn(ρ n+1 j -c) F n j+1/2 (c) -F n j-1/2 (c) ϕ n j /∆x ≥ 0.
(5.3) By Lebesgue's dominated convergence theorem, as ∆x → 0, we have

(5.1) → ∞ 0 R |ρ -c|ϕ t dxdt + ∞ -∞ |ρ 0 (x) -c|ϕ(0, x)dx.
As ∆x → 0, the sums in (5.2) converge by standard arguments, see [START_REF] Betancourt | On nonlocal conservation laws modelling sedimentation[END_REF], [START_REF] Blandin | Well-posedness of a conservation law with non-local flux arising in traffic flow modeling[END_REF]Sec. 4 Proof of Theorem 1], [START_REF] Karlsen | L 1 -stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients[END_REF], to

∞ 0 R sgn(ρ -c)(f (t, x, ρ) -f (t, x, c))ϕ x dxdt.
Now let us study the sum (5.3) and we have

(5.3) = -∆x∆t n≥0 j∈Z sgn(ρ n+1 j -c) cV 1 j + g(c)V 2 j -cV 1 j-1 -g(c)V 2 j-1 ϕ n j /∆x = -∆x∆t n≥0 j∈Z sgn(ρ n+1 j -c) c V 1 j -V 1 j-1 ∆x + g(c) V 2 j -V 2 j-1 ∆x ϕ n j = -∆x∆t n≥0 j∈Z (sgn(ρ n+1 j -c) -sgn(ρ n j -c)) c V 1 j -V 1 j-1 ∆x + g(c) V 2 j -V 2 j-1 ∆x ϕ n j -∆x∆t n≥0 j∈Z sgn(ρ n j -c) c V 1 j -V 1 j-1 ∆x + g(c) V 2 j -V 2 j-1 ∆x ϕ n j .
The second term in the last equality clearly converges to

- ∞ 0 ∞ -∞ sgn(ρ -c) c(V 1 ) x + g(c)(V 2 ) x ϕ)dxdt.
We will show now that the first term vanishes as ∆x → 0. We follow here [START_REF] Betancourt | On nonlocal conservation laws modelling sedimentation[END_REF][START_REF] Blandin | Well-posedness of a conservation law with non-local flux arising in traffic flow modeling[END_REF] and we perform a summation by parts, which gives us:

∆t n≥0 j∈Z sgn(ρ n+1 j -c)ϕ n j c V 1,n+1 j -V 1,n+1 j-1 -V 1,n j -V 1,n j-1 +g(c) V 2,n+1 j -V 2,n+1 j-1 -V 2,n j -V 2,n j-1 + ∆t∆t∆x n≥0 j<0 sgn(ρ n+1 j -c)   c V 1,n+1 j -V 1,n+1 j-1 ∆x + g(c) V 2,n+1 j -V 2,n+1 j-1 ∆x    ϕ n+1 j -ϕ n j ∆t .
As can be seen in (4.6) and (4.7)

V i,n+1 j -V i,n+1 j-1
≤ ∆xω η (0) v holds and due to the compactness of the support function the second term vanishes as ∆x, ∆t → 0. For the first term we first obtain that

V 1,n+1 j -V 1,n+1 j-1 -V 1,n j -V 1,n j-1 =                        Nη-1 k=1 (γ k-1 -γ k )(v 1 (ρ n+1 j+k -v 1 (ρ n j+k )) + γ Nη-1 (v 1 (ρ n+1 j+Nη ) -v 1 (ρ n j+Nη )) -γ 0 (v 1 (ρ n+1 j ) -v 1 (ρ n j )), j ≤ -N η -1, -j-1 k=1 (γ k-1 -γ k )(v 1 (ρ n+1 j+k ) -v 1 (ρ n j+k )) -γ 0 (v 1 (ρ n+1 j ) -v 1 (ρ n j )), -N η ≤ j ≤ -2, γ 0 (v 1 (ρ n+1 -1 ) -v 1 (ρ n -1 )), j = -1, and
V 2,n+1 j -V 2,n+1 j-1 -V 2,n j -V 2,n j-1 =              0, j ≤ -N η -1, γ Nη-1 (v 2 (ρ n+1 j+Nη ) -v 2 (ρ n j+Nη )), j = -N η , Nη-1 k=-j (γ k-1 -γ k )(v 2 (ρ n+1 j+k ) -v 2 (ρ n j+k )) + γ Nη-1 (v 2 (ρ n+1 j+Nη ) -v 2 (ρ n j+Nη )), -N η + 1 ≤ j ≤ -1.
Now we use the compact support of the test function. There exist T > 0 and R > 0 such that ϕ(t, x) = 0 for t > T and |x| > R. Let N T ∈ N and j 0 , j 1 ∈ Z be such that

T ∈ ]n T ∆t, (n T + 1)∆t], -R ∈]x j 0 -1 2 , x j 0 + 1 2 ], R ∈]x j 1 -1 2 , x j 1 + 1 2
]. We only consider j 0 < 0, since otherwise the term is already 0. In addition, similar to [START_REF] Friedrich | A Godunov type scheme for a class of LWR traffic flow models with non-local flux[END_REF]Theorem 3.3], the following estimate is derived during the proof of Lemma 3:

N T n=0 j ∆x|ρ n+1 j -ρ n j | ≤ K,
By plugging in the equality obtained before, using the mean value theorem, the above mentioned estimate and g(c) ≤ c we obtain

∆t n≥0 j<0 sgn(ρ n+1 j -c)ϕ n j c V 1,n+1 j -V 1,n+1 j-1 -V 1,n j -V 1,n j-1 +g(c) V 2,n+1 j -V 2,n+1 j-1 -V 2,n j -V 2,n j-1 ≤ ∆t ∆x ϕ v c   γ Nη-1 N T n=0 min{-1,j 1 } j=j 0 ∆x|ρ n+1 j+Nη -ρ n j+Nη |+ Nη-1 k=1 (γ k-1 -γ k ) N T n=0 min{-1,j 1 } j=j 0 ∆x|ρ n+1 j+k -ρ n j+k | + γ 0 N T n=0 min{-1,j 1 } j=j 0 ∆x|ρ n+1 j -ρ n j |   ≤ ∆t ϕ v c K2ω η (0),
which goes to zero as ∆x → 0 (and then ∆t → 0). This concludes the proof.

Proof of Theorem 1.

Similar to [8, Theorem 1], [17, Theorem 2.3] or [6, Theorem 1], the convergence of the approximate solutions constructed by the upwind scheme (4.4) to the unique weak entropy solution can be proven by applying Helly's theorem, see [START_REF] Eymard | Finite volume methods[END_REF]Lemma 5.6]. Due to Lemma 1 and Lemma 3, there exists a sub-sequence of approximate solutions that converges to some ρ ∈ (L ∞ ∩ BV)([0, +∞) × R; [0, max{ρ 1 max , ρ 2 max }]). Lemma 5 shows that the limit function ρ is a weak entropy solution of (2.1)-(2.6) in the sense of Definition 1. Adding the uniqueness result in Theorem 2, we conclude the proof of Theorem 1.

Numerical simulations

In this section, we show some simulation results to illustrate the numerical solutions of the non-local model (2.1). The behaviour of solutions is also studied as the look-ahead distance η tends to zero. To this end, we will consider Riemann initial data of the type

ρ 0 (x) = ρ L , if x < 0, ρ R , if x > 0. (6.1)
We take a spatial step size of ∆x = 10 -3 . Due to the CFL condition (4.5) the time step size ∆t is given by ∆t ≈ 0.9∆x/(γ 0 v ρ + v ).

We divide this section into three parts. In the first part we analyze how our model behaves for a fixed look ahead distance η > 0. For non-local conservation laws, it is still an open question whether the model tends to the corresponding local equation for η tending to zero (see for example [START_REF] Colombo | Recent results on the singular local limit for nonlocal conservation laws[END_REF] for a recent overview). For this reason, we will investigate the limit question as η → 0 from the numerical point of view in Section 6.2. Overall, we will consider the following settings: The first two settings are used to show that the obtained solutions are reasonable also for nonlinear velocity functions, while the last two settings turn out to be interesting in Section 6.2. For all the tests, the kernel function is given by ω η (x) = 2(η -x)/η 2 and the final simulation time is T = 1. Finally, in Section 6.3, we will show that our model can be easily extended to more than two stretches and therefore to a sequence of 1-to-1 junctions to simulate traffic.

Test 1: v i (ρ) = v i max 1 - ρ ρ i max 2 for i ∈ {1, 2}, with v 1 max = 1, v 2 max = 2, ρ 1 max = ρ 2 max = 1, ρ L = 0.75, ρ R = 0.5; Test 2: as in Test 1, but with v 1 max = 2, v 2 max = 1; Test 3: v i (ρ) = v i max 1 -ρ ρ i max for i ∈ {1, 2}, with v 1 max = 2, v 2 max = 1, ρ 1 max = 0.5, ρ 2 max = 1, ρ L = 0.

Fixed look-ahead distance η

We set η = 0.1. Let us consider the first test. Here we start with a congested situation on the first road segment. In addition, the maximum velocity on the first road is lower than the one on the second road segment. Therefore, the traffic jam resolves over time as can be seen in Figure 2, left. In contrast to Test 1, Test 2 presents the opposite situation: the velocity on the first road segment is now higher than the second one. Hence, the traffic jam can not resolve and we get a backward traveling increase of the density (see Figure 2, right).

In the last two settings we can see that the presence of the look ahead distance results in a smoothing of the density close to the end of the first and the beginning of the second road segment, see Figure 3.

Look-ahead distance η tending to zero

As mentioned above, the behaviour of solutions for η tending to zero is of special interest for non-local conservation laws. Concerning non-local LWR traffic flow models as in [START_REF] Chiarello | Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel[END_REF][START_REF] Friedrich | A Godunov type scheme for a class of LWR traffic flow models with non-local flux[END_REF], or model (2.1) with v 1 ≡ v 2 , so far the convergence to the classical LWR traffic flow model [START_REF] Lighthill | On kinematic waves. II. A theory of traffic flow on long crowded roads[END_REF][START_REF] Richards | Shock waves on the highway[END_REF] can only be proven for monotone initial data (see [START_REF] Colombo | Recent results on the singular local limit for nonlocal conservation laws[END_REF][START_REF] Keimer | On approximation of local conservation laws by nonlocal conservation laws[END_REF]), since the solution is monotonicity preserving and therefore has a strict maximum principle and a bounded total variation, uniformly in η. Unfortunately, similar results do not hold for model (2.1) with v 1 = v 2 , since the model is, in general, not monotonicity preserving even for constant initial data. Therefore, we just investigate the limit numerically.

The local (discontinuous) conservation law corresponding to model (2.1) is given by:

ρ t + f (x, ρ) x = 0, with f (x, ρ) := H(-x)ρv 1 (ρ) + H(x)ρv 2 (ρ), (6.2)
where H(x) is the Heaviside function. As pointed out in [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF][START_REF] Bürger | An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections[END_REF], (6.2) admits many L 1 contraction semigroups, one for each so-called (A, B)-connection. The two most common connections are the one corresponding to the supply-demand approach [START_REF] Lebacque | The Godunov scheme and what it means for first order traffic flow models[END_REF], and the vanishing viscosity solution (see [START_REF] Karlsen | Convergence of a Godunov scheme for conservation laws with a discontinuous flux lacking the crossing condition[END_REF]Definition 3.1]), which is a weak solution satisfying, besides the Kruzkov entropy inequalities for x < 0 and x > 0, the Γ-condition of [START_REF] Diehl | On scalar conservation laws with point source and discontinuous flux function[END_REF][START_REF] Diehl | A uniqueness condition for nonlinear convection-diffusion equations with discontinuous coefficients[END_REF], see also [START_REF] Karlsen | Convergence of a Godunov scheme for conservation laws with a discontinuous flux lacking the crossing condition[END_REF]Definition 3.1] and [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF].

For instance, the vanishing viscosity solution can be obtained by a Godunov scheme considering a grid where x = 0 is a cell midpoint, see [START_REF] Karlsen | Convergence of a Godunov scheme for conservation laws with a discontinuous flux lacking the crossing condition[END_REF].

In the following, we will consider η ∈ {0.05, 0.01, 0.005, 0.001} and ∆x = 10 -4 to keep the nonlocal impact. We compare it to the solution of (6.2)-(6.1), which will be computed by the Godunov scheme as presented in [START_REF] Karlsen | Convergence of a Godunov scheme for conservation laws with a discontinuous flux lacking the crossing condition[END_REF], since we are interested in the vanishing viscosity solution. Note that, due to the different grids, we do not compute L 1 -errors between the different solutions.

We will now investigate the previous four test cases. In the first two settings, as η → 0 the solution of (2.1) with initial conditions (6.1) is very similar to the vanishing viscosity solution of the corresponding local problem, see Figure 4. We also remark that, in the parameters settings Test 1 and Test 2, the solution obtained by the supply-demand approach is equal to the vanishing viscosity solution.

Let us now consider Tests 3 and 4. The initial datum in both of them is exactly the density corresponding to the maximum fluxes attainable on each road segment. Therefore, the solution of the supply and demand approach is given by a stationary discontinuity coinciding with the initial datum. As can be seen in Figure 5, in both tests the limit of model (2.1) behaves as the vanishing viscosity solution. In Test 4, the numerical results also coincide with supply-demand solution. The most interesting case is Test 3. For these parameters, the vanishing viscosity solution differs from the supply-demand solution and, as can be seen in Figure 5 (left picture) the solution of the model (2.1) seems to converge to the vanishing viscosity solution for η tending to zero.

Linear network scenario

Finally, we show that the model can be extended to more than two stretches of a road. We consider the case of road works on a highway, modeled by the segment [0, L], with L = 2, where the road capacity and the maximal speed are smaller. Therefore, we have three different As in Section 6.1, the look ahead distance is η = 0.1, and as in Section 6.2 we also present the vanishing viscosity solution obtained by the Godunov scheme of [START_REF] Karlsen | Convergence of a Godunov scheme for conservation laws with a discontinuous flux lacking the crossing condition[END_REF] to get an impression of the corresponding local problem. As can be seen in Figure 6, the presence of the road works results in a traffic jam upstream and a decrease of the density downstream. As noticed in Section 6.2, the numerical solution of the non-local problem tends for small η towards the vanishing viscosity solution.

Conclusion

In this work we have presented a non-local flux model, which can handle different maximum velocities and capacities, i.e. different velocity functions, on the road and therefore models a 1-to-1 junction. The model considers a non-local mean downstream velocity on both road segments and satisfies a maximum principle on each road segment. We have proven its wellposedness, i.e. existence, uniqueness and continuous dependence of solutions with respect to the initial data, via an upwind numerical scheme. Numerical examples suggest that the solution tends to the vanishing viscosity solution of the corresponding local conservation law as the look-ahead distance goes to 0. We intend to further investigate this question in future work.

In addition, the model can be extended to more than two stretches to model traffic behavior on a more complex road segment, as shown in Section 6.3. Hence, this model can be seen as a first step towards non-local traffic flow models on networks. In the future, we aim to extend this model from the current simple network structure to a more general network formulation.

Lemma 4 .

 4 Let ρ ∆x be constructed by (4.1)-(4.4). If the CFL condition (4.5) holds, then for c ∈ R we have the following discrete entropy inequality

  [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF], ρ R = 0.5; Test 4: v i as in Test 3, but with v 1 max = 1, v 2 max = 2, ρ 1 max = 1, ρ 2 max = 0.5, ρ L = 0.5, ρ R = 0.25.

2 Figure 2 :

 22 Figure 2: Numerical solutions at T = 1 corresponding to Test 1 (left) and Test 2 (right).

4 Figure 3 :

 43 Figure 3: Numerical solutions at T = 1 for the Test 3 (left) and Test 4 (right).

2 Figure 4 : 4 Figure 5 :

 2445 Figure 4: Numerical solutions at T = 1 corresponding to Test 1 (left) and Test 2 (right) and different values of η.

Figure 6 : 1 Test 3 , 4 ,

 6134 Figure 6: Numerical solution for three road segments at T = 1

Inria Sophia Antipolis -Méditerranée, Université Côte d'Azur, Inria, CNRS, LJAD,

route des Lucioles -BP 93, 06902 Sophia Antipolis Cedex, France. E-mail: {felisia.chiarello, paola.goatin}@inria.fr 2 University of Mannheim, Department of Mathematics, 68131 Mannheim, Germany E-mail: {jan.friedrich, goettlich, kolb}@uni-mannheim.de

Acknowledgements

This work was financially supported by the project "Non-local conservation laws for engineering applications" co-funded by DAAD (Project-ID 57445223) and the PHC Procope (Projectno. 42503RM).