On the accuracy of the finite volume approximations to nonlocal conservation laws

In this article, we discuss the error analysis for a certain class of monotone finite volume schemes approximating nonlocal scalar conservation laws, modeling traffic flow and crowd dynamics, without any additional assumptions on monotonicity or linearity of the kernel $\mu$ or the flux $f$. We first prove a novel Kuznetsov-type lemma for this class of PDEs and thereby show that the finite volume approximations converge to the entropy solution at the rate of $\sqrt{\Delta t}$ in $L^1(\mathbb{R})$. To the best of our knowledge, this is the first proof of any type of convergence rate for this class of conservation laws. We also present numerical experiments to illustrate this result.


Introduction
The celebrated Lighthill-Whitham-Richards (LWR) model [29,31] given by u t + (uν(u)) x = 0, (t, x) ∈ (0, ∞) × R, with u being the mean traffic density and ν the mean traffic speed, is one of the widely used models in traffic flow modeling.However, being a non-linear hyperbolic conservation law, it can have solutions with discontinuities and infinite accelerations, adversely impacting its capability to capture physical traffic phenomena effectively.Thus, over last decade, the parallel class of conservation laws with nonlocal parts in the flux is gaining particular interest in the modeling as well as mathematical community, where a convolution is introduced in the flux to produce Lipschitz-continuous velocities ensuring bounded accelerations.The two most popular strategies are to evaluate the traffic speed, from either averaging the traffic density, or averaging over the velocity, leading to the following two conservation laws, where ν, µ ∈ (C 2 ∩ W 2,∞ )(R), and (µ * u)(t, x) = R µ(x − ξ) u(t, ξ) dξ .
The wellposedness of this class has been of interest in the last few years.The local counterpart of (1.4)-(1.5)enjoys a rich literature, with [27] as one of the pioneering papers to fix the wellposedness of the entropy solutions for such PDEs.Similar to its local counterpart, since f can be possibly nonlinear, there can be multiple weak solutions of IVP (1.4)- (1.5).Hence, an additional entropy condition is required to single out a unique solution.where U(t, x) = ν(µ * β(u(t))(x)).
With some appropriate modifications, the proof of [5] can be adapted to prove that any two entropy solutions satisfying Definition 1.1 are equal, while existence of these solutions has been proven in [4,1] for non linear f and in [8] for linear f , via the convergence of finite volume approximations.These articles dealing with existence of solutions establish that the schemes converge to the entropy solution u ∈ C [0, T ]; L 1 (R) .
What remains unexplored is to analyze the rate of convergence, i.e., how fast the error ||u ∆ (T, • )−u(T, • )|| L 1 (R) made by the numerical solution u ∆ in approximating the exact solution u goes to zero as the mesh size ∆x goes to zero.That is the precise aim of this article.In other words, we look for an (optimal) α satisfying with C being an appropriate positive constant.To achieve this, we first prove a Kuznetsovtype lemma using the entropy formulation (1.6).We further estimate the relative entropy functional involving the solution u and numerical approximation u ∆ to obtain (1.7) with an optimal α = 1/2, same as the one obtained in [32,28] for local fluxes (homogeneous).To the best of our knowledge, this is the first result in this direction for such nonlocal conservation laws.It is to be noted that the results of the article hold under no additional assumptions on monotonicity/linearity of the kernel µ or the flux f or ν.
The paper is organized as follows.In Section 2, we discuss the wellposedness of (1.4)-(1.5)via convergence of a general class of monotone finite volume approximations.In Section 3, we prove the Kuznetsov-type lemma for (1.4)-(1.5)and obtain the rate of convergence as 1/2.In Section 4, we also briefly comment on the extensions to higher dimensions.In Section 5, we present some numerical experiments which illustrate the theory.

Finite volume approximations and wellposedness
We now introduce the notations to be used in the article: TV(u(t, • )). ii.

Uniqueness of the entropy solution
Any two entropy solutions of the IVP (1.4)-(1.5)are equal.More precisely, we have the following result: be two entropy solutions of the IVP (1.4)-(1.5)corresponding to the initial data u 0 and v 0 respectively.Then, there exists a constant In particular, if u 0 = v 0 , then u = v a.e. in Q T .
Proof.Note that the nonlocal coefficient of the flux function considered in this article is more general than the ones in all the previous results on the uniqueness of nonlinear-nonlocal conservation laws.However, the continuous dependence estimates for the entropy solution of the conservation laws (local) derived in [23,Thm. 1.3] can still be invoked to prove the desired weighted contraction estimates as in [5,Thm. 4.1].Alternatively, this can also be seen as a consequence of the Kuznetsov-type estimates derived in the sequel (see Lemma 3.2), by sending ε, ε 0 → 0.

Existence via numerical approximations
For ∆x, ∆t > 0, and λ := ∆t/∆x, consider equidistant spatial grid points x i := i∆x for i ∈ Z and temporal grid points t n := n∆t for non-negative integers n ≤ N , such that T = N ∆t.
and let χ n (t) denote the indicator function of C n := [t n , t n+1 ).We approximate the initial data according to: We define a piecewise constant approximate solution u ∆ to (1.4) by through the following marching formula: where the convolution term µ * β(u) is computed through a standard quadrature formula using the same space mesh, i.e., with u n p+1/2 being any convex combination of u n p and u n p+1 , and µ i+1/2 = µ(x i+1/2 ).Further, F(ν(c n i+1/2 ), u n i , u n i+1 ) denotes the numerical approximation of the flux f (u)ν(µ * β(u)) at the interface x = x i+1/2 for i ∈ Z, with H being increasing in the last three arguments.The approximations generated by the scheme, namely u n i ≈ u(t n , x i ) are extended to a function defined on In general, F can be defined as an appropriate nonlocal extension of any monotone numerical flux, meant for local conservation laws.Here, we present examples of celebrated Lax-Friedrichs flux and Godunov flux.

Time continuity:
(2.10) 6. Discrete entropy inequality: For any k ∈ R we have where Furthermore, the finite volume approximations converge to the unique entropy solution u of the IVP (1.4)-(1.5).
Proof.The proof follows by invoking the monotonicity of the scheme and writing it in the incremental form.The details can be worked out exactly on the similar lines of [ The above theorem implies that the entropy solution satisfies the following regularity estimates.

Error estimate
We define Φ : where Furthermore, we assume that ˆR ω a (x) dx = 1 and We now state and prove the Kuznetsov-type lemma for nonlocal conservation laws.
Lemma 3.2.Let u be the entropy solution of (1.4)-(1.5)and v ∈ K.Then, where K is constant that depends on In other words, we have Since u is the entropy solution of (1.4)-(1.5),we have that Λ ε,ε 0 (u, v) ≥ 0, and hence, The terms I 0 and I T appear in the local case as well so they can be estimated on the similar lines of [22,19] to get: where ).Now, we estimate the other terms one by one.Using integration by parts I Φ ′ can be written as, Consequently,

Consider the term
the sense of measures, see [7, Lem.A2.1] for details), we have, Note that the term Consequently we get: U , and I 3 U satisfy the following estimates.
Collectively, we have for some appropriate constants K 2 and K 3 .Now, we consider, Note that Now, adding and subtracting Furthermore, . Collecting all terms, we have . Now, it can be observed that I Ux can be handled like (3.4), leading to the following estimate: for some appropriate constants K 4 and K 5 .Substituting the above estimates in (3.1), we get Now, the result follows by Gronwall's inequality.
The remaining section is dedicated to estimating the relative entropy functional Λ ε,ε 0 (u ∆ , u) for which we follow the following notations: Lemma 3.3.The relative entropy functional Λ ε,ε 0 (u ∆ , u) satisfies: where C is a constant independent of ∆x, ∆t.
Proof.Let  Applying the fundamental theorem of calculus, followed by summation by parts, we get x)Φ(s, y, t, x) dt dx ds dy We have ) dt dx ds dy ) .Also for t ∈ C n , using (2.10), we have Now using the Lipschitz continuity of U ∆ in the space variable, we have Furthermore, Finally, combining all the above estimates we get, Thus, we have Now, we consider )) Ĉn Φ(s, y, t, x i+1/2 ) dt ds dy x) Φ(s, y, t, x) − Φ(s, y, t, x i+1/2 ) dx dt dy ds ) Φ(s, y, t, x i+1/2 ) dx dt dy ds (3.12) The error terms can be estimated as below, using Theorem 2.2, equation (2.8), and . Furthermore, we have, by applying fundamental theorem of calculus and rearrange the terms, )) dx dt dy ds Apply summation by parts in i to get, Using (2.10), for t ∈ C n , we have Consequently, Finally, combining the above estimates, we get: Thus, so far we have proved )) Ĉi Φ(s, y, t n+1 , x) dx ds dy by applying the discrete entropy inequality (2.11).Furthermore, we can rewrite λ ′ 2 , cf. (3.7), as follows ) dt ds dy by using the fundamental theorem of calculus, followed by summation by parts.

Apply summation by parts to get
where for the following estimates hold: ) (3.17) Using (3.16)-(3.17),we have Now, since µ has bounded variation, the claim follows because of the following estimates, which is true because of (2.8) and the estimates ˆQT Ĉn Φ(s, y, t, x i+ 1 2 the proofs of which can be found in [22,Ex. 3.17].
Finally, estimates on the remaining boundary terms Ẽ23 and Ẽ24 , easily follow from (3.18).Specifically, )) Ĉi Φ(s, y, t 1 , x) dx ds dy Substituting the assertions of Claim 1 and Claim 2 in (3.14) we get Now, we state and prove the main result of this paper.
Theorem 3.4.[Rate of Convergence] Let u be the entropy solution of (1.4)-(1.5)and u ∆ be the numerical solution given by (2.2).Then we have the following convergence rate: Proof.The CFL condition implies that ∆x = O(∆t).Furthermore, the initial approximation (2.1) implies u ∆ 0 − u 0 = O(∆t).Now, the desired error estimate follows from Lemma 3.3 and Lemma 3.

Extension to Multi Dimensions
We consider the case of two space dimensions and denote the space variables by (x, y) ∈ R 2 , and consider the following PDE: Further, for numerical scheme, fix a rectangular grid with sizes ∆x and ∆y in R 2 and choose a time step ∆t.For later use, we also introduce the usual notation (t n , x i , y j ) = (n∆t, i∆x, j∆y), n ∈ N, i, j ∈ Z, λ x = ∆t ∆x , λ y = ∆t ∆y .
We define a piecewise constant approximate solution u ∆ by )×[y j−1/2 ,y j+1/2 ) (t, x, y), n ∈ N, i, j ∈ Z, where χ A denotes the indicator function of a set A, through the following marching formula based on dimensional splitting, (see [15,Sec. 3] and [21,Sec. 5] for details): ) , where F x,n i+1/2,j and F y,n i,j+1/2 denote the numerical approximations of the fluxes f 1 (u)ν 1 (µ 1 * β 1 (u)) and f 2 (u)ν 2 (µ 2 * β 2 (u)) at the interfaces (x i+1/2 , y j ) and (x i , y j+1/2 ), respectively, for i, j ∈ Z.The convolution terms are computed through quadrature formula, i.e., ), (4.3) where, u n l+1/2,p is any convex combination of u n l,p and u n l+1,p , with µ 1 i+1/2,j = µ 1 (x i+1/2 , y j ) and µ 2 i+1/2,j = µ 2 (x i+1/2 , y j ) .Throughout, we require that ∆t is chosen in order to satisfy the CFL conditions ) and with numerical fluxes F x or F y chosen as Lax-Friedrichs flux and Godunov flux, respectively.Extension to other monotone fluxes and for higher dimensions is similar.The numerical scheme can now be shown to converge to entropy solution, see, for example, [1].The Kuznetsov Lemma and the theorem on error estimate presented in Section 3 can now be extended to several space dimensions using dimension splitting arguments (see [22,Sec. 4.3]) with appropriate modifications throughout the proof.

Numerical Experiments
We now present some numerical experiments to illustrate the theory presented in the previous section.We show the results for the Lax-Friedrichs scheme.The results obtained by Godunov scheme are similar, and are not shown here.Throughout the section, θ = θ x = θ y is chosen to be 0.3333, and λ and λ x = λ y are chosen to be 0.1286 and 0.2857, respectively, so as to satisfy the CFL condition (2.5) and (4.4), respectively, in one and two dimensions, for any grid size ∆x or ∆y used in this section.
(5.1) Figure 4 displays the numerical approximations of (1.4), (5.1) generated by the numerical scheme (2.2), using space mesh ∆x = 0.0015625.The numerical solution of the corresponding local version (1.1), (5.1), obtained using standard Godunov scheme, has also been plotted at the respective times alongside the nonlocal solutions, as a reference to indicate the difference between the solutions of these two PDEs.Figure 1 shows the the numerical scheme is able to capture both shocks and rarefactions well, and that the density does not cross the maximal density 1.
To compute the convergence rate of the scheme (2.2), we obtain numerical approximations to (1.4), (5.1) with decreasing grid sizes ∆x, starting with ∆x = 0.00625.The convergence rate α is then calculated at time T = 0.5 by computing the L 1 distance between the numerical solutions u ∆x (T, • ) and u ∆x/2 (T, • ) obtained for the grid size ∆x and 1 2 ∆x, for each grid size ∆x.The results recorded in Table 1 and Figure 2 show that the observed convergence rates lie strictly between 0.5 and 1.
The present numerical integration resonates well with the  theoretical convergence rate obtained in Theorem 3.4 in this article.Figure 3 illustrates the nonlocal to local limit, see [9,11,25] and references therein, namely that the entropy solutions of the nonlocal conservation laws converge to the entropy solution of the corresponding local conservation law as the radius of the kernel goes to zero.This numerical example indicates that also the present model has singular solutions.

Two Dimensions
To illustrate our results in two dimensions, we employ the model introduced in [1], modeling crowd dynamics in two dimensions, which fits in the framework of the article.Assume that a group of pedestrians in a square room [−4, 4] 2 , can be described through the density u = u(t, x, y) that satisfies the nonlocal conservation law where the smooth, non-negative and compactly supported function µ models the way in which each individual averages the density around her/his position to adjust her/his speed.As initial data, we consider two cases: 1. Annular initial data, where the crowd is concentrated in an annulus: (x, y). (5.5) The system (5.2) fits into the setting of (4.2) with The numerical integrations of (5.2), (5.4) and (5.2), (5.  2 and Figure 5 show that the observed convergence rates lie strictly between 0.5 and 1.The present numerical integration resonate well with theoretical convergence rate obtained

Conclusions
In this article, we have established the convergence rate estimates for scalar nonlocal nonlinear conservation laws, modeling traffic and crowd dynamics, with no additional assumptions on

2 (
5) are obtained by the the algorithm described in Section 4 and are shown in Figure4.The figures depict that the density does not cross the maximal density 1 and the numerical simulations are able to capture the physical properties well.To compute the convergence rate of the scheme (4.2) with Lax-Friedrichs flux, we apply the algorithm to problem (5.2), (5.4) and (5.2), (5.5) on the domain [−4, 4] 2 on the time interval [0, 0.5] with different grid sizes with λ x = λ y = 0.2857.The convergence rate α is then calculated at time T = 0.5 by computing the L 1 distance between the numerical solutions u ∆x (T, • ) and u∆x T, • ) obtained for the grid size ∆x and ∆x 2 , for each grid size ∆x.The results recorded in Table

Table 1 :
Convergence rate α for the numerical scheme