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Review

Avocado/Soybean Unsaponifiables (Piacledine®300) show beneficial effect on the metabolism of osteoarthritic cartilage, synovium and subchondral bone: An overview of the mechanisms

  • Received: 02 September 2017 Accepted: 11 December 2017 Published: 26 December 2017
  • Objectives: The aim of this narrative review of the literature was to synthesize and comment the mechanisms of action of avocado/soybean unsaponifiable mixture (ASU-E, Piascledine®300) on articular tissues involved in the OA pathogenesis. Materials and methods: The search was performed in Pubmed and Scopus between January 1981 and December 2016. Keywords used were—any field—(Cartilage OR Bone OR Synovium) AND Avocado AND Soybean. 32 articles out-off 35 found have been considered. The review has included eleven in vitro and animal studies investigating Avocado Soybean Unsaponifiables (ASU) from Laboratoires Expanscience (Piascledine®300) used separately or in combination. Only research articles published in English and French have been taken into account. Results: ASU-E stimulated proteoglycans synthesis in chondrocytes cultures and counteracted the effects of IL-1 on metalloproteases and inflammatory mediators. Some of these effects were associated with inhibition of NF-kB nuclear translocation and stimulation of TGF-synthesis. ASU-E also positively modulated the altered phenotype of OA subchondral bone osteoblasts and reduced the production of collagenases by synovial cells. Conclusions: ASU-E has positive effects on the metabolic changes of synovium, subchondral bone and cartilage which are the main tissues involved in the pathophysiology of OA. These findings contribute to explain the beneficial effects of ASU-E in clinical trials.

    Citation: Yves Edgard Henrotin. Avocado/Soybean Unsaponifiables (Piacledine®300) show beneficial effect on the metabolism of osteoarthritic cartilage, synovium and subchondral bone: An overview of the mechanisms[J]. AIMS Medical Science, 2018, 5(1): 33-52. doi: 10.3934/medsci.2018.1.33

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  • Objectives: The aim of this narrative review of the literature was to synthesize and comment the mechanisms of action of avocado/soybean unsaponifiable mixture (ASU-E, Piascledine®300) on articular tissues involved in the OA pathogenesis. Materials and methods: The search was performed in Pubmed and Scopus between January 1981 and December 2016. Keywords used were—any field—(Cartilage OR Bone OR Synovium) AND Avocado AND Soybean. 32 articles out-off 35 found have been considered. The review has included eleven in vitro and animal studies investigating Avocado Soybean Unsaponifiables (ASU) from Laboratoires Expanscience (Piascledine®300) used separately or in combination. Only research articles published in English and French have been taken into account. Results: ASU-E stimulated proteoglycans synthesis in chondrocytes cultures and counteracted the effects of IL-1 on metalloproteases and inflammatory mediators. Some of these effects were associated with inhibition of NF-kB nuclear translocation and stimulation of TGF-synthesis. ASU-E also positively modulated the altered phenotype of OA subchondral bone osteoblasts and reduced the production of collagenases by synovial cells. Conclusions: ASU-E has positive effects on the metabolic changes of synovium, subchondral bone and cartilage which are the main tissues involved in the pathophysiology of OA. These findings contribute to explain the beneficial effects of ASU-E in clinical trials.



    Macroscopic traffic flow models based on fluid-dynamics equations have been introduced in the transport engineering literature since the mid-fifties of last century, with the celebrated Lighthill, Whitham [11] and Richards [13] (LWR) model. Since then, the engineering and applied mathematical literature on the subject has considerably grown, addressing the need for more sophisticated models better capturing traffic flow characteristics. Indeed, the LWR model is based on the assumption that the mean traffic speed is a function of the traffic density, which is not experimentally verified in congested regimes. To overcome this issue, the so-called "second order" models (e.g. Payne-Whitham [12,15] and Aw-Rascle-Zhang [3,16]) consist of a mass conservation equation for the density and an acceleration balance law for the speed, thus considering the two quantities as independent.

    More recently, "non-local" versions of the LWR model have been proposed in [5,14], where the speed function depends on a weighted mean of the downstream vehicle density to better represent the reaction of drivers to downstream traffic conditions.

    Another limitation of the standard LWR model is the first-in first-out rule, not allowing faster vehicles to overtake slower ones. To address this and other traffic heterogeneities, "multi-class" models consist of a system of conservation equations, one for each vehicle class, coupled in the speed terms, see [4] and references therein for more details.

    In this paper, we consider the following class of non-local systems of $ M $ conservation laws in one space dimension:

    $ tρi(t,x)+x(ρi(t,x)vi((rωi)(t,x)))=0,i=1,...,M, $ (1)

    where

    $ r(t,x):=Mi=1ρi(t,x), $ (2)
    $ vi(ξ):=vmaxiψ(ξ), $ (3)
    $ (rωi)(t,x):=x+ηixr(t,y)ωi(yx)dy, $ (4)

    and we assume:

    $ \mathbf{(H1)} $ The convolution kernels $ \omega_i\in {{\mathbf{C}}^{\mathbf{1}}}([0, \eta_i]; \mathbb{R}^+) $, $ \eta_i>0 $, are non-increasing functions such that $ \int_{0}^{\eta_i}\omega_i(y)\mathinner{{\rm{d}}{y}} = J_i $. We set $ W_0: = \max_{i = 1, \ldots, M}\omega_i(0) $.

    $ \mathbf{(H2)} $ $ v_i^{\max} $ are the maximal velocities, with $ 0<v_1^{\max}\leq v_2^{\max}\leq \ldots\leq v_M^{\max} $.

    $ \mathbf{(H3)} $ $ \psi: \mathbb{R}^+\rightarrow \mathbb{R}^+ $ is a smooth non-increasing function such that $ \psi(0) = 1 $ and $ \psi(r) = 0 $ for $ r\geq 1 $ (for simplicity, we can consider the function $ \psi(r) = \max\{1-r, 0\} $).

    We couple (1) with an initial datum

    $ ρi(0,x)=ρ0i(x),i=1,,M. $ (5)

    Model (1) is obtained generalizing the $ n $-populations model for traffic flow described in [4] and it is a multi-class version of the one dimensional scalar conservation law with non-local flux proposed in [5], where $ \rho_i $ is the density of vehicles belonging to the $ i $-th class, $ \eta_i $ is proportional to the look-ahead distance and $ J_i $ is the interaction strength. In our setting, the non-local dependence of the speed functions $ v_i $ describes the reaction of drivers that adapt their velocity to the downstream traffic, assigning greater importance to closer vehicles, see also [7,9]. We allow different anisotropic kernels for each equation of the system. The model takes into account the distribution of heterogeneous drivers and vehicles characterized by their maximal speeds and look-ahead visibility in a traffic stream.

    Due to the possible presence of jump discontinuities, solutions to (1), (5) are intended in the following weak sense.

    Definition 1.1. A function $ \boldsymbol{\rho} = (\rho_1, \ldots, \rho_M)\in ({{\mathbf{L}}^{\mathbf{1}}}\cap{{\mathbf{L}}^\infty })([0, T[\, \times \mathbb{R}; \mathbb{R}^M) $, $ T>0 $, is a weak solution of (1), (5) if

    $ \int_0^T\!\!\int_{-\infty}^\infty \left(\rho_i \partial_t \varphi +\rho_i v_i(r\ast\omega_i) \partial_x \varphi \right)(t, x) \mathinner{{\rm{d}}{x}} \mathinner{{\rm{d}}{t}} +\int_{-\infty}^\infty \rho_i^0 (x)\varphi(0, x) \mathinner{{\rm{d}}{x}} = 0 $

    for all $ \varphi\in\mathbf{C}_{c}^{\mathbf{1}} (]-\infty, T[\, \times \mathbb{R}; \mathbb{R}) $, $ i = 1, \ldots, M $.

    The main result of this paper is the proof of existence of weak solutions to (1), (5), locally in time. We remark that, since the convolution kernels $ \omega_i $ are not smooth on $ \mathbb{R} $, the results in [1] cannot be applied due to the lack of $ {{\mathbf{L}}^\infty } $-bounds on their derivatives.

    Theorem 1.2. Let $ \rho_i^0(x) \in \left( \mathbf{BV}\cap{{\mathbf{L}}^\infty }\right)( \mathbb{R}; \mathbb{R}^+) $, for $ i = 1, \ldots, M $, and assumptions (H1) - (H3) hold. Then the Cauchy problem (1), (5) admits a weak solution on $ [0, T[\, \times \mathbb{R} $, for some $ T>0 $ sufficiently small.

    In this work, we do not address the question of uniqueness of the solutions to (1). Indeed, even if discrete entropy inequalities can be derived as in [5,Proposition 3], in the case of systems this is in general not sufficient to single out a unique solution.

    The paper is organized as follows. Section 2 is devoted to prove uniform $ {{\mathbf{L}}^\infty } $ and BV estimates on the approximate solutions obtained through an approximation argument based on a Godunov type numerical scheme, see [8]. We have to point out that these estimates heavily rely on the monotonicity properties of the kernel functions $ \omega_i. $ In Section 3 we prove the existence in finite time of weak solutions applying Helly's theorem and a Lax-Wendroff type argument, see [10]. In Section 4 we present some numerical simulations for $ M = 2 $. In particular, we consider the case of a mixed flow of cars and trucks on a stretch of road, and the flow of mixed autonomous and non-autonomous vehicles on a circular road. In this latter case, we analyze two cost functionals measuring the traffic congestion, depending on the penetration ratio of autonomous vehicles. The final Appendix contains alternative $ {{\mathbf{L}}^\infty } $ and $ \mathbf{BV} $ estimates, based on approximate solutions constructed via a Lax-Friedrichs type scheme, which is commonly used in the framework of non-local equations, see [1,2,5].

    First of all, we extend $ \omega_i(x) = 0 $ for $ x>\eta_i $. For $ j\in \mathbb{Z} $ and $ n\in \mathbb{N}, $ let $ x_{j+1/2} = j \Delta x $ be the cell interfaces, $ x_j = (j-1/2)\Delta x $ the cells centers and $ t^n = n\Delta t $ the time mesh. We aim at constructing a finite volume approximate solution $ \boldsymbol{\rho}^{\Delta x} = \left(\rho_1^{\Delta x}, \ldots, \rho_M^{\Delta x}\right) $, with $ \rho_i^{\Delta x}(t, x) = \rho_{i, j}^n $ for $ (t, x)\in C^n_j = [t^n, t^{n+1}[\times ]x_{j-1/2}, x_{j+1/2}] $ and $ i = 1, ..., M. $

    To this end, we approximate the initial datum $ \rho_i^0 $ for $ i = 1, ..., M $ with a piecewise constant function

    $ ρ0i,j=1Δxxj+1/2xj1/2ρ0i(x)dx,jZ. $

    Similarly, for the kernel, we set

    $ ωki:=1Δx(k+1)ΔxkΔxω0i(x)dx,kN, $

    so that $ \Delta x \sum_{k = 0}^{+\infty}\omega_i^k = \int_0^{\eta_i}\omega_i(x) \mathinner{{\rm{d}}{x}} = J_i $ (the sum is indeed finite since $ \omega_i^k = 0 $ for $ k\geq N_i $ sufficiently large). Moreover, we set $ r^n_{j+k} = \mathop \sum \limits_{i = 1}^M \rho_{i, j+k}^n $ for $ k\in \mathbb{N} $ and

    $ Vni,j:=vmaxiψ(Δx+k=0ωkirnj+k),i=1,,M,jZ. $ (6)

    We consider the following Godunov-type scheme adapted to (1), which was introduced in [8] in the scalar case:

    $ ρn+1i,j=ρni,jλ(ρni,jVni,j+1ρni,j1Vni,j) $ (7)

    where we have set $ \lambda = \frac{\Delta t}{\Delta x} $.

    We provide here the necessary estimates to prove the convergence of the sequence of approximate solutions constructed via the Godunov scheme (7).

    Lemma 2.1. (Positivity) For any $ T>0, $ under the CFL condition

    $ λ1vmaxMψ, $ (8)

    the scheme (7) is positivity preserving on $ [0, T]\times \mathbb{R}. $

    Proof. Let us assume that $ \rho_{i, j}^n\geq 0 $ for all $ j\in \mathbb{Z} $ and $ i\in{1, ..., M}. $ It suffices to prove that $ \rho_{i, j}^{n+1} $ in (7) is non-negative. We compute

    $ ρn+1i,j=ρni,j(1λVni,j+1)+λρni,j1Vni,j0 $ (9)

    under assumption (8).

    Corollary 1. ($ \mathbf{L^{1}} $-bound) For any $ n\in \mathbb{N} $, under the CFL condition (8) the approximate solutions constructed via the scheme (7) satisfy

    $ ρni1=ρ0i1,i=1,,M, $ (10)

    where $ {\left\|{\rho_i^n}\right\|}_{1}: = \Delta x \sum_j {\left|{\rho_{i, j}^n}\right|} $ denotes the $ \mathbf{L^{1}} $ norm of the $ i $-th component of $ \rho^{\Delta x} $.

    Proof. Thanks to Lemma 2.1, for all $ i\in\{1, ..., M\} $ we have

    $ ρn+1i1=Δxjρn+1i,j=Δxj(ρni,jλρni,jVni,j+1+λρni,j1Vni,j)=Δxjρni,j, $

    proving (10).

    Lemma 2.2. ($ {{\mathbf{L}}^\infty } $-bound) If $ \rho_{i, j}^0\geq 0 $ for all $ j\in \mathbb{Z} $ and $ i = 1, ..., M $, and (8) holds, then the approximate solution $ \boldsymbol{\rho}^{\Delta x} $ constructed by the algorithm (7) is uniformly bounded on $ [0, T]\times \mathbb{R} $ for any $ T $ such that

    $ T < \left( M {\left\|{{ \boldsymbol{\rho}}^0}\right\|}_\infty v_M^{\max} {\left\|{\psi'}\right\|}_\infty W_0\right)^{-1}. $

    Proof. Let $ \bar\rho = \max\{\rho_{i, j-1}^n, \rho_{i, j}^n\}. $ Then we get

    $ ρn+1i,j=ρni,j(1λVni,j+1)+λρni,j1Vni,jˉρ(1+λ(Vni,jVni,j+1)) $ (11)

    and

    $ |Vni,jVni,j+1|=vmaxi|ψ(Δx+k=0ωkirnj+k)ψ(Δx+k=0ωkirnj+k+1)|vmaxiψΔx|+k=0ωki(rnj+k+1rnj+k)|=vmaxiψΔx|ω0irnj++k=1(ωk1iωki)rnj+k|vmaxiψΔxMρnωi(0) $ (12)

    where $ {\left\|{{ \boldsymbol{\rho}}}\right\|}_\infty = {\left\|{(\rho_1, \ldots, \rho_M)}\right\|}_\infty = \max_{i, j}{\left|{\rho_{i, j}}\right|} $. Let now $ K>0 $ be such that $ {\left\|{{{ \boldsymbol{\rho}}}^{\ell}}\right\|}_\infty\leq K $, $ \ell = 0, \ldots, n $. From (11) and (12) we get

    $ {\left\|{{{ \boldsymbol{\rho}}}^{n+1}}\right\|}_\infty \leq {\left\|{{{ \boldsymbol{\rho}}}^{n}}\right\|}_\infty \left( 1+ M K v_M^{\max} {\left\|{\psi'}\right\|}_\infty W_0 \Delta t \right), $

    which implies

    $ ρnρ0eCnΔt, $

    with $ C = M K v_M^{\max} {\left\|{\psi'}\right\|}_\infty W_0 $. Therefore we get that $ {\left\|{{ \boldsymbol{\rho}}(t, \cdot)}\right\|}_\infty \leq K $ for

    $ t\leq \frac{1}{ M K v_M^{\max} {\left\|{\psi'}\right\|}_\infty W_0}\ln \left(\frac{K}{{\left\|{{ \boldsymbol{\rho}}^0}\right\|}_\infty}\right) \leq \frac{1}{ M e {\left\|{{ \boldsymbol{\rho}}^0}\right\|}_\infty v_M^{\max} {\left\|{\psi'}\right\|}_\infty W_0}\, , $

    where the maximum is attained for $ K = e {\left\|{{ \boldsymbol{\rho}}^0}\right\|}_\infty $.

    Iterating the procedure, at time $ t^m $, $ m\geq 1 $ we set $ K = e^m {\left\|{{ \boldsymbol{\rho}}^0}\right\|}_\infty $ and we get that the solution is bounded by $ K $ until $ t^{m+1} $ such that

    $ t^{m+1}\leq t^m + \frac{m}{ M e^m {\left\|{{ \boldsymbol{\rho}}^0}\right\|}_\infty v_M^{\max} {\left\|{\psi'}\right\|}_\infty W_0}. $

    Therefore, the approximate solution remains bounded, uniformly in $ \Delta x $, at least for $ t\leq T $ with

    $ T\leq \frac{1}{ M {\left\|{{ \boldsymbol{\rho}}^0}\right\|}_\infty v_M^{\max} {\left\|{\psi'}\right\|}_\infty W_0} \sum\limits_{m = 1}^{+\infty} \frac{m}{e^m} \leq \frac{1}{ M {\left\|{{ \boldsymbol{\rho}}^0}\right\|}_\infty v_M^{\max} {\left\|{\psi'}\right\|}_\infty W_0 }\, . $

    Remark 1. Figure 1 shows that the simplex

    Figure 1.  Numerical simulation illustrating that the simplex $\mathcal{S}$ is not an invariant domain for (1). We take $M = 2$ and we consider the initial conditions $\rho_1(0, x) = 0.9 \chi_{[-0.5, -0.3]}$ and $\rho_2(0, x) = 0.1 \chi_{]-\infty, 0]}+\chi_{]0, +\infty[}$ depicted in (a), the constant kernels $\omega_1(x) = \omega_2(x) = 1/\eta$, $\eta = 0.5$, and the speed functions given by $v^{max}_1 = 0.2$, $v^{max}_2 = 1$, $\psi(\xi) = \max\{1-\xi, 0\}$ for $\xi\geq 0$. The space and time discretization steps are $\Delta x = 0.001$ and $\Delta t = 0.4 \Delta x$. Plots (b) and (c) show the density profiles of $\rho_1$, $\rho_2$ and their sum $r$ at times $t = 1.8, ~2.8$. The function $\max_{x\in\mathbb{R}} r(t, x)$ is plotted in (d), showing that $r$ can take values greater than 1, even if $r(0, x) = \rho_1(0, x)+\rho_2(0, x)\leq1$.
    $ \mathcal{S}: = \left\{ \boldsymbol{\rho}\in \mathbb{R}^M \colon \sum\limits_{i = 1}^M \rho_i \leq 1, \; \rho_i \geq 0\; \mbox{for}\; i = 1, \ldots, M\right\} $

    is not an invariant domain for (1), unlike the classical multi-population model [4]. Indeed, let us consider the system

    $ tρi(t,x)+x(ρi(t,x)vi(r(t,x)))=0,i=1,...,M, $ (13)

    where $ r $ and $ v_i $ are as in (2) and (3), respectively. We have the following:

    Lemma 2.3. Under the CFL condition

    $ λ1vmaxM(ψ+ψ), $

    for any initial datum $ \boldsymbol{\rho}_0\in \mathcal{S} $ the approximate solutions to (13) computed by the upwind scheme

    $ ρn+1j=ρnjλ[F(ρnj,ρnj+1)F(ρnj1,ρnj)], $ (14)

    with $ \mathbf{F} ( \boldsymbol{\rho}_j^n, \boldsymbol{\rho}_{j+1}^n) = \boldsymbol{\rho}^n_j \psi(r^n_{j+1}) $, satisfy the following uniform bounds:

    $ ρnjSjZ,nN. $ (15)

    Proof. Assuming that $ \boldsymbol{\rho}_j^n\in \mathcal{S} $ for all $ j\in \mathbb{Z} $, we want to prove that $ \boldsymbol{\rho}_j^{n+1}\in \mathcal{S} $. Rewriting (14), we get

    $ ρn+1i,j=ρni,jλ[vmaxiρni,jψ(rnj+1)vmaxiρni,j1ψ(rnj)]. $

    Summing on the index $ i = 1, \ldots, M $, gives

    $ rn+1j=Mi=1ρn+1i,j=Mi=1ρni,jλMi=1[vmaxiρni,jψ(rnj+1)vmaxiρni,j1ψ(rnj)]=rnj+λψ(rnj)Mi=1vmaxiρni,j1λψ(rnj+1)Mi=1vmaxiρni,j. $

    Defining the following function of $ \boldsymbol{\rho}_j^n $

    $ \Phi(\rho^n_{1, j}, \ldots, \rho^n_{M, j}) = r^n_j+\lambda {\psi(r^n_j)\sum\limits_{i = 1}^{M}} v_i^{\max}\rho^n_{i, j-1}-\lambda {\psi(r^n_{j+1})\sum\limits_{i = 1}^{M}} v_i^{\max}\rho^n_{i, j}, $

    we observe that

    $ \Phi(0, \ldots, 0) = \lambda \psi(0)\sum\limits_{i}^M v_i^{\max}\rho^n_{i, j-1}\leq \lambda {\left\|{\psi}\right\|}_\infty v_{M}^{\max} \leq 1 $

    if $ \lambda \leq 1/v_{M}^{\max} {\left\|{\psi}\right\|}_\infty $ and

    $ \Phi(\rho^n_{1, j}, ..., \rho^n_{M, j}) = 1-\lambda \psi(r^n_{j+1})\sum\limits_{i = 1}^{M}v_i^{\max}\rho^n_{i, j}\leq 1 $

    for $ \boldsymbol{\rho}_j^n\in \mathcal{S} $ such that $ r^n_j = \sum_{i = 1}^M \rho^n_{i, j} = 1 $. Moreover

    $ \frac{ \partial\Phi}{ \partial\rho^n_{i, j}} ( \boldsymbol{\rho}_j^n) = 1 + \lambda \psi'(r^n_j) \sum\limits_{i = 1}^M v_i^{\max} \rho^n_{i, j-1} - \lambda\psi(r^n_{j+1}) v_i^{\max} \geq 0 $

    if $ \lambda\leq 1/ v_M^{\max}\left({\left\|{\psi}\right\|}_\infty+{\left\|{\psi'}\right\|}_\infty\right) $. This proves that $ r_j^{n+1}\leq 1 $. To prove the positivity of (14), we observe that

    $ ρn+1i,j=ρni,j(1λvmaxiψ(rnj+1))+λvmaxiρni,j1ψ(rnj)0 $

    if $ \lambda\leq 1/ v_M^{\max} {\left\|{\psi}\right\|}_\infty $.

    Lemma 2.4. (Spatial $ \mathbf{BV} $-bound) Let $ \rho_i^0\in \left( \mathbf{BV}\cap{{\mathbf{L}}^\infty }\right) ( \mathbb{R}, \mathbb{R}^+) $ for all $ i = 1, ..., M. $ If (8) holds, then the approximate solution $ \boldsymbol{\rho}^{\Delta x}(t, \cdot) $ constructed by the algorithm (7) has uniformly bounded total variation for $ t\in [0, T] $, for any $ T $ such that

    $ Tmini=1,,M 1H(TV(ρ0i)+1), $ (16)

    where $ \mathcal{H} = {\left\|{{ \boldsymbol{\rho}}}\right\|}_\infty\, v_M^{\max} W_0 M\left(6 M J_0 {\left\|{{ \boldsymbol{\rho}}}\right\|}_\infty{\left\|{\psi''}\right\|}_\infty +{\left\|{\psi'}\right\|}_\infty\right) $.

    Proof. Subtracting the identities

    $ ρn+1i,j+1=ρni,j+1λ(ρni,j+1Vni,j+2ρni,jVni,j+1), $ (17)
    $ ρn+1i,j=ρni,jλ(ρni,jVni,j+1ρni,j1Vni,j), $ (18)

    and setting $ \Delta^{n}_{i, j+1/2} = \rho_{i, j+1}^n-\rho_{i, j}^n $, we get

    $ Δn+1i,j+1/2=Δni,j+1/2λ(ρni,j+1Vni,j+22ρni,jVni,j+1+ρni,j1Vni,j). $

    Now, we can write

    $ Δn+1i,j+1/2=(1λVni,j+2)Δni,j+1 $ (19)
    $ +λVni,jΔni,j1/2λρni,j(Vni,j+22Vni,j+1+Vni,j). $ (20)

    Observe that assumption (8) guarantees the positivity of (19). The term (20) can be estimated as

    $ Vni,j+22Vni,j+1+Vni,j==vmaxi(ψ(Δx+k=0ωkirnj+k+2)2ψ(Δx+k=0ωkirnj+k+1)+ψ(Δx+k=0ωkirnj+k))=vmaxiψ(ξj+1)Δx(+k=0ωkirnj+k+2+k=0ωkirnj+k+1)+vmaxiψ(ξj)Δx(+k=0ωkirnj+k+k=0ωkirnj+k+1)=vmaxiψ(ξj+1)Δx(+k=1(ωk1iωki)rnj+k+1ω0irnj+1)+vmaxiψ(ξj)Δx(+k=1(ωkiωk1i)rnj+k+ω0irnj)=vmaxi(ψ(ξj+1)ψ(ξj))Δx(+k=1(ωk1iωki)rnj+k+1ω0irnj+1)+vmaxiψ(ξj)Δx(+k=1(ωk1iωki)(rnj+k+1rnj+k)+ω0i(rnjrnj+1))=vmaxiψ(˜ξj+1/2)(ξj+1ξj)Δx(+k=1Mβ=1ωkiΔnβ,j+k+3/2)+vmaxiψ(ξj)Δx(Mβ=1N1k=1(ωk1iωki)Δnβ,j+k+1/2ω0iΔnβ,j+1/2), $

    with $ \xi_j \in {\mathcal I}\left( \Delta x\sum_{k = 0}^{+\infty} \omega_i^k r_{j+k}^n, \Delta x\sum_{k = 0}^{+\infty}\omega_i^k r_{j+k+1}^n\right) $ and $ \tilde\xi_{j+1/2}\in {\mathcal I}\left(\xi_{j}, \xi_{j+1}\right) $, where we set $ {\mathcal I} (a, b) = \left[\min\{a, b\}, \max\{a, b\}\right] $. For some $ \vartheta, \mu\in [0, 1] $, we compute

    $ ξj+1ξj=ϑΔx+k=0ωkiMβ=1ρnβ,j+k+2+(1ϑ)Δx+k=0ωkiMβ=1ρnβ,j+k+1μΔx+k=0ωkiMβ=1ρnβ,j+k+1(1μ)Δx+k=0ωkiMβ=1ρnβ,j+k=ϑΔx+k=1ωk1iMβ=1ρnβ,j+k+1+(1ϑ)Δx+k=0ωkiMβ=1ρnβ,j+k+1μΔx+k=0ωkiMβ=1ρnβ,j+k+1(1μ)Δx+k=1ωk+1iMβ=1ρnβ,j+k+1=Δx+k=1[ϑωk1i+(1ϑ)ωkiμωki(1μ)ωk+1i]Mβ=1ρnβ,j+k+1+(1ϑ)Δxω0iMβ=1ρnβ,j+1μΔxω0iMβ=1ρnβ,j+1(1μ)Δx(ω0iMβ=1ρnβ,j+ω1iMβ=1ρnβ,j+1). $

    By monotonicity of $ \omega_i $ we have

    $ \vartheta \omega_i^{k-1} + (1- \vartheta)\omega_i^{k} -\mu \omega_i^{k} -(1-\mu)\omega_i^{k+1} \geq 0\, . $

    Taking the absolute values we get

    $ |ξj+1ξj|Δx{+k=2[ϑωk1i+(1ϑ)ωkiμωki(1μ)ωk+1i]+4ω0i}MρnΔx{+k=2[ωk1iωk+1i]+4ω0i}MρnΔx6W0Mρn. $

    Let now $ K_1>0 $ be such that $ \sum_{j}{\left|{\Delta^\ell_{\beta, j}}\right|}\leq K_1 \text{ for } \beta = 1, \ldots, M $, $ \ell = 0, \ldots, n $. Taking the absolute values and rearranging the indexes, we have

    $ j|Δn+1i,j+1/2|j|Δni,j+1/2|(1λ(Vni,j+2Vni,j+1))+ΔtHK1, $

    where $ \mathcal{H} = {\left\|{{ \boldsymbol{\rho}}}\right\|}_\infty\, v_M^{\max} W_0 M\left(6 M J_0 {\left\|{{ \boldsymbol{\rho}}}\right\|}_\infty{\left\|{\psi"}\right\|}_\infty +{\left\|{\psi'}\right\|}_\infty\right). $ Therefore, by (12) we get

    $ j|Δn+1i,j+1/2|j|Δni,j+1/2|(1+ΔtG)+ΔtHK1, $

    with $ \mathcal{G} = \, v_M^{\max} {\left\|{\psi'}\right\|}_\infty W_0 M {\left\|{{ \boldsymbol{\rho}}}\right\|}_\infty. $ We thus obtain

    $ j|Δni,j+1/2|eGnΔtj|Δ0i,j+1/2|+eHK1nΔt1, $

    that we can rewrite as

    $ TV(ρΔxi)(nΔt,)eGnΔtTV(ρ0i)+eHK1nΔt1eHK1nΔt(TV(ρ0i)+1)1, $

    since $ \mathcal{H}\geq \mathcal{G} $ and it is not restrictive to assume $ K_1 \geq 1 $. Therefore, we have that $ \mathinner{{\rm{TV}}}(\rho_i^{\Delta x})\leq K_1 $ for

    $ t1HK1ln(K1+1TV(ρ0i)+1), $

    where the maximum is attained for some $ K_1 < e \left( \mathinner{{\rm{TV}}}(\rho^0_i)+1\right) -1 $ such that

    $ \ln\left(\frac{K_1+1}{ \mathinner{{\rm{TV}}}(\rho^0_i)+1}\right) = \frac{K_1}{K_1+1}\, . $

    Therefore the total variation is uniformly bounded for

    $ t1He(TV(ρ0i)+1). $

    Iterating the procedure, at time $ t^m $, $ m\geq 1 $ we set $ K_1 = e^{m } \left( \mathinner{{\rm{TV}}}(\rho^0_i)+1\right)-1 $ and we get that the solution is bounded by $ K_1 $ until $ t^{m+1} $ such that

    $ tm+1tm+mHem(TV(ρ0i)+1). $ (21)

    Therefore, the approximate solution has bounded total variation for $ t\leq T $ with

    $ T1H(TV(ρ0i)+1). $

    Corollary 2. Let $ \rho_i^0\in \left( \mathbf{BV}\cap{{\mathbf{L}}^\infty }\right)( \mathbb{R}; \mathbb{R}^+) $. If (8) holds, then the approximate solution $ \boldsymbol{\rho}^{\Delta x} $ constructed by the algorithm (7) has uniformly bounded total variation on $ [0, T]\times \mathbb{R} $, for any $ T $ satisfying (16).

    Proof. If $ T\leq \Delta t, $ then $ \mathinner{{\rm{TV}}} (\rho_i^{\Delta x}; [0, T] \times \mathbb{R} )\leq T \mathinner{{\rm{TV}}}(\rho_i^0). $ Let us assume now that $ T> \Delta t. $ Let $ n_T\in \mathbb{N} \backslash \{0\} $ such that $ n_T \Delta t< T \leq (n_T+1) \Delta t $. Then

    $ TV(ρΔxi;[0,T]×R)=nT1n=0jZΔt|ρni,j+1ρni,j|+(TnTΔt)jZ|ρnTi,j+1ρnTi,j|Tsupt[0,T]TV(ρΔxi)(t,)+nT1n=0jZΔx|ρn+1i,jρni,j|. $

    We then need to bound the term

    $ \sum\limits_{n = 0}^{n_T-1}\sum\limits_{j\in \mathbb{Z}}\Delta x {\left|{\rho_{i, j}^{n+1}-\rho_{i, j}^n}\right|}. $

    From the definition of the numerical scheme (7), we obtain

    $ ρn+1i,jρni,j=λ(ρni,j1Vni,jρni,jVni,j+1)=λ(ρni,j1(Vni,jVni,j+1)+Vni,j+1(ρni,j1ρni,j)). $

    Taking the absolute values and using (12) we obtain

    $ |ρn+1i,jρni,j|λ(vmaxiψMρnωi(0)Δx|ρni,j1|+vmaxiψ|ρni,j1ρni,j|). $

    Summing on $ j $, we get

    $ jZΔx|ρn+1i,jρni,j|=vmaxiψMρnωi(0)ΔtjZΔx|ρni,j1|+vmaxiψΔtjZ|ρni,j1ρni,j|, $

    which yields

    $ nT1n=0jZΔx|ρn+1i,jρni,j|vmaxMψTsupt[0,T]TV(ρΔxi)(t,)+vmaxMψMW0Tsupt[0,T]ρΔxi(t,)1ρΔxi(t,) $

    that is bounded by Corollary 1, Lemma 2.2 and Lemma 2.4.

    To complete the proof of the existence of solutions to the problem (1), (5), we follow a Lax-Wendroff type argument as in [5], see also [10], to show that the approximate solutions constructed by scheme (7) converge to a weak solution of (1). By Lemma 2.2, Lemma 2.4 and Corollary 2, we can apply Helly's theorem, stating that for $ i = 1, \ldots, M $, there exists a subsequence, still denoted by $ \rho_i^{\Delta x} $, which converges to some $ \rho_i\in ({{\mathbf{L}}^{\mathbf{1}}}\cap \mathbf{BV})([0, T]\times \mathbb{R}; \mathbb{R}^+) $ in the $ \mathbf{L^{{1}}_{loc}} $-norm. Let us fix $ i\in\{1, \ldots, M\}. $ Let $ \varphi\in\mathbf{C}_c^{{1}}([0, T[\, \times \mathbb{R}) $ and multiply (7) by $ \varphi(t^n, x_j). $ Summing over $ j\in \mathbb{Z} $ and $ n\in\{0, \ldots, n_T\} $ we get

    $ nT1n=0jφ(tn,xj)(ρn+1i,jρni,j)=λnT1n=0jφ(tn,xj)(ρni,jVni,j+1ρni,j1Vni,j). $

    Summing by parts we obtain

    $ jφ((nT1)Δt,xj)ρnTi,j+jφ(0,xj)ρ0i,j+nT1n=1j(φ(tn,xj)φ(tn1,xj))ρni,j+λnT1n=0j(φ(tn,xj+1)φ(tn,xj))Vni,j+1ρni,j=0. $ (22)

    Multiplying by $ \Delta x $ we get

    $ Δxjφ((nT1)Δt,xj)ρnTi,j+Δxjφ(0,xj)ρ0i,j $ (23)
    $ +ΔxΔtnT1n=1j(φ(tn,xj)φ(tn1,xj))Δtρni,j $ (24)
    $ +ΔxΔtnT1n=0j(φ(tn,xj+1)φ(tn,xj))ΔxVni,j+1ρni,j=0. $ (25)

    By $ \mathbf{L^{{1}}_{loc}} $ convergence of $ \rho_i^{\Delta x}\to \rho_i, $ it is straightforward to see that the terms in (23), (24) converge to

    $ R(ρ0i(x)φ(0,x)ρi(T,x)φ(T,x))dx+T0Rρi(t,x)tφ(t,x)dxdt, $ (26)

    as $ \Delta x \to 0. $ Concerning the last term (25), we can rewrite

    $ ΔxΔtnT1n=0jφ(tn,xj+1)φ(tn,xj)ΔxVni,j+1ρni,j=ΔxΔtnT1n=0jφ(tn,xj+1)φ(tn,xj)Δx(ρni,jVni,j+1ρni,jVni,j)+ΔxΔtnT1n=0jφ(tn,xj+1)φ(tn,xj)Δxρni,jVni,j. $ (27)

    By (12) we get the estimate

    $ ρni,jVni,j+1ρni,jVni,jvmaxiψΔxMρ2ωi(0). $

    Set $ R>0 $ such that $ \varphi(t, x) = 0 $ for $ {\left|{x}\right|}>R $ and $ j_0, \, j_1\in \mathbb{Z} $ such that $ -R\in \, ]x_{j_0-\frac{1}{2}}, x_{j_0+\frac{1}{2}}] $ and $ R\in\, ]x_{j_1-\frac{1}{2}}, x_{j_1+\frac{1}{2}}], $ then

    $ ΔxΔtnTn=0jφ(tn,xj+1)φ(tn,xj)Δx(ρni,jVni,j+1ρni,jVni,j)ΔxΔtxφnTn=0j1j=j0vmaxiψMρ2ωi(0)ΔxxφvmaxiψMρ2ωi(0)Δx2RT, $

    which goes to zero as $ \Delta x \to 0. $

    Finally, again by the $ \mathbf{L^{{1}}_{loc}} $ convergence of $ \rho_i^{\Delta x} \to \rho_i $, we have that

    $ ΔxΔtnT1n=0j(φ(tn,xj+1)φ(tn,xj))Δxρni,jVni,j12T0Rxφ(t,x)ρi(t,x)vi(rωi)dxdt. $

    In this section we perform some numerical simulations to illustrate the behaviour of solutions to (1) for $ M = 2 $ modeling two different scenarios. In the following, the space mesh is set to $ \Delta x = 0.001 $.

    In this example, we consider a stretch of road populated by cars and trucks. The space domain is given by the interval $ [-2, 3] $ and we impose absorbing conditions at the boundaries, adding $ N_1 = \eta_1/\Delta x $ ghost cells for the first population and $ N_2 = \eta_2/\Delta x $ for the second one at the right boundary, and just one ghost cell for both populations at the left boundary, where we extend the solution constantly equal to the last value inside the domain. The dynamics is described by the following $ 2\times 2 $ system

    $ {tρ1(t,x)+x(ρ1(t,x)vmax1ψ((rω1)(t,x)))=0,tρ2(t,x)+x(ρ2(t,x)vmax2ψ((rω2)(t,x)))=0, $ (28)

    with

    $ ω1(x)=2η1(1xη1),η1=0.3,ω2(x)=2η2(1xη2),η2=0.1,ψ(ξ)=max{1ξ,0},ξ0,vmax1=0.8,vmax2=1.3. $

    In this setting, $ \rho_1 $ represents the density of trucks and $ \rho_2 $ is the density of cars on the road. Trucks moves at lower maximal speed than cars and have grater view horizon, but of the same order of magnitude. Figure 2 describes the evolution in time of the two population densities, correspondent to the initial configuration

    Figure 2.  Density profiles of cars and trucks at increasing times corresponding to the non-local model (28).
    $ {ρ1(0,x)=0.5χ[1.1,1.6],ρ2(0,x)=0.5χ[1.6,1.9], $

    in which a platoon of trucks precedes a group of cars. Due to their higher speed, cars overtake trucks, in accordance with what observed in the local case [4].

    The aim of this test is to study the possible impact of the presence of Connected Autonomous Vehicles (CAVs) on road traffic performances. Let us consider a circular road modeled by the space interval $ [-1, 1] $ with periodic boundary conditions at $ x = \pm 1 $. In this case, we assume that autonomous and non-autonomous vehicles have the same maximal speed, but the interaction radius of CAVs is two orders of magnitude grater than the one of human-driven cars. Moreover, we assume CAVs have constant convolution kernel, modeling the fact that they have the same degree of accuracy on information about surrounding traffic, independent from the distance. In this case, model (1) reads

    $ {tρ1(t,x)+x(ρ1(t,x)vmax1ψ((rω1)(t,x)))=0,tρ2(t,x)+x(ρ2(t,x)vmax2ψ((rω2)(t,x)))=0,ρ1(0,x)=β(0.5+0.3sin(5πx)),ρ2(0,x)=(1β)(0.5+0.3sin(5πx)), $ (29)

    with

    $ ω1(x)=1η1,η1=1,ω2(x)=2η2(1xη2),η2=0.01,ψ(ξ)=max{1ξ,0},ξ0,vmax1=vmax2=1. $

    Above $ \rho_1 $ represents the density of autonomous vehicles, $ \rho_2 $ the density of non-autonomous vehicles and $ \beta\in [0, 1] $ is the penetration rate of autonomous vehicle. Figure 3 displays the traffic dynamics in the case $ \beta = 0.9 $.

    Figure 3.  Density profiles corresponding to the non-local problem (29) with $\beta = 0.9$ at different times.

    As a metric of traffic congestion, given a time horizon $ T>0 $, we consider the two following functionals:

    $ J(β)=T0d|xr|dt, $ (30)
    $ Ψ(β)=T0[ρ1(t,ˉx)vmax1ψ((rω1)(t,ˉx))+ρ2(t,ˉx)vmax2ψ((rω2)(t,ˉx))]dt, $ (31)

    where $ \bar x = x_0\approx 0. $ The functional $ J $ measures the integral with respect to time of the spatial total variation of the total traffic density, see [6]. The functional $ \Psi $ measures the integral with respect to time of the traffic flow at a given point $ \bar x $, corresponding to the number of cars that have passed through $ \bar x $ in the studied time interval. Figure 4 displays the values of the functionals $ J $ and $ \Psi $ for different values of $ \beta = 0, \; 0.1, \; 0.2, \ldots, \; 1. $ We can notice that the functionals are not monotone and present minimum and maximum values. The traffic evolution patterns corresponding these stationary values are reported in Figure 5, showing the $ (t, x) $-plots of the total traffic density $ r(t, x) $ corresponding to these values of $ \beta $.

    Figure 4.  Functional $J$ (left) and $\Psi$ (right).
    Figure 5.  $(t, x)$-plots of the total traffic density $r(t, x) = \rho_1(t, x)+\rho_2(t, x)$ in (29) corresponding to different values of $\beta$: (a) no autonomous vehicles are present; (b) point of minimum for $\Psi$; (c) point of minimum for $J$; (d) point of maximum for $J$.

    The authors are grateful to Luis M. Villada for suggesting the non-local multi-class traffic model studied in this paper.

    We provide here alternative estimates for (1), based on approximate solutions constructed via the following adapted Lax-Friedrichs scheme:

    $ ρn+1i,j=ρni,jλ(Fni,j+1/2Fni,j1/2), $ (32)

    with

    $ Fni,j+1/2:=12ρni,jVni,j+12ρni,j+1Vni,j+1+α2(ρni,jρni,j+1), $ (33)

    where $ \alpha\geq 1 $ is the viscosity coefficient and $ \lambda = \frac{\Delta t}{\Delta x}. $ The proofs are very similar to those exposed for Godunov approximations.

    Lemma A.1. For any $ T>0, $ under the CFL conditions

    $ λα<1, $ (34)
    $ αvmaxMψ, $ (35)

    the scheme (33)-(32) is positivity preserving on $ [0, T]\times \mathbb{R}. $

    Lemma A.2. ($ {{\mathbf{L}}^\infty } $-bound) If $ \rho_{i, j}^0\geq 0 $ for all $ j\in \mathbb{Z} $ and $ i = 1, ..., M $, and the CFL conditions (34)- (35) hold, the approximate solution $ \boldsymbol{\rho}^{\Delta x} $ constructed by the algorithm (33)- (32) is uniformly bounded on $ [0, T]\times \mathbb{R} $ for any $ T $ such that

    $ T<(Mρ0vmaxMψW0)1. $ (36)

    Lemma A.3. ($ \mathbf{BV} $ estimates) Let $ \rho_{i}^0\in \left( \mathbf{BV}\cap{{\mathbf{L}}^\infty }\right)( \mathbb{R}, \mathbb{R}^+) $ for all $ i = 1, ..., M $. If (35) holds and

    $ \begin{align} \Delta t &\leq \frac{2}{2 \alpha +\Delta x \, {\left\|{\psi'}\right\|}_\infty \, W_0 \, v_M^{\max}{\left\|{ \boldsymbol{\rho}}\right\|}_\infty} \, \Delta x , \end{align} $ (37)

    then the solution constructed by the algorithm (33)-(32) has uniformly bounded total variation for any $ T $ such that

    $ \begin{equation} T \leq {\min\limits_{i = 1, ..., M}}\, \frac{1}{\mathcal{D} \left( \mathinner{{\rm{TV}}}(\rho^0_i)+1\right)\, }, \end{equation} $ (38)

    where $ \mathcal{D} = {\left\|{{ \boldsymbol{\rho}}}\right\|}_\infty\, \, v_M^{\max} W_0 M\left(3M J_0 {\left\|{{ \boldsymbol{\rho}}}\right\|}_\infty{\left\|{\psi"}\right\|}_\infty +2{\left\|{\psi'}\right\|}_\infty\right) $.

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