
Citation: George P. Stamou, Sotiris Konstadinou, Nikolaos Monokrousos, Anna Mastrogianni, Michalis Orfanoudakis, Christos Hassiotis, Urania Menkissoglu-Spiroudi, Despoina Vokou, Efimia M. Papatheodorou. The effects of arbuscular mycorrhizal fungi and essential oil on soil microbial community and N-related enzymes during the fungal early colonization phase[J]. AIMS Microbiology, 2017, 3(4): 938-959. doi: 10.3934/microbiol.2017.4.938
[1] | Paola Goatin, Chiara Daini, Maria Laura Delle Monache, Antonella Ferrara . Interacting moving bottlenecks in traffic flow. Networks and Heterogeneous Media, 2023, 18(2): 930-945. doi: 10.3934/nhm.2023040 |
[2] | Felisia Angela Chiarello, Paola Goatin . Non-local multi-class traffic flow models. Networks and Heterogeneous Media, 2019, 14(2): 371-387. doi: 10.3934/nhm.2019015 |
[3] | Jan Friedrich, Oliver Kolb, Simone Göttlich . A Godunov type scheme for a class of LWR traffic flow models with non-local flux. Networks and Heterogeneous Media, 2018, 13(4): 531-547. doi: 10.3934/nhm.2018024 |
[4] | Abraham Sylla . Influence of a slow moving vehicle on traffic: Well-posedness and approximation for a mildly nonlocal model. Networks and Heterogeneous Media, 2021, 16(2): 221-256. doi: 10.3934/nhm.2021005 |
[5] | Christophe Chalons, Paola Goatin, Nicolas Seguin . General constrained conservation laws. Application to pedestrian flow modeling. Networks and Heterogeneous Media, 2013, 8(2): 433-463. doi: 10.3934/nhm.2013.8.433 |
[6] | Caterina Balzotti, Simone Göttlich . A two-dimensional multi-class traffic flow model. Networks and Heterogeneous Media, 2021, 16(1): 69-90. doi: 10.3934/nhm.2020034 |
[7] | Dong Li, Tong Li . Shock formation in a traffic flow model with Arrhenius look-ahead dynamics. Networks and Heterogeneous Media, 2011, 6(4): 681-694. doi: 10.3934/nhm.2011.6.681 |
[8] | Raimund Bürger, Kenneth H. Karlsen, John D. Towers . On some difference schemes and entropy conditions for a class of multi-species kinematic flow models with discontinuous flux. Networks and Heterogeneous Media, 2010, 5(3): 461-485. doi: 10.3934/nhm.2010.5.461 |
[9] | Raimund Bürger, Christophe Chalons, Rafael Ordoñez, Luis Miguel Villada . A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function. Networks and Heterogeneous Media, 2021, 16(2): 187-219. doi: 10.3934/nhm.2021004 |
[10] | Alexander Kurganov, Anthony Polizzi . Non-oscillatory central schemes for traffic flow models with Arrhenius look-ahead dynamics. Networks and Heterogeneous Media, 2009, 4(3): 431-451. doi: 10.3934/nhm.2009.4.431 |
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
∂tρi(t,x)+∂x(ρi(t,x)vi((r∗ωi)(t,x)))=0,i=1,...,M, | (1) |
where
r(t,x):=M∑i=1ρi(t,x), | (2) |
vi(ξ):=vmaxiψ(ξ), | (3) |
(r∗ωi)(t,x):=∫x+ηixr(t,y)ωi(y−x)dy, | (4) |
and we assume:
We couple (1) with an initial datum
ρi(0,x)=ρ0i(x),i=1,…,M. | (5) |
Model (1) is obtained generalizing the
Due to the possible presence of jump discontinuities, solutions to (1), (5) are intended in the following weak sense.
Definition 1.1. A function
∫T0∫∞−∞(ρi∂tφ+ρivi(r∗ωi)∂xφ)(t,x)dxdt+∫∞−∞ρ0i(x)φ(0,x)dx=0 |
for all
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
Theorem 1.2. Let
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
First of all, we extend
To this end, we approximate the initial datum
ρ0i,j=1Δx∫xj+1/2xj−1/2ρ0i(x)dx,j∈Z. |
Similarly, for the kernel, we set
ωki:=1Δx∫(k+1)ΔxkΔxω0i(x)dx,k∈N, |
so that
Vni,j:=vmaxiψ(Δx+∞∑k=0ωkirnj+k),i=1,…,M,j∈Z. | (6) |
We consider the following Godunov-type scheme adapted to (1), which was introduced in [8] in the scalar case:
ρn+1i,j=ρni,j−λ(ρni,jVni,j+1−ρni,j−1Vni,j) | (7) |
where we have set
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
λ≤1vmaxM‖ψ‖∞, | (8) |
the scheme (7) is positivity preserving on
Proof. Let us assume that
ρn+1i,j=ρni,j(1−λVni,j+1)+λρni,j−1Vni,j≥0 | (9) |
under assumption (8).
Corollary 1. (
‖ρni‖1=‖ρ0i‖1,i=1,…,M, | (10) |
where
Proof. Thanks to Lemma 2.1, for all
‖ρn+1i‖1=Δx∑jρn+1i,j=Δx∑j(ρni,j−λρni,jVni,j+1+λρni,j−1Vni,j)=Δx∑jρni,j, |
proving (10).
Lemma 2.2. (
T<(M‖ρ0‖∞vmaxM‖ψ′‖∞W0)−1. |
Proof. Let
ρn+1i,j=ρni,j(1−λVni,j+1)+λρni,j−1Vni,j≤ˉρ(1+λ(Vni,j−Vni,j+1)) | (11) |
and
|Vni,j−Vni,j+1|=vmaxi|ψ(Δx+∞∑k=0ωkirnj+k)−ψ(Δx+∞∑k=0ωkirnj+k+1)|≤vmaxi‖ψ′‖∞Δx|+∞∑k=0ωki(rnj+k+1−rnj+k)|=vmaxi‖ψ′‖∞Δx|−ω0irnj++∞∑k=1(ωk−1i−ωki)rnj+k|≤vmaxi‖ψ′‖∞ΔxM‖ρn‖∞ωi(0) | (12) |
where
‖ρn+1‖∞≤‖ρn‖∞(1+MKvmaxM‖ψ′‖∞W0Δt), |
which implies
‖ρn‖∞≤‖ρ0‖∞eCnΔt, |
with
t≤1MKvmaxM‖ψ′‖∞W0ln(K‖ρ0‖∞)≤1Me‖ρ0‖∞vmaxM‖ψ′‖∞W0, |
where the maximum is attained for
Iterating the procedure, at time
tm+1≤tm+mMem‖ρ0‖∞vmaxM‖ψ′‖∞W0. |
Therefore, the approximate solution remains bounded, uniformly in
T≤1M‖ρ0‖∞vmaxM‖ψ′‖∞W0+∞∑m=1mem≤1M‖ρ0‖∞vmaxM‖ψ′‖∞W0. |
Remark 1. Figure 1 shows that the simplex
S:={ρ∈RM:M∑i=1ρi≤1,ρi≥0fori=1,…,M} |
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
Lemma 2.3. Under the CFL condition
λ≤1vmaxM(‖ψ‖∞+‖ψ′‖∞), |
for any initial datum
ρn+1j=ρnj−λ[F(ρnj,ρnj+1)−F(ρnj−1,ρnj)], | (14) |
with
ρnj∈S∀j∈Z,n∈N. | (15) |
Proof. Assuming that
ρn+1i,j=ρni,j−λ[vmaxiρni,jψ(rnj+1)−vmaxiρni,j−1ψ(rnj)]. |
Summing on the index
rn+1j=M∑i=1ρn+1i,j=M∑i=1ρni,j−λM∑i=1[vmaxiρni,jψ(rnj+1)−vmaxiρni,j−1ψ(rnj)]=rnj+λψ(rnj)M∑i=1vmaxiρni,j−1−λψ(rnj+1)M∑i=1vmaxiρni,j. |
Defining the following function of
Φ(ρn1,j,…,ρnM,j)=rnj+λψ(rnj)M∑i=1vmaxiρni,j−1−λψ(rnj+1)M∑i=1vmaxiρni,j, |
we observe that
Φ(0,…,0)=λψ(0)M∑ivmaxiρni,j−1≤λ‖ψ‖∞vmaxM≤1 |
if
Φ(ρn1,j,...,ρnM,j)=1−λψ(rnj+1)M∑i=1vmaxiρni,j≤1 |
for
∂Φ∂ρni,j(ρnj)=1+λψ′(rnj)M∑i=1vmaxiρni,j−1−λψ(rnj+1)vmaxi≥0 |
if
ρn+1i,j=ρni,j(1−λvmaxiψ(rnj+1))+λvmaxiρni,j−1ψ(rnj)≥0 |
if
Lemma 2.4. (Spatial
T≤mini=1,…,M 1H(TV(ρ0i)+1), | (16) |
where
Proof. Subtracting the identities
ρn+1i,j+1=ρni,j+1−λ(ρni,j+1Vni,j+2−ρni,jVni,j+1), | (17) |
ρn+1i,j=ρni,j−λ(ρni,jVni,j+1−ρni,j−1Vni,j), | (18) |
and setting
Δn+1i,j+1/2=Δni,j+1/2−λ(ρni,j+1Vni,j+2−2ρni,jVni,j+1+ρni,j−1Vni,j). |
Now, we can write
Δn+1i,j+1/2=(1−λVni,j+2)Δni,j+1 | (19) |
+λVni,jΔni,j−1/2−λρni,j(Vni,j+2−2Vni,j+1+Vni,j). | (20) |
Observe that assumption (8) guarantees the positivity of (19). The term (20) can be estimated as
Vni,j+2−2Vni,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(ωk−1i−ωki)rnj+k+1−ω0irnj+1)+vmaxiψ′(ξj)Δx(+∞∑k=1(ωki−ωk−1i)rnj+k+ω0irnj)=vmaxi(ψ′(ξj+1)−ψ′(ξj))Δx(+∞∑k=1(ωk−1i−ωki)rnj+k+1−ω0irnj+1)+vmaxiψ′(ξj)Δx(+∞∑k=1(ωk−1i−ωki)(rnj+k+1−rnj+k)+ω0i(rnj−rnj+1))=vmaxiψ″(˜ξj+1/2)(ξj+1−ξj)Δx(+∞∑k=1M∑β=1ωkiΔnβ,j+k+3/2)+vmaxiψ′(ξj)Δx(M∑β=1N−1∑k=1(ωk−1i−ωki)Δnβ,j+k+1/2−ω0iΔnβ,j+1/2), |
with
ξj+1−ξj=ϑΔx+∞∑k=0ωkiM∑β=1ρnβ,j+k+2+(1−ϑ)Δx+∞∑k=0ωkiM∑β=1ρnβ,j+k+1−μΔx+∞∑k=0ωkiM∑β=1ρnβ,j+k+1−(1−μ)Δx+∞∑k=0ωkiM∑β=1ρnβ,j+k=ϑΔx+∞∑k=1ωk−1iM∑β=1ρnβ,j+k+1+(1−ϑ)Δx+∞∑k=0ωkiM∑β=1ρnβ,j+k+1−μΔx+∞∑k=0ωkiM∑β=1ρnβ,j+k+1−(1−μ)Δx+∞∑k=−1ωk+1iM∑β=1ρnβ,j+k+1=Δx+∞∑k=1[ϑωk−1i+(1−ϑ)ωki−μωki−(1−μ)ωk+1i]M∑β=1ρnβ,j+k+1+(1−ϑ)Δxω0iM∑β=1ρnβ,j+1−μΔxω0iM∑β=1ρnβ,j+1−(1−μ)Δx(ω0iM∑β=1ρnβ,j+ω1iM∑β=1ρnβ,j+1). |
By monotonicity of
ϑωk−1i+(1−ϑ)ωki−μωki−(1−μ)ωk+1i≥0. |
Taking the absolute values we get
|ξj+1−ξj|≤Δx{+∞∑k=2[ϑωk−1i+(1−ϑ)ωki−μωki−(1−μ)ωk+1i]+4ω0i}M‖ρn‖∞≤Δx{+∞∑k=2[ωk−1i−ωk+1i]+4ω0i}M‖ρn‖∞≤Δx6W0M‖ρn‖∞. |
Let now
∑j|Δn+1i,j+1/2|≤∑j|Δni,j+1/2|(1−λ(Vni,j+2−Vni,j+1))+ΔtHK1, |
where
∑j|Δn+1i,j+1/2|≤∑j|Δni,j+1/2|(1+ΔtG)+ΔtHK1, |
with
∑j|Δni,j+1/2|≤eGnΔt∑j|Δ0i,j+1/2|+eHK1nΔt−1, |
that we can rewrite as
TV(ρΔxi)(nΔt,⋅)≤eGnΔtTV(ρ0i)+eHK1nΔt−1≤eHK1nΔt(TV(ρ0i)+1)−1, |
since
t≤1HK1ln(K1+1TV(ρ0i)+1), |
where the maximum is attained for some
ln(K1+1TV(ρ0i)+1)=K1K1+1. |
Therefore the total variation is uniformly bounded for
t≤1He(TV(ρ0i)+1). |
Iterating the procedure, at time
tm+1≤tm+mHem(TV(ρ0i)+1). | (21) |
Therefore, the approximate solution has bounded total variation for
T≤1H(TV(ρ0i)+1). |
Corollary 2. Let
Proof. If
TV(ρΔxi;[0,T]×R)=nT−1∑n=0∑j∈ZΔt|ρni,j+1−ρni,j|+(T−nTΔt)∑j∈Z|ρnTi,j+1−ρnTi,j|⏟≤Tsupt∈[0,T]TV(ρΔxi)(t,⋅)+nT−1∑n=0∑j∈ZΔx|ρn+1i,j−ρni,j|. |
We then need to bound the term
nT−1∑n=0∑j∈ZΔx|ρn+1i,j−ρni,j|. |
From the definition of the numerical scheme (7), we obtain
ρn+1i,j−ρni,j=λ(ρni,j−1Vni,j−ρni,jVni,j+1)=λ(ρni,j−1(Vni,j−Vni,j+1)+Vni,j+1(ρni,j−1−ρni,j)). |
Taking the absolute values and using (12) we obtain
|ρn+1i,j−ρni,j|≤λ(vmaxi‖ψ′‖∞M‖ρn‖∞ωi(0)Δx|ρni,j−1|+vmaxi‖ψ‖∞|ρni,j−1−ρni,j|). |
Summing on
∑j∈ZΔx|ρn+1i,j−ρni,j|=vmaxi‖ψ′‖∞M‖ρn‖∞ωi(0)Δt∑j∈ZΔx|ρni,j−1|+vmaxi‖ψ‖∞Δt∑j∈Z|ρni,j−1−ρni,j|, |
which yields
nT−1∑n=0∑j∈ZΔx|ρn+1i,j−ρni,j|≤vmaxM‖ψ‖∞Tsupt∈[0,T]TV(ρΔxi)(t,⋅)+vmaxM‖ψ′‖∞MW0Tsupt∈[0,T]‖ρΔxi(t,⋅)‖1‖ρΔxi(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
nT−1∑n=0∑jφ(tn,xj)(ρn+1i,j−ρni,j)=−λnT−1∑n=0∑jφ(tn,xj)(ρni,jVni,j+1−ρni,j−1Vni,j). |
Summing by parts we obtain
−∑jφ((nT−1)Δt,xj)ρnTi,j+∑jφ(0,xj)ρ0i,j+nT−1∑n=1∑j(φ(tn,xj)−φ(tn−1,xj))ρni,j+λnT−1∑n=0∑j(φ(tn,xj+1)−φ(tn,xj))Vni,j+1ρni,j=0. | (22) |
Multiplying by
−Δx∑jφ((nT−1)Δt,xj)ρnTi,j+Δx∑jφ(0,xj)ρ0i,j | (23) |
+ΔxΔtnT−1∑n=1∑j(φ(tn,xj)−φ(tn−1,xj))Δtρni,j | (24) |
+ΔxΔtnT−1∑n=0∑j(φ(tn,xj+1)−φ(tn,xj))ΔxVni,j+1ρni,j=0. | (25) |
By
∫R(ρ0i(x)φ(0,x)−ρi(T,x)φ(T,x))dx+∫T0∫Rρi(t,x)∂tφ(t,x)dxdt, | (26) |
as
ΔxΔtnT−1∑n=0∑jφ(tn,xj+1)−φ(tn,xj)ΔxVni,j+1ρni,j=ΔxΔtnT−1∑n=0∑jφ(tn,xj+1)−φ(tn,xj)Δx(ρni,jVni,j+1−ρni,jVni,j)+ΔxΔtnT−1∑n=0∑jφ(tn,xj+1)−φ(tn,xj)Δxρni,jVni,j. | (27) |
By (12) we get the estimate
ρni,jVni,j+1−ρni,jVni,j≤vmaxi‖ψ′‖∞ΔxM‖ρ‖2∞ωi(0). |
Set
ΔxΔtnT∑n=0∑jφ(tn,xj+1)−φ(tn,xj)Δx(ρni,jVni,j+1−ρni,jVni,j)≤ΔxΔt‖∂xφ‖∞nT∑n=0j1∑j=j0vmaxi‖ψ′‖∞M‖ρ‖2∞ωi(0)Δx≤‖∂xφ‖∞vmaxi‖ψ′‖∞M‖ρ‖2∞ωi(0)Δx2RT, |
which goes to zero as
Finally, again by the
ΔxΔtnT−1∑n=0∑j(φ(tn,xj+1)−φ(tn,xj))Δxρni,jVni,j−12→∫T0∫R∂xφ(t,x)ρi(t,x)vi(r∗ωi)dxdt. |
In this section we perform some numerical simulations to illustrate the behaviour of solutions to (1) for
In this example, we consider a stretch of road populated by cars and trucks. The space domain is given by the interval
{∂tρ1(t,x)+∂x(ρ1(t,x)vmax1ψ((r∗ω1)(t,x)))=0,∂tρ2(t,x)+∂x(ρ2(t,x)vmax2ψ((r∗ω2)(t,x)))=0, | (28) |
with
ω1(x)=2η1(1−xη1),η1=0.3,ω2(x)=2η2(1−xη2),η2=0.1,ψ(ξ)=max{1−ξ,0},ξ≥0,vmax1=0.8,vmax2=1.3. |
In this setting,
{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
{∂tρ1(t,x)+∂x(ρ1(t,x)vmax1ψ((r∗ω1)(t,x)))=0,∂tρ2(t,x)+∂x(ρ2(t,x)vmax2ψ((r∗ω2)(t,x)))=0,ρ1(0,x)=β(0.5+0.3sin(5πx)),ρ2(0,x)=(1−β)(0.5+0.3sin(5πx)), | (29) |
with
ω1(x)=1η1,η1=1,ω2(x)=2η2(1−xη2),η2=0.01,ψ(ξ)=max{1−ξ,0},ξ≥0,vmax1=vmax2=1. |
Above
As a metric of traffic congestion, given a time horizon
J(β)=∫T0d|∂xr|dt, | (30) |
Ψ(β)=∫T0[ρ1(t,ˉx)vmax1ψ((r∗ω1)(t,ˉx))+ρ2(t,ˉx)vmax2ψ((r∗ω2)(t,ˉx))]dt, | (31) |
where
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,j−λ(Fni,j+1/2−Fni,j−1/2), | (32) |
with
Fni,j+1/2:=12ρni,jVni,j+12ρni,j+1Vni,j+1+α2(ρni,j−ρni,j+1), | (33) |
where
Lemma A.1. For any
λα<1, | (34) |
α≥vmaxM‖ψ‖∞, | (35) |
the scheme (33)-(32) is positivity preserving on
Lemma A.2. (
T<(M‖ρ0‖∞vmaxM‖ψ′‖∞W0)−1. | (36) |
Lemma A.3. (
Δt≤22α+Δx‖ψ′‖∞W0vmaxM‖ρ‖∞Δx, | (37) |
then the solution constructed by the algorithm (33)-(32) has uniformly bounded total variation for any
T≤mini=1,...,M1D(TV(ρ0i)+1), | (38) |
where
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