April  2022, 17(2): 203-226. doi: 10.3934/nhm.2022003

Nonlocal reaction traffic flow model with on-off ramps

1. 

Department of Mathematical Sciences "G. L. Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

2. 

GIMNAP-Departamento de Matemáticas, Universidad del Bío-Bío, Concepción, Chile

3. 

GIMNAP-Departamento de Matemáticas, Universidad del Bío-Bío, Concepción, Chile, and CI2MA, Universidad de Concepción, Casilla 160-C, Concepción, Chile

* Corresponding author: Luis Miguel Villada

Received  August 2021 Revised  December 2021 Published  April 2022 Early access  February 2022

We present a non-local version of a scalar balance law modeling traffic flow with on-ramps and off-ramps. The source term is used to describe the inflow and output flow over the on-ramp and off-ramps respectively. We approximate the problem using an upwind-type numerical scheme and we provide $ \mathbf{L^{\infty}} $ and $ \mathbf{BV} $ estimates for the sequence of approximate solutions. Together with a discrete entropy inequality, we also show the well-posedness of the considered class of scalar balance laws. Some numerical simulations illustrate the behaviour of solutions in sample cases.

Citation: Felisia Angela Chiarello, Harold Deivi Contreras, Luis Miguel Villada. Nonlocal reaction traffic flow model with on-off ramps. Networks and Heterogeneous Media, 2022, 17 (2) : 203-226. doi: 10.3934/nhm.2022003
References:
[1]

P. AmorimR. M. Colombo and A. Teixeira, On the numerical integration of scalar nonlocal conservation laws, ESAIM: Mathematical Modelling and Numerical Analysis, 49 (2015), 19-37.  doi: 10.1051/m2an/2014023.

[2]

A. Bayen, A. Keimer, L. Pflug and T. Veeravalli, Modeling multi-lane traffic with moving obstacles by nonlocal balance laws, Preprint, (2020).

[3]

S. Blandin and P. Goatin, Well-posedness of a conservation law with non-local flux arising in traffic flow modeling, Numerische Mathematik, 132 (2016), 217-241.  doi: 10.1007/s00211-015-0717-6.

[4]

F. A. ChiarelloJ. FriedrichP. GoatinS. Göttlich and O. Kolb, A non-local traffic flow model for 1-to-1 junctions, European Journal of Applied Mathematics, 31 (2020), 1029-1049.  doi: 10.1017/S095679251900038X.

[5]

F. A. Chiarello and P. Goatin, Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel, ESAIM: Mathematical Modelling and Numerical Analysis, 52 (2018), 163-180.  doi: 10.1051/m2an/2017066.

[6]

F. A. Chiarello and P. Goatin, Non-local multi-class traffic flow models, Networks & Heterogeneous Media, 14 (2019), 371-380.  doi: 10.3934/nhm.2019015.

[7]

M. L. Delle MonacheJ. ReillyS. SamaranayakeW. KricheneP. Goatin and A. M. Bayen, A PDE-ODE model for a junction with ramp buffer, SIAM Journal on Applied Mathematics, 74 (2014), 22-39.  doi: 10.1137/130908993.

[8]

J. Friedrich, S. Göttlich and E. Rossi, Nonlocal approaches for multilane traffic models, Commun. Math. Sci., 19 (2021), 2291–2317, arXiv preprint, arXiv: 2012.05794, (2020). doi: 10.4310/CMS.2021.v19.n8.a10.

[9]

J. FriedrichO. Kolb and S. Göttlich, A godunov type scheme for a class of lwr traffic flow models with non-local flux, Networks & Heterogeneous Media, 13 (2018), 531-547.  doi: 10.3934/nhm.2018024.

[10]

P. Goatin and E. Rossi, A multilane macroscopic traffic flow model for simple networks, SIAM Journal on Applied Mathematics, 79 (2019), 1967-1989.  doi: 10.1137/19M1254386.

[11]

P. Goatin and S. Scialanga, Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity, Netw. Heterog. Media, 11 (2016), 107-121.  doi: 10.3934/nhm.2016.11.107.

[12]

Y. HanM. RamezaniA. HegyiY. Yuan and S. Hoogendoorn, Hierarchical ramp metering in freeways: An aggregated modeling and control approach, Transportation Research Part C: Emerging Technologies, 110 (2020), 1-19.  doi: 10.1016/j.trc.2019.09.023.

[13]

D. HelbingA. HenneckeV. Shvetsov and M. Treiber, Master: Macroscopic traffic simulation based on a gas-kinetic, non-local traffic model, Transportation Research Part B: Methodological, 35 (2001), 183-211.  doi: 10.1016/S0191-2615(99)00047-8.

[14]

H. Holden and N. H. Risebro, Models for dense multilane vehicular traffic, SIAM Journal on Mathematical Analysis, 51 (2019), 3694-3713.  doi: 10.1137/19M124318X.

[15]

D. JacquetC. C. De Wit and D. Koenig, Optimal ramp metering strategy with extended lwr model, analysis and computational methods, IFAC Proceedings Volumes, 38 (2005), 99-104.  doi: 10.3182/20050703-6-CZ-1902.00877.

[16]

G. Lipták, M. Pereira, B. Kulcsár, M. Kovács and G. Szederkényi, Traffic reaction model, arXiv preprint, arXiv: 2101.10190, (2021).

[17]

G. LiuA. S. Lyrintzis and P. G. Michalopoulos, Modelling of freeway merging and diverging flow dynamics, Applied Mathematical Modelling, 20 (1996), 459-469.  doi: 10.1016/0307-904X(95)00165-G.

[18]

A. Sopasakis and M. A. Katsoulakis, Stochastic modeling and simulation of traffic flow: Asymmetric single exclusion process with arrhenius look-ahead dynamics, SIAM Journal on Applied Mathematics, 66 (2006), 921-944.  doi: 10.1137/040617790.

[19]

J. SunZ. Li and J. Sun, Study on traffic characteristics for a typical expressway on-ramp bottleneck considering various merging behaviors, Physica A: Statistical Mechanics and its Applications, 440 (2015), 57-67.  doi: 10.1016/j.physa.2015.08.007.

[20]

T.-Q. TangH. J. Huang and H.-Y. Shang, Effects of the number of on-ramps on the ring traffic flow, Chinese Physics B, 19 (2010), 050517.  doi: 10.1088/1674-1056/19/5/050517.

[21]

T.-Q. TangH. J. HuangS. C. WongZ.-Y. Gao and Y. Zhang, A new macro model for traffic flow on a highway with ramps and numerical tests, Communications in Theoretical Physics, 51 (2009), 71.  doi: 10.1088/0253-6102/51/1/15.

[22]

T. WangJ. ZhangZ. GaoW. Zhang and S. Li, Congested traffic patterns of two-lane lattice hydrodynamic model with on-ramp, Nonlinear Dynamics, 88 (2017), 1345-1359.  doi: 10.1007/s11071-016-3314-z.

show all references

References:
[1]

P. AmorimR. M. Colombo and A. Teixeira, On the numerical integration of scalar nonlocal conservation laws, ESAIM: Mathematical Modelling and Numerical Analysis, 49 (2015), 19-37.  doi: 10.1051/m2an/2014023.

[2]

A. Bayen, A. Keimer, L. Pflug and T. Veeravalli, Modeling multi-lane traffic with moving obstacles by nonlocal balance laws, Preprint, (2020).

[3]

S. Blandin and P. Goatin, Well-posedness of a conservation law with non-local flux arising in traffic flow modeling, Numerische Mathematik, 132 (2016), 217-241.  doi: 10.1007/s00211-015-0717-6.

[4]

F. A. ChiarelloJ. FriedrichP. GoatinS. Göttlich and O. Kolb, A non-local traffic flow model for 1-to-1 junctions, European Journal of Applied Mathematics, 31 (2020), 1029-1049.  doi: 10.1017/S095679251900038X.

[5]

F. A. Chiarello and P. Goatin, Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel, ESAIM: Mathematical Modelling and Numerical Analysis, 52 (2018), 163-180.  doi: 10.1051/m2an/2017066.

[6]

F. A. Chiarello and P. Goatin, Non-local multi-class traffic flow models, Networks & Heterogeneous Media, 14 (2019), 371-380.  doi: 10.3934/nhm.2019015.

[7]

M. L. Delle MonacheJ. ReillyS. SamaranayakeW. KricheneP. Goatin and A. M. Bayen, A PDE-ODE model for a junction with ramp buffer, SIAM Journal on Applied Mathematics, 74 (2014), 22-39.  doi: 10.1137/130908993.

[8]

J. Friedrich, S. Göttlich and E. Rossi, Nonlocal approaches for multilane traffic models, Commun. Math. Sci., 19 (2021), 2291–2317, arXiv preprint, arXiv: 2012.05794, (2020). doi: 10.4310/CMS.2021.v19.n8.a10.

[9]

J. FriedrichO. Kolb and S. Göttlich, A godunov type scheme for a class of lwr traffic flow models with non-local flux, Networks & Heterogeneous Media, 13 (2018), 531-547.  doi: 10.3934/nhm.2018024.

[10]

P. Goatin and E. Rossi, A multilane macroscopic traffic flow model for simple networks, SIAM Journal on Applied Mathematics, 79 (2019), 1967-1989.  doi: 10.1137/19M1254386.

[11]

P. Goatin and S. Scialanga, Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity, Netw. Heterog. Media, 11 (2016), 107-121.  doi: 10.3934/nhm.2016.11.107.

[12]

Y. HanM. RamezaniA. HegyiY. Yuan and S. Hoogendoorn, Hierarchical ramp metering in freeways: An aggregated modeling and control approach, Transportation Research Part C: Emerging Technologies, 110 (2020), 1-19.  doi: 10.1016/j.trc.2019.09.023.

[13]

D. HelbingA. HenneckeV. Shvetsov and M. Treiber, Master: Macroscopic traffic simulation based on a gas-kinetic, non-local traffic model, Transportation Research Part B: Methodological, 35 (2001), 183-211.  doi: 10.1016/S0191-2615(99)00047-8.

[14]

H. Holden and N. H. Risebro, Models for dense multilane vehicular traffic, SIAM Journal on Mathematical Analysis, 51 (2019), 3694-3713.  doi: 10.1137/19M124318X.

[15]

D. JacquetC. C. De Wit and D. Koenig, Optimal ramp metering strategy with extended lwr model, analysis and computational methods, IFAC Proceedings Volumes, 38 (2005), 99-104.  doi: 10.3182/20050703-6-CZ-1902.00877.

[16]

G. Lipták, M. Pereira, B. Kulcsár, M. Kovács and G. Szederkényi, Traffic reaction model, arXiv preprint, arXiv: 2101.10190, (2021).

[17]

G. LiuA. S. Lyrintzis and P. G. Michalopoulos, Modelling of freeway merging and diverging flow dynamics, Applied Mathematical Modelling, 20 (1996), 459-469.  doi: 10.1016/0307-904X(95)00165-G.

[18]

A. Sopasakis and M. A. Katsoulakis, Stochastic modeling and simulation of traffic flow: Asymmetric single exclusion process with arrhenius look-ahead dynamics, SIAM Journal on Applied Mathematics, 66 (2006), 921-944.  doi: 10.1137/040617790.

[19]

J. SunZ. Li and J. Sun, Study on traffic characteristics for a typical expressway on-ramp bottleneck considering various merging behaviors, Physica A: Statistical Mechanics and its Applications, 440 (2015), 57-67.  doi: 10.1016/j.physa.2015.08.007.

[20]

T.-Q. TangH. J. Huang and H.-Y. Shang, Effects of the number of on-ramps on the ring traffic flow, Chinese Physics B, 19 (2010), 050517.  doi: 10.1088/1674-1056/19/5/050517.

[21]

T.-Q. TangH. J. HuangS. C. WongZ.-Y. Gao and Y. Zhang, A new macro model for traffic flow on a highway with ramps and numerical tests, Communications in Theoretical Physics, 51 (2009), 71.  doi: 10.1088/0253-6102/51/1/15.

[22]

T. WangJ. ZhangZ. GaoW. Zhang and S. Li, Congested traffic patterns of two-lane lattice hydrodynamic model with on-ramp, Nonlinear Dynamics, 88 (2017), 1345-1359.  doi: 10.1007/s11071-016-3314-z.

Figure 1.  Illustration of the model setting
Figure 2.  Example 1. Numerical approximations of the problem (2.1). Dynamic of Model 1 vs. Model 2 at (a)$ T = 0.5 $, (b)$ T = 2 $, (c)$ T = 5 $, (d)$ T = 7 $
Figure 3.  Example 2. Numerical approximations of the problem (2.1) at $ T = 5 $. Comparison of local and non-local versions of the model (2.1) with $ \delta = 0 $ and different values for $ \eta $
Figure 4.  Example 3. Numerical approximation at time $ T = 0.3 $. (a) Model 1, Model 2 satisfying a maximum principle and Model 0 not satisfying a maximum principle. (b) Zoom of a part of (a)
Figure 5.  Example 4. Dynamic of the model (2.1). Behavior of the numerical solution computed with Algorithm 3.1 by means of Model 1 and Model 2 at time (a)$ T = 1 $, (b)$ T = 2 $, (c)$ T = 5 $, (d)$ T = 7 $
Table 1.  Example 2. $ \mathbf{L^{1}} $ distance between the approximate solutions to the non-local problem and the local problem for different values of $ \eta $ at $ T = 5 $ with $ \Delta x = 1/1000 $
$ \eta $ 0.1 0.05 0.01 0.004
$ \mathbf{L^{1}} $ distance 2.8e-1 1.6e-1 3.6e-2 1.1e-2
$ \eta $ 0.1 0.05 0.01 0.004
$ \mathbf{L^{1}} $ distance 2.8e-1 1.6e-1 3.6e-2 1.1e-2
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