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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 |
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.
Keywords:
Lighthill-Whitham-Richards traffic model,
nonlocal conservation laws,
ramps,
upwind scheme,
well-posedness.
Mathematics Subject Classification:
Primary: 65M08; Secondary: 35L45, 90B20.
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
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References:
| [1] |
P. Amorim, R. 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. Chiarello, J. Friedrich, P. Goatin, S. 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 Monache, J. Reilly, S. Samaranayake, W. Krichene, P. 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. Friedrich, O. 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. Han, M. Ramezani, A. Hegyi, Y. 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. Helbing, A. Hennecke, V. 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. Jacquet, C. 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. Liu, A. 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. Sun, Z. 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. Tang, H. 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. Tang, H. J. Huang, S. C. Wong, Z.-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. Wang, J. Zhang, Z. Gao, W. 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. Amorim, R. 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. Chiarello, J. Friedrich, P. Goatin, S. 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 Monache, J. Reilly, S. Samaranayake, W. Krichene, P. 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. Friedrich, O. 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. Han, M. Ramezani, A. Hegyi, Y. 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. Helbing, A. Hennecke, V. 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. Jacquet, C. 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. Liu, A. 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. Sun, Z. 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. Tang, H. 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. Tang, H. J. Huang, S. C. Wong, Z.-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. Wang, J. Zhang, Z. Gao, W. 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 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 $
| 0.1 | 0.05 | 0.01 | 0.004 | |
| 2.8e-1 | 1.6e-1 | 3.6e-2 | 1.1e-2 |
| 0.1 | 0.05 | 0.01 | 0.004 | |
| 2.8e-1 | 1.6e-1 | 3.6e-2 | 1.1e-2 |
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