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## Implementation of the vehicular occupancy-emission relation using a cubic B-splines collocation method

 1 Department of Computer Sciences, Faculty of Sciences and Techniques, University Moulay Ismail, BP 509 Boutalamine Errachidia, Morocco 2 Department of Mathematics, Laboratory LMPA, University Littoral Cote d'Opale, France

* Corresponding author: Sofiya Chergui

Received  July 2018 Revised  August 2018 Published  March 2019

The complexity and non-linearity of flow phenomena are explained by numerous criteria, including the interactions of the large number of vehicles occupying the road, which influence the road density. This density under certain conditions, leads to traffic congestion which has dangerous effects on the environment such as; resources consumption; noise and the effect caused by greenhouse gas emissions of the $CO_{2}$ and other pollutants. In this paper we consider working in an uniform, homogeneous road where the traffic is described by the Lighthill Whitham-Richard (LWR) model resolved using a cubic B-spline collocation scheme in space and an implicit Runge Kutta scheme in time. We also shed light on the relation between vehicle occupancy and vehicle emissions.

Citation: Said Agoujil, Abderrahman Bouhamidi, Sofiya Chergui, Youssef Qaraai. Implementation of the vehicular occupancy-emission relation using a cubic B-splines collocation method. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020022
##### References:
 [1] W. F. Adams, Road traffic considered as a random series, J. Inst. Civil Engineers, 4 (1936), 121-130. [2] S. Ardekani, E. Hauer and B. Jamei, Traffic impact models, In: Traffic Flow Theory. US Federal Highway Administration, Washington, DC, (1996), p1. [3] T. Bektas and G. Laporte, The pollution-routing problem, Transportation Research Part B, 45 (2011), 1232-1250. doi: 10.1016/j.trb.2011.02.004. [4] G. Bharti and V. K. Kukreja, Numerical approach for solving diffusion problems using cubic B-spline collocation method, Applied Mathematics and Computation, 219 (2012), 2087-2099. doi: 10.1016/j.amc.2012.08.053. [5] A. Bressan, Hyperbolic Systems of Conservation Laws, The One Dimensional Cauchy Problem. Oxford University Press, 2000. [6] D. Catalin, Contributions à la Modélisation et la Commande des Réseaux de Trafic Routier, Ph.D thesis, Ecole Centrale de Lille et le Departement AIS, Universit Politehnica de Bucarest, 2013. [7] D. Catalin, D. T. Genevive and D. Popescu, Macroscopic modeling of road traffic by using hydrodynamic flow models, 20th Mediterranean Conference on Control and Automation, (2012). doi: 10.1109/MED.2012.6265612. [8] D. Catalin, D. Popescu and D. Stefanoiu, Fuzzy modeling and control for a road section, 18th International Conference on System Theory, (2014). [9] R. E. Chandler, R. Herman and E. W. Montroll, Traffic dynamics: Studies in car following, Operation Research, 6 (1958), 165-184. doi: 10.1287/opre.6.2.165. [10] C. F. Daganzo, Fundamentals of Transportation and Traffic Operations, Pergamon, 1997. doi: 10.1108/9780585475301. [11] S. Darbha, K. R. Rajagopal and V. Tyagi, A review of mathematical models for the flow of traffic and some recent results, Nonlinear Analysis, 69 (2008), 950-970. doi: 10.1016/j.na.2008.02.123. [12] C. De Boor, A Practical Guide to Splines, Springer-Velay, Berlin, 1978. [13] J. S. Drake, J. L. Schfer and A. May, A Statistical Analysis of Speed Density Hypotheses, Proceedings of the Third International Symposium on the Theory of Traffic Flow, Elsevier North-Holland, New York, 1967. [14] A. Esen and O. Tasbozan, Cubic B-spline collocation method for solving time fractional gas dynamics equation, Tbilisi Math. J., 8 (2015), 221-231. doi: 10.1515/tmj-2015-0024. [15] K. Fagerholt, G. Laporte and I. Norstad, Reducing fuel emissions by optimizing speed on shipping routes, Journal of the Operational Research Society, 61 (2010), 523-529. doi: 10.1057/jors.2009.77. [16] A. Franceschetti, D. Honhon, T. V. Woensel, T. Bektas and G. Laporte, The time-dependent pollution-routing problem, Transportation Research Part B: Methodological, 56 (2013), 265-293. [17] M. Gholamian and N. J. Saberi, Cubic B-splines collocation method for a class of partial integro-differential equation, Alexandria Engineering Journal, 57 (2018), 2157-2165. doi: 10.1016/j.aej.2017.06.004. [18] S. K. Godunov, A difference method for numerical calculations of discontinuous solutions of the equations of hydrodynamics, Matematicheskii Sbornik, 47 (1959), 271-306. [19] S. Gottlich, U. Ziegler and M. Herty, Numerical discretization of Hamilton-Jacobi equation on networks, Netw. Heterog. Media, 8 (2013), 685-705. doi: 10.3934/nhm.2013.8.685. [20] B. D. Greenshields, A study of traffic capacity, Proceedings Highway Research Board, 14 (1934), 448-477. [21] K. Han, H. Liu, V. V. Gayah, T. L. Friesz and T. Yao, A robust optimization approach for dynamic traffic signal control with emission considerations, Transportation Research Part C, 70 (2016), 3-26. doi: 10.1016/j.trc.2015.04.001. [22] O. Jabali, T. Van Woensel and A. G. de Kok, Analysis of travel times and CO2 emissions in time-dependent vehicle routing. Tech. rep., Eindhoven University of Technology; (2009). [23] M. Koshi, M. Iwasaki and I. Ohkura, Some findings and an overview on vehicular flow characteristics, In Proceedings of the 8th International Symposium on Transportation and Traffic Theory, Univ. of Toronto Press, Toronto, (1981), 403-426. [24] J. P. Lebacque, The godunov scheme and what it means for first order traffic flow models, In The International Symposium on Transportation and Traffic Theory, Lyon, France, (1996). [25] L. Leclercq, J. A. Laval and E. Chevallier, The lagrangian coordinates and what it means for first order traffic flow models, In R. Allsop and B. Heydecker (Eds), Transportation and traffic theory, (2007), 735-753. [26] M. Lighthill and G. Whitham, On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, Proceedings of the Royal Society of London, Series A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. [27] W. Maden, R. W. Eglese and D. Black, Vehicle routing and scheduling with time varying data: A case study, Journal of the Operational Research Society, 61 (2010), 515-522. doi: 10.1057/jors.2009.116. [28] R. C. Mittal and R. Bhatia, Numerical solution of second order one dimensional hyperbolic telegraph equation by cubic B-spline collocation method, Applied Mathematics and Computation, 220 (2013), 496-506. doi: 10.1016/j.amc.2013.05.081. [29] R. C. Mittal and R. K. Jain, Redefined cubic B-splines collocation method for solving convection-diffusion equations, Applied Mathematical Modelling, 36 (2012), 5555-5573. doi: 10.1016/j.apm.2012.01.009. [30] R. C. Mittal and R. K. Jain, Cubic B-splines collocation method for solving nonlinear parabolic partial differential equations with Neumann boundary conditions, Commun Nonlinear Sci Numer Simulat, 17 (2012), 4616-4625. doi: 10.1016/j.cnsns.2012.05.007. [31] R. Mohammadi, Quintic B-spline collocation approach for solving generalized Black Scholes equation governing option pricing, Computers and Mathematics with Applications, 69 (2015), 777-797. doi: 10.1016/j.camwa.2015.02.018. [32] A. Palmer, The Development of an Integrated Routing and Carbon Dioxide Emissions Model for Goods Vehicles, Ph.D, thesis, Cranfield University, School of Management, 2007. [33] L. A. Pipes, An operational analysis of traffic dynamics, Journal of Applied Physics, 24 (1953), 274-281. doi: 10.1063/1.1721265. [34] K. Post, J. H. Kent, J. Tomlin and N. Carruthers, Fuel consumption and emission modelling by power demand and a comparison with other models, Transport. Res. Part A: Policy Pract, 18 (1984), 191-213. doi: 10.1016/0191-2607(84)90126-2. [35] P. I. Richards, Shock waves on the highway, Oper. Res, 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. [36] B. Saka and I. Dag, Quartic B-spline collocation method to the numerical solutions of the Burgers'equation, Chaos, Solitons and Fractals, 32 (2007), 1125-1137. doi: 10.1016/j.chaos.2005.11.037. [37] J. G. Wardrop, Some theoretical aspects of road traffic research, Proceedings of the Institution of Civil Engineers, Part II, 1 (1952), 325-362. [38] J. Wenlong, Traffic Flow Models and Their Numerical Solutions, University of Science and Technology of China, 1998. [39] J. Wenlong, Traffic Flow Models and Their Numerical Solutions, University of California Davis, 2000. [40] G. C. K. Wong and S. C. Wong, A multi-class traffic flow model-an extension of LWR model with heterogeneous drivers, Transportation Research Part A, Policy and Practice, 36 (2013), 827-841. [41] N. Wu, A new approach for modeling of Fundamental Diagrams, Transportation Research Part A: Policy and Practice, 36 (2002), 867-884. doi: 10.1016/S0965-8564(01)00043-X.

show all references

##### References:
 [1] W. F. Adams, Road traffic considered as a random series, J. Inst. Civil Engineers, 4 (1936), 121-130. [2] S. Ardekani, E. Hauer and B. Jamei, Traffic impact models, In: Traffic Flow Theory. US Federal Highway Administration, Washington, DC, (1996), p1. [3] T. Bektas and G. Laporte, The pollution-routing problem, Transportation Research Part B, 45 (2011), 1232-1250. doi: 10.1016/j.trb.2011.02.004. [4] G. Bharti and V. K. Kukreja, Numerical approach for solving diffusion problems using cubic B-spline collocation method, Applied Mathematics and Computation, 219 (2012), 2087-2099. doi: 10.1016/j.amc.2012.08.053. [5] A. Bressan, Hyperbolic Systems of Conservation Laws, The One Dimensional Cauchy Problem. Oxford University Press, 2000. [6] D. Catalin, Contributions à la Modélisation et la Commande des Réseaux de Trafic Routier, Ph.D thesis, Ecole Centrale de Lille et le Departement AIS, Universit Politehnica de Bucarest, 2013. [7] D. Catalin, D. T. Genevive and D. Popescu, Macroscopic modeling of road traffic by using hydrodynamic flow models, 20th Mediterranean Conference on Control and Automation, (2012). doi: 10.1109/MED.2012.6265612. [8] D. Catalin, D. Popescu and D. Stefanoiu, Fuzzy modeling and control for a road section, 18th International Conference on System Theory, (2014). [9] R. E. Chandler, R. Herman and E. W. Montroll, Traffic dynamics: Studies in car following, Operation Research, 6 (1958), 165-184. doi: 10.1287/opre.6.2.165. [10] C. F. Daganzo, Fundamentals of Transportation and Traffic Operations, Pergamon, 1997. doi: 10.1108/9780585475301. [11] S. Darbha, K. R. Rajagopal and V. Tyagi, A review of mathematical models for the flow of traffic and some recent results, Nonlinear Analysis, 69 (2008), 950-970. doi: 10.1016/j.na.2008.02.123. [12] C. De Boor, A Practical Guide to Splines, Springer-Velay, Berlin, 1978. [13] J. S. Drake, J. L. Schfer and A. May, A Statistical Analysis of Speed Density Hypotheses, Proceedings of the Third International Symposium on the Theory of Traffic Flow, Elsevier North-Holland, New York, 1967. [14] A. Esen and O. Tasbozan, Cubic B-spline collocation method for solving time fractional gas dynamics equation, Tbilisi Math. J., 8 (2015), 221-231. doi: 10.1515/tmj-2015-0024. [15] K. Fagerholt, G. Laporte and I. Norstad, Reducing fuel emissions by optimizing speed on shipping routes, Journal of the Operational Research Society, 61 (2010), 523-529. doi: 10.1057/jors.2009.77. [16] A. Franceschetti, D. Honhon, T. V. Woensel, T. Bektas and G. Laporte, The time-dependent pollution-routing problem, Transportation Research Part B: Methodological, 56 (2013), 265-293. [17] M. Gholamian and N. J. Saberi, Cubic B-splines collocation method for a class of partial integro-differential equation, Alexandria Engineering Journal, 57 (2018), 2157-2165. doi: 10.1016/j.aej.2017.06.004. [18] S. K. Godunov, A difference method for numerical calculations of discontinuous solutions of the equations of hydrodynamics, Matematicheskii Sbornik, 47 (1959), 271-306. [19] S. Gottlich, U. Ziegler and M. Herty, Numerical discretization of Hamilton-Jacobi equation on networks, Netw. Heterog. Media, 8 (2013), 685-705. doi: 10.3934/nhm.2013.8.685. [20] B. D. Greenshields, A study of traffic capacity, Proceedings Highway Research Board, 14 (1934), 448-477. [21] K. Han, H. Liu, V. V. Gayah, T. L. Friesz and T. Yao, A robust optimization approach for dynamic traffic signal control with emission considerations, Transportation Research Part C, 70 (2016), 3-26. doi: 10.1016/j.trc.2015.04.001. [22] O. Jabali, T. Van Woensel and A. G. de Kok, Analysis of travel times and CO2 emissions in time-dependent vehicle routing. Tech. rep., Eindhoven University of Technology; (2009). [23] M. Koshi, M. Iwasaki and I. Ohkura, Some findings and an overview on vehicular flow characteristics, In Proceedings of the 8th International Symposium on Transportation and Traffic Theory, Univ. of Toronto Press, Toronto, (1981), 403-426. [24] J. P. Lebacque, The godunov scheme and what it means for first order traffic flow models, In The International Symposium on Transportation and Traffic Theory, Lyon, France, (1996). [25] L. Leclercq, J. A. Laval and E. Chevallier, The lagrangian coordinates and what it means for first order traffic flow models, In R. Allsop and B. Heydecker (Eds), Transportation and traffic theory, (2007), 735-753. [26] M. Lighthill and G. Whitham, On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, Proceedings of the Royal Society of London, Series A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. [27] W. Maden, R. W. Eglese and D. Black, Vehicle routing and scheduling with time varying data: A case study, Journal of the Operational Research Society, 61 (2010), 515-522. doi: 10.1057/jors.2009.116. [28] R. C. Mittal and R. Bhatia, Numerical solution of second order one dimensional hyperbolic telegraph equation by cubic B-spline collocation method, Applied Mathematics and Computation, 220 (2013), 496-506. doi: 10.1016/j.amc.2013.05.081. [29] R. C. Mittal and R. K. Jain, Redefined cubic B-splines collocation method for solving convection-diffusion equations, Applied Mathematical Modelling, 36 (2012), 5555-5573. doi: 10.1016/j.apm.2012.01.009. [30] R. C. Mittal and R. K. Jain, Cubic B-splines collocation method for solving nonlinear parabolic partial differential equations with Neumann boundary conditions, Commun Nonlinear Sci Numer Simulat, 17 (2012), 4616-4625. doi: 10.1016/j.cnsns.2012.05.007. [31] R. Mohammadi, Quintic B-spline collocation approach for solving generalized Black Scholes equation governing option pricing, Computers and Mathematics with Applications, 69 (2015), 777-797. doi: 10.1016/j.camwa.2015.02.018. [32] A. Palmer, The Development of an Integrated Routing and Carbon Dioxide Emissions Model for Goods Vehicles, Ph.D, thesis, Cranfield University, School of Management, 2007. [33] L. A. Pipes, An operational analysis of traffic dynamics, Journal of Applied Physics, 24 (1953), 274-281. doi: 10.1063/1.1721265. [34] K. Post, J. H. Kent, J. Tomlin and N. Carruthers, Fuel consumption and emission modelling by power demand and a comparison with other models, Transport. Res. Part A: Policy Pract, 18 (1984), 191-213. doi: 10.1016/0191-2607(84)90126-2. [35] P. I. Richards, Shock waves on the highway, Oper. Res, 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. [36] B. Saka and I. Dag, Quartic B-spline collocation method to the numerical solutions of the Burgers'equation, Chaos, Solitons and Fractals, 32 (2007), 1125-1137. doi: 10.1016/j.chaos.2005.11.037. [37] J. G. Wardrop, Some theoretical aspects of road traffic research, Proceedings of the Institution of Civil Engineers, Part II, 1 (1952), 325-362. [38] J. Wenlong, Traffic Flow Models and Their Numerical Solutions, University of Science and Technology of China, 1998. [39] J. Wenlong, Traffic Flow Models and Their Numerical Solutions, University of California Davis, 2000. [40] G. C. K. Wong and S. C. Wong, A multi-class traffic flow model-an extension of LWR model with heterogeneous drivers, Transportation Research Part A, Policy and Practice, 36 (2013), 827-841. [41] N. Wu, A new approach for modeling of Fundamental Diagrams, Transportation Research Part A: Policy and Practice, 36 (2002), 867-884. doi: 10.1016/S0965-8564(01)00043-X.
Approximate density (veh/m)
Approximate density. Vs exact density (veh/m)
The variations of Link occupancy in time
The evolution of hydrocarbon emission rate in time
Aggregate emission rate vs. link occupancy
Flow.Vs density (fundamental diagram)
Travel speed
coefficients of $B_{j}$ and its derivatives
 $x$ $x_{j-1}$ $x_{j}$ $x_{j+1}$ $B_{j}$ $\frac{1}{6}$ $\frac{4}{6}$ $\frac{1}{6}$ $B^{'}_{j}$ $\frac{-1}{2h}$ $0$ $\frac{1}{2h}$ $B^{''}_{j}$ $\frac{1}{h^{2}}$ $\frac{-2}{h^{2}}$ $\frac{1}{h^{2}}$
 $x$ $x_{j-1}$ $x_{j}$ $x_{j+1}$ $B_{j}$ $\frac{1}{6}$ $\frac{4}{6}$ $\frac{1}{6}$ $B^{'}_{j}$ $\frac{-1}{2h}$ $0$ $\frac{1}{2h}$ $B^{''}_{j}$ $\frac{1}{h^{2}}$ $\frac{-2}{h^{2}}$ $\frac{1}{h^{2}}$
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