doi: 10.3934/nhm.2021021
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An explicit finite volume algorithm for vanishing viscosity solutions on a network

MiraCosta College, 3333 Manchester Avenue, Cardiff-by-the-Sea, CA 92007-1516, USA

Received  April 2021 Revised  July 2021 Early access September 2021

In [Andreianov, Coclite, Donadello, Discrete Contin. Dyn. Syst. A, 2017], a finite volume scheme was introduced for computing vanishing viscosity solutions on a single-junction network, and convergence to the vanishing viscosity solution was proven. This problem models $ m $ incoming and $ n $ outgoing roads that meet at a single junction. On each road the vehicle density evolves according to a scalar conservation law, and the requirements for joining the solutions at the junction are defined via the so-called vanishing viscosity germ. The algorithm mentioned above processes the junction in an implicit manner. We propose an explicit version of the algorithm. It differs only in the way that the junction is processed. We prove that the approximations converge to the unique entropy solution of the associated Cauchy problem.

Citation: John D. Towers. An explicit finite volume algorithm for vanishing viscosity solutions on a network. Networks & Heterogeneous Media, doi: 10.3934/nhm.2021021
References:
[1]

B. Andreianov and C. Can$\mathop {\text{c}}\limits^、 $es, On interface transmission conditions for conservation laws with discontinuous flux of general shape, J. Hyberbolic Differ. Equ., 12 (2015), 343-384.  doi: 10.1142/S0219891615500101.  Google Scholar

[2]

B. P. AndreianovG. M. Coclite and C. Donadello, Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network, Discrete Contin. Dyn. Syst. - A, 37 (2017), 5913-5942.  doi: 10.3934/dcds.2017257.  Google Scholar

[3]

B. AndreianovK. H. Karlsen and N. H. Risebro, A theory of L1-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., 201 (2011), 27-86.  doi: 10.1007/s00205-010-0389-4.  Google Scholar

[4]

A. BressanS. ČanićM. GaravelloM. Herty and B. Piccoli, Flows on networks: Recent results and perspectives, EMS Surv. Math. Sci., 1 (2014), 47-111.  doi: 10.4171/EMSS/2.  Google Scholar

[5]

G. BrettiR. Natalini and B. Piccoli, Numerical approximations of a traffic flow model on networks, Netw. Heterog. Media, 1 (2006), 57-84.  doi: 10.3934/nhm.200A.1.57.  Google Scholar

[6]

G. M. Coclite and C. Donadello, Vanishing viscosity on a star-shaped graph under general transmission conditions at the node, Netw. Heterog. Media, 15 (2020), 197-213.  doi: 10.3934/nhm.2020009.  Google Scholar

[7]

G. M. Coclite and M. Garavello, Vanishing viscosity for traffic flow on networks, SIAM J. Math. Anal., 42 (2010), 1761-1783.  doi: 10.1137/090771417.  Google Scholar

[8]

G. M. CocliteM. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886.  doi: 10.1137/S0036141004402683.  Google Scholar

[9]

M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp., 34 (1980), 1-21.  doi: 10.1090/S0025-5718-1980-0551288-3.  Google Scholar

[10]

S. Diehl, On scalar conservation laws with point source and discontinuous flux function, SIAM J. Math. Anal., 26 (1995), 1425-1451.  doi: 10.1137/S0036141093242533.  Google Scholar

[11]

U. S. Fjordholm, M. Musch and N. H. Risebro, Well-posedness theory for nonlinear scalar conservation laws on networks, preprint, https://arXiv.org/pdf/2102.06400.pdf. Google Scholar

[12]

M. Garavello and B. Piccoli, Conservation laws on complex networks, Ann. Inst. H. Poincaré Anal. Non Linéare, 26 (2009), 1925-1951.  doi: 10.1016/j.anihpc.2009.04.001.  Google Scholar

[13]

P. GoatinS. Göttlich and O. Kolb, Speed limit and ramp meter control for traffic flow networks, Eng. Optim., 48 (2016), 1121-1144.  doi: 10.1080/0305215X.2015.1097099.  Google Scholar

[14]

P. Goatin and E. Rossi, Comparative study of macroscopic traffic flow models at road junctions, Netw. Heterog. Media, 15 (2020), 216-279.  doi: 10.3934/nhm.2020012.  Google Scholar

[15]

M. Hilliges and W. Weidlich, Phenomenological model for dynamic traffic flow in networks, Transp. Res. B, 29 (1995), 407-431.  doi: 10.1016/0191-2615(95)00018-9.  Google Scholar

[16]

H. Holden and N. H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 26 (1995), 999-1017.  doi: 10.1137/S0036141093243289.  Google Scholar

[17]

K. H. Karlsen and J. D. Towers, Convergence of a Godunov scheme for for conservation laws with a discontinuous flux lacking the crossing condition, J. Hyperbolic Differ. Equ., 14 (2017), 671-701.  doi: 10.1142/S0219891617500229.  Google Scholar

[18]

J. Lebacque, The Godunov scheme and what it means for first order traffic flow models, in Proceedings of the 13th International Symposium of Transportation and Traffic Theory (ed. J. Lesort), Elsevier, (1996), 647–677. Google Scholar

[19]

J. P. Lebacque and M. M. Khoshyaran, First order macroscopic traffic flow models for networks in the context of dynamic assignment, Transportation Planning and Applied Optimization, 64 (2004), 119-140.  doi: 10.1007/0-306-48220-7_8.  Google Scholar

[20]

E. Yu. Panov, Existence of strong traces for quasi-solutions of multidimensional scalar conservation laws, J. Hyperbolic Differ. Equ., 4 (2007), 729-770.  doi: 10.1142/S0219891607001343.  Google Scholar

[21]

S. Pellegrino, On the implementation of a finite volumes scheme with monotone transmission conditions for scalar conservation laws on a star-shaped network, Appl. Numer. Math., 155 (2020), 181-191.  doi: 10.1016/j.apnum.2019.09.011.  Google Scholar

show all references

References:
[1]

B. Andreianov and C. Can$\mathop {\text{c}}\limits^、 $es, On interface transmission conditions for conservation laws with discontinuous flux of general shape, J. Hyberbolic Differ. Equ., 12 (2015), 343-384.  doi: 10.1142/S0219891615500101.  Google Scholar

[2]

B. P. AndreianovG. M. Coclite and C. Donadello, Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network, Discrete Contin. Dyn. Syst. - A, 37 (2017), 5913-5942.  doi: 10.3934/dcds.2017257.  Google Scholar

[3]

B. AndreianovK. H. Karlsen and N. H. Risebro, A theory of L1-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., 201 (2011), 27-86.  doi: 10.1007/s00205-010-0389-4.  Google Scholar

[4]

A. BressanS. ČanićM. GaravelloM. Herty and B. Piccoli, Flows on networks: Recent results and perspectives, EMS Surv. Math. Sci., 1 (2014), 47-111.  doi: 10.4171/EMSS/2.  Google Scholar

[5]

G. BrettiR. Natalini and B. Piccoli, Numerical approximations of a traffic flow model on networks, Netw. Heterog. Media, 1 (2006), 57-84.  doi: 10.3934/nhm.200A.1.57.  Google Scholar

[6]

G. M. Coclite and C. Donadello, Vanishing viscosity on a star-shaped graph under general transmission conditions at the node, Netw. Heterog. Media, 15 (2020), 197-213.  doi: 10.3934/nhm.2020009.  Google Scholar

[7]

G. M. Coclite and M. Garavello, Vanishing viscosity for traffic flow on networks, SIAM J. Math. Anal., 42 (2010), 1761-1783.  doi: 10.1137/090771417.  Google Scholar

[8]

G. M. CocliteM. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886.  doi: 10.1137/S0036141004402683.  Google Scholar

[9]

M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp., 34 (1980), 1-21.  doi: 10.1090/S0025-5718-1980-0551288-3.  Google Scholar

[10]

S. Diehl, On scalar conservation laws with point source and discontinuous flux function, SIAM J. Math. Anal., 26 (1995), 1425-1451.  doi: 10.1137/S0036141093242533.  Google Scholar

[11]

U. S. Fjordholm, M. Musch and N. H. Risebro, Well-posedness theory for nonlinear scalar conservation laws on networks, preprint, https://arXiv.org/pdf/2102.06400.pdf. Google Scholar

[12]

M. Garavello and B. Piccoli, Conservation laws on complex networks, Ann. Inst. H. Poincaré Anal. Non Linéare, 26 (2009), 1925-1951.  doi: 10.1016/j.anihpc.2009.04.001.  Google Scholar

[13]

P. GoatinS. Göttlich and O. Kolb, Speed limit and ramp meter control for traffic flow networks, Eng. Optim., 48 (2016), 1121-1144.  doi: 10.1080/0305215X.2015.1097099.  Google Scholar

[14]

P. Goatin and E. Rossi, Comparative study of macroscopic traffic flow models at road junctions, Netw. Heterog. Media, 15 (2020), 216-279.  doi: 10.3934/nhm.2020012.  Google Scholar

[15]

M. Hilliges and W. Weidlich, Phenomenological model for dynamic traffic flow in networks, Transp. Res. B, 29 (1995), 407-431.  doi: 10.1016/0191-2615(95)00018-9.  Google Scholar

[16]

H. Holden and N. H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 26 (1995), 999-1017.  doi: 10.1137/S0036141093243289.  Google Scholar

[17]

K. H. Karlsen and J. D. Towers, Convergence of a Godunov scheme for for conservation laws with a discontinuous flux lacking the crossing condition, J. Hyperbolic Differ. Equ., 14 (2017), 671-701.  doi: 10.1142/S0219891617500229.  Google Scholar

[18]

J. Lebacque, The Godunov scheme and what it means for first order traffic flow models, in Proceedings of the 13th International Symposium of Transportation and Traffic Theory (ed. J. Lesort), Elsevier, (1996), 647–677. Google Scholar

[19]

J. P. Lebacque and M. M. Khoshyaran, First order macroscopic traffic flow models for networks in the context of dynamic assignment, Transportation Planning and Applied Optimization, 64 (2004), 119-140.  doi: 10.1007/0-306-48220-7_8.  Google Scholar

[20]

E. Yu. Panov, Existence of strong traces for quasi-solutions of multidimensional scalar conservation laws, J. Hyperbolic Differ. Equ., 4 (2007), 729-770.  doi: 10.1142/S0219891607001343.  Google Scholar

[21]

S. Pellegrino, On the implementation of a finite volumes scheme with monotone transmission conditions for scalar conservation laws on a star-shaped network, Appl. Numer. Math., 155 (2020), 181-191.  doi: 10.1016/j.apnum.2019.09.011.  Google Scholar

Figure 1.  Example 1. Left panel: $ P^0 = 1/5 $. Right panel: $ P^0 $ computed by the fixed point iteration (2.9), or by choosing $ P^0 = 4/5 $. Solid line: $ u_1 $, dashed line: $ u_2 $, dot-dashed line: $ u_3 $. In the left panel a spurious bump in $ u_2 $ is visible, due to a bad choice of $ P^0 $ (which does not affect convergence). In the right panel there is no spurious bump
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