February  2022, 17(1): 101-128. doi: 10.3934/nhm.2021025

Well-posedness theory for nonlinear scalar conservation laws on networks

Department of Mathematics, University of Oslo, Postboks 1053, Blindern, 0316 Oslo, Norway

* Corresponding author: Markus Musch

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

Fund Project: USF was partially supported by the Research Council of Norway project INICE, project no. 301538

We consider nonlinear scalar conservation laws posed on a network. We define an entropy condition for scalar conservation laws on networks and establish $L^1$ stability, and thus uniqueness, for weak solutions satisfying the entropy condition. We apply standard finite volume methods and show stability and convergence to the unique entropy solution, thus establishing existence of a solution in the process. Both our existence and stability/uniqueness theory is centred around families of stationary states for the equation. In one important case – for monotone fluxes with an upwind difference scheme – we show that the set of (discrete) stationary solutions is indeed sufficiently large to suit our general theory. We demonstrate the method's properties through several numerical experiments.

Citation: Markus Musch, Ulrik Skre Fjordholm, Nils Henrik Risebro. Well-posedness theory for nonlinear scalar conservation laws on networks. Networks and Heterogeneous Media, 2022, 17 (1) : 101-128. doi: 10.3934/nhm.2021025
References:
[1]

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., 37 (2017), 5913-5942.  doi: 10.3934/dcds.2017257.

[2]

B. P. Andreianov, G. M. Coclite and C. Donadello, Well-posedness for a monotone solver for traffic junctions, preprint, arXiv: 1605.01554.

[3]

B. AndreianovK. H. Karlsen and N. H. Risebro, A Theory of $L^1$-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., 201 (2011), 27-86.  doi: 10.1007/s00205-010-0389-4.

[4]

E. Audusse and B. Perthame, Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 253-265.  doi: 10.1017/S0308210500003863.

[5]

J. Badwaik and A. M. Ruf, Convergence rates of monotone schemes for conservation laws with discontinuous flux, SIAM J. Numer. Anal., 58 (2020), 607-629.  doi: 10.1137/19M1283276.

[6]

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

[7]

R. BürgerK. H. Karlsen and J. D. Towers, An Engquist–Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections, SIAM J. Numer. Anal., 47 (2009), 1684-1712.  doi: 10.1137/07069314X.

[8]

G. M. Coclite and L. di Ruvo, Vanishing viscosity for traffic on networks with degenerate diffusivity, Mediterr. J. Math., 16 (2019), Paper No. 110, 21 pp. doi: 10.1007/s00009-019-1391-1.

[9]

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.

[10]

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

[11]

M. G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings, Proc. Amer. Math. Soc., 78 (1980), 385-390.  doi: 10.1090/S0002-9939-1980-0553381-X.

[12]

B. Engquist and S. Osher, One-sided difference approximations for nonlinear conservation laws, Math. Comp., 36 (1981), 321-351.  doi: 10.1090/S0025-5718-1981-0606500-X.

[13]

U. S. Fjordholm, M. Musch and N. H. Risebro, Well-posedness of traffic flow models on networks, Submitted to SIAM Journal on Numerical Analysis, 2021.

[14]

M. Garavello, A review of conservation laws on networks, Netw. Heterog. Media, 5 (2010), 565-581.  doi: 10.3934/nhm.2010.5.565.

[15]

M. Garavello and B. Piccoli, Entropy type conditions for Riemann solvers at nodes, Adv. Differential Equations, 16 (2011), 113-144. 

[16]

S. K. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics., Mat. Sb., 47 (1959), 271-306. 

[17]

H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, 2$^{nd}$ edition, Springer-Verlag, Berlin, Heidelberg, 2015. doi: 10.1007/978-3-662-47507-2.

[18]

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.

[19]

K. H. Karlsen, N. H. Risebro and J. D. Towers, $L^1$-stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients, Skr. K. Nor. Vidensk. Selsk., (2003), 1–49.

[20]

K. H. KarlsenN. H. Risebro and J. D. Towers, Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient, IMA J. Numer. Anal., 22 (2002), 623-664.  doi: 10.1093/imanum/22.4.623.

[21]

S. N. Kružkov, First order quasilinear equaitons in several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255. 

[22]

N. N. Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation, USSR Computational Mathematics and Mathematical Physics, 16 (1976), 105-119.  doi: 10.1016/0041-5553(76)90046-X.

[23]

M. J. Lighthill and G. B. Whitham, On kinematic waves Ⅱ. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London Ser. A, 229 (1955), 317-345.  doi: 10.1098/rspa.1955.0089.

[24]

S. Osher, Riemann solvers, the entropy condition, and difference approximations, SIAM J. Numer. Anal., 21 (1984), 217-235.  doi: 10.1137/0721016.

[25]

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

[26]

P. I. Richards, Waves on the highway, Operations Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.

[27]

J. Ridder and A. M. Ruf, A convergent finite difference scheme for the Ostrovsky–Hunter equation with Dirichlet boundary conditions, BIT Numerical Mathematics, 59 (2019), 775-796.  doi: 10.1007/s10543-019-00746-7.

[28]

B. Temple, Global solution of the Cauchy problem for a class of $2\times 2$ nonstrictly hyperbolic conservation laws, Adv. in Appl. Math., 3 (1982), 335-375.  doi: 10.1016/S0196-8858(82)80010-9.

[29]

J. D. Towers, An explicit finite volume algorithm for vanishing viscosity solutions on a network, Netw. Heterog. Media, 2021. doi: 10.3934/nhm.2021021.

[30]

J. D. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal., 38 (2000), 681-698.  doi: 10.1137/S0036142999363668.

show all references

References:
[1]

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., 37 (2017), 5913-5942.  doi: 10.3934/dcds.2017257.

[2]

B. P. Andreianov, G. M. Coclite and C. Donadello, Well-posedness for a monotone solver for traffic junctions, preprint, arXiv: 1605.01554.

[3]

B. AndreianovK. H. Karlsen and N. H. Risebro, A Theory of $L^1$-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., 201 (2011), 27-86.  doi: 10.1007/s00205-010-0389-4.

[4]

E. Audusse and B. Perthame, Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 253-265.  doi: 10.1017/S0308210500003863.

[5]

J. Badwaik and A. M. Ruf, Convergence rates of monotone schemes for conservation laws with discontinuous flux, SIAM J. Numer. Anal., 58 (2020), 607-629.  doi: 10.1137/19M1283276.

[6]

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

[7]

R. BürgerK. H. Karlsen and J. D. Towers, An Engquist–Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections, SIAM J. Numer. Anal., 47 (2009), 1684-1712.  doi: 10.1137/07069314X.

[8]

G. M. Coclite and L. di Ruvo, Vanishing viscosity for traffic on networks with degenerate diffusivity, Mediterr. J. Math., 16 (2019), Paper No. 110, 21 pp. doi: 10.1007/s00009-019-1391-1.

[9]

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.

[10]

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

[11]

M. G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings, Proc. Amer. Math. Soc., 78 (1980), 385-390.  doi: 10.1090/S0002-9939-1980-0553381-X.

[12]

B. Engquist and S. Osher, One-sided difference approximations for nonlinear conservation laws, Math. Comp., 36 (1981), 321-351.  doi: 10.1090/S0025-5718-1981-0606500-X.

[13]

U. S. Fjordholm, M. Musch and N. H. Risebro, Well-posedness of traffic flow models on networks, Submitted to SIAM Journal on Numerical Analysis, 2021.

[14]

M. Garavello, A review of conservation laws on networks, Netw. Heterog. Media, 5 (2010), 565-581.  doi: 10.3934/nhm.2010.5.565.

[15]

M. Garavello and B. Piccoli, Entropy type conditions for Riemann solvers at nodes, Adv. Differential Equations, 16 (2011), 113-144. 

[16]

S. K. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics., Mat. Sb., 47 (1959), 271-306. 

[17]

H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, 2$^{nd}$ edition, Springer-Verlag, Berlin, Heidelberg, 2015. doi: 10.1007/978-3-662-47507-2.

[18]

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.

[19]

K. H. Karlsen, N. H. Risebro and J. D. Towers, $L^1$-stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients, Skr. K. Nor. Vidensk. Selsk., (2003), 1–49.

[20]

K. H. KarlsenN. H. Risebro and J. D. Towers, Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient, IMA J. Numer. Anal., 22 (2002), 623-664.  doi: 10.1093/imanum/22.4.623.

[21]

S. N. Kružkov, First order quasilinear equaitons in several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255. 

[22]

N. N. Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation, USSR Computational Mathematics and Mathematical Physics, 16 (1976), 105-119.  doi: 10.1016/0041-5553(76)90046-X.

[23]

M. J. Lighthill and G. B. Whitham, On kinematic waves Ⅱ. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London Ser. A, 229 (1955), 317-345.  doi: 10.1098/rspa.1955.0089.

[24]

S. Osher, Riemann solvers, the entropy condition, and difference approximations, SIAM J. Numer. Anal., 21 (1984), 217-235.  doi: 10.1137/0721016.

[25]

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

[26]

P. I. Richards, Waves on the highway, Operations Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.

[27]

J. Ridder and A. M. Ruf, A convergent finite difference scheme for the Ostrovsky–Hunter equation with Dirichlet boundary conditions, BIT Numerical Mathematics, 59 (2019), 775-796.  doi: 10.1007/s10543-019-00746-7.

[28]

B. Temple, Global solution of the Cauchy problem for a class of $2\times 2$ nonstrictly hyperbolic conservation laws, Adv. in Appl. Math., 3 (1982), 335-375.  doi: 10.1016/S0196-8858(82)80010-9.

[29]

J. D. Towers, An explicit finite volume algorithm for vanishing viscosity solutions on a network, Netw. Heterog. Media, 2021. doi: 10.3934/nhm.2021021.

[30]

J. D. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal., 38 (2000), 681-698.  doi: 10.1137/S0036142999363668.

Figure 1.  A star shaped network with two ingoing and three outgoing edges
Figure 2.  A network with a periodic edge
Figure 3.  Initial state and state at $ t = 0.7 $ of a {Burgers-type equation} with travelling shock wave which hits the vertex at time $ t = 1-\frac{1}{\sqrt{2}} $. Here, the graph includes a periodic edge
Figure 4.  Initial state at $ t = 0 $ and state at $ t = 0.2 $ of a traffic flow problem with an initial shock at the vertex developing a different elementary wave on each outgoing edge
Table 1.  $ L^1 $ errors and estimated orders of convergence (EOC) for a selection of examples
Example 7.3 Example 7.5 Example 7.4 Example 7.1 Example 7.2
Grid level $ L^1 $ error EOC $ L^1 $ error EOC $ L^1 $ error EOC $ L^1 $ error EOC $ L^1 $ error EOC
3 0.10877 - 0.11630 - 0.14459 - 0.07087 - 0.09904 -
4 0.05496 0.98 0.07136 0.70 0.08016 0.85 0.0546 0.38 0.04913 1.01
5 0.03649 0.59 0.04372 0.71 0.04651 0.79 0.03117 0.81 0.02844 0.79
6 0.02629 0.47 0.02255 0.96 0.02711 0.78 0.01903 0.71 0.01627 0.81
7 0.01830 0.52 0.01360 0.73 0.01495 0.86 0.01115 0.77 0.00919 0.82
8 0.01255 0.54 0.00653 1.06 0.00925 0.69 0.00644 0.79 0.00527 0.80
9 0.00883 0.51 0.00325 1.01 0.00480 0.95 0.00330 0.96 0.00268 0.98
10 0.00625 0.50 0.00160 1.02 0.00295 0.70 0.00173 0.93 0.00150 0.84
11 0.00442 0.50 0.00086 0.90 0.00152 0.96 0.00085 1.03 0.00084 0.84
12 0.00312 0.50 0.00040 1.10 0.00081 0.91 0.00042 1.02 0.00047 0.84
Example 7.3 Example 7.5 Example 7.4 Example 7.1 Example 7.2
Grid level $ L^1 $ error EOC $ L^1 $ error EOC $ L^1 $ error EOC $ L^1 $ error EOC $ L^1 $ error EOC
3 0.10877 - 0.11630 - 0.14459 - 0.07087 - 0.09904 -
4 0.05496 0.98 0.07136 0.70 0.08016 0.85 0.0546 0.38 0.04913 1.01
5 0.03649 0.59 0.04372 0.71 0.04651 0.79 0.03117 0.81 0.02844 0.79
6 0.02629 0.47 0.02255 0.96 0.02711 0.78 0.01903 0.71 0.01627 0.81
7 0.01830 0.52 0.01360 0.73 0.01495 0.86 0.01115 0.77 0.00919 0.82
8 0.01255 0.54 0.00653 1.06 0.00925 0.69 0.00644 0.79 0.00527 0.80
9 0.00883 0.51 0.00325 1.01 0.00480 0.95 0.00330 0.96 0.00268 0.98
10 0.00625 0.50 0.00160 1.02 0.00295 0.70 0.00173 0.93 0.00150 0.84
11 0.00442 0.50 0.00086 0.90 0.00152 0.96 0.00085 1.03 0.00084 0.84
12 0.00312 0.50 0.00040 1.10 0.00081 0.91 0.00042 1.02 0.00047 0.84
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