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Telegraph systems on networks and port-Hamiltonians. Ⅱ. Network realizability
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 |
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.
Keywords:
Hyperbolic conservation laws,
networks,
finite volume methods,
well-posedness,
convergence.
Mathematics Subject Classification:
Primary: 65M12, 35L65; Secondary: 65M08.
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. Andreianov, G. 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. Andreianov, K. 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. Bressan, S. Čanič, M. Garavello, M. 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. Bürger, K. 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. Karlsen, N. 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. Andreianov, G. 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. Andreianov, K. 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. Bressan, S. Čanič, M. Garavello, M. 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. Bürger, K. 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. Karlsen, N. 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 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 | EOC | EOC | EOC | EOC | 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 | EOC | EOC | EOC | EOC | 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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