We provide a complete study of the model investigated in [Coclite, Garavello, SIAM J. Math. Anal., 2010]. We prove well-posedness of solutions obtained as vanishing viscosity limits for the Cauchy problem for scalar conservation laws $ ρ_{h, t} + f_h(ρ_h)_x = 0$, for $h∈ \{1, ..., m+n\}$, on a junction where $m$ incoming and $n$ outgoing edges meet. Our analysis and the definition of the admissible solution rely upon the complete description of the set of edge-wise constant solutions and its properties, which is of some interest on its own. The Riemann solver at the junction is characterized. In order to prove uniqueness, we introduce a family of Kruzhkov-type adapted entropies at the junction. Existence is justified both by the vanishing viscosity method and via the proof of convergence of a monotone well-balanced finite volume discretization. Beyond the classical vanishing viscosity framework, the numerical procedure and the uniqueness argument can be applied to general junction solvers enjoying the crucial order-preservation property.
Citation: |
B. Andreianov
and C. Cancés
, On interface transmission conditions for conservation laws with discontinuous flux of general shape, J. Hyperbolic Differ. Equ., 12 (2015)
, 343-384.
doi: 10.1142/S0219891615500101.![]() ![]() ![]() |
|
B. Andreianov
, K. H. Karlsen
and N. H. Risebro
, On vanishing viscosity approximation of conservation laws with discontinuous flux, Netw. Heterog. Media, 5 (2010)
, 617-633.
doi: 10.3934/nhm.2010.5.617.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
B. Andreianov
and D. Mitrović
, Entropy conditions for scalar conservation laws with discontinuous flux revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015)
, 1307-1335.
doi: 10.1016/j.anihpc.2014.08.002.![]() ![]() ![]() |
|
B. Andreianov
and K. Sbihi
, Well-posedness of general boundary-value problems for scalar conservation laws, Trans. Amer. Math. Soc., 367 (2015)
, 3763-3806.
doi: 10.1090/S0002-9947-2015-05988-1.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
P. Baiti
and H. K. Jenssen
, Well-posedness for a class of $2×2$ conservation laws with $L^∞$ data, J. Differential Equations, 140 (1997)
, 161-185.
doi: 10.1006/jdeq.1997.3308.![]() ![]() ![]() |
|
C. Bardos
, A. Y. Leroux
and J.-C. Nedelec
, First order quasilinear equations with boundary conditions, Communications in partial differential equations, 4 (1979)
, 1017-1034.
doi: 10.1080/03605307908820117.![]() ![]() ![]() |
|
A. Bressan
, S. Čanić
, M. Garavello
, M. Herty
and B. Piccoli
, Flows on networks: Recent results and perspectives, EMS Surv. Math. Sci., 1 (2014)
, 47-111.
doi: 10.4171/EMSS/2.![]() ![]() ![]() |
|
R. Bürger
, A. García
, K. H. Karlsen
and J. D. Towers
, A family of numerical schemes for kinematic flows with discontinuous flux, J. Engrg. Math., 60 (2008)
, 387-425.
doi: 10.1007/s10665-007-9148-4.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
G. M. Coclite
and M. Garavello
, Vanishing viscosity for traffic on networks, SIAM J. Math. Anal., 42 (2010)
, 1761-1783.
doi: 10.1137/090771417.![]() ![]() ![]() |
|
G. M. Coclite
, M. Garavello
and B. Piccoli
, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005)
, 1862-1886.
doi: 10.1137/S0036141004402683.![]() ![]() ![]() |
|
M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp., 34 (1980), 1-21, URL http://dx.doi.org/10.2307/2006218.
doi: 10.1090/S0025-5718-1980-0551288-3.![]() ![]() ![]() |
|
S. Diehl
, A uniqueness condition for nonlinear convection-diffusion equations with discontinuous coefficients, J. Hyperbolic Differ. Equ., 6 (2009)
, 127-159.
doi: 10.1142/S0219891609001794.![]() ![]() ![]() |
|
F. Dubois
and P. LeFloch
, Boundary conditions for nonlinear hyperbolic systems of conservation laws, J. Differential Equations, 71 (1988)
, 93-122.
doi: 10.1016/0022-0396(88)90040-X.![]() ![]() ![]() |
|
R. Eymard
, T. Gallouët
and R. Herbin
, Finite volume methods, in Handbook of numerical analysis, Vol. Ⅶ, Handb. Numer. Anal., Ⅶ, North-Holland, Amsterdam, (2000)
, 713-1020.
![]() ![]() |
|
M. Garavello and B. Piccoli,
Traffic Flow on Networks, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
H. Holden and N. H. Risebro,
Front Tracking for Hyperbolic Conservation Laws, vol. 152 of Applied Mathematical Sciences, 2nd edition, Springer, Heidelberg, 2015.
doi: 10.1007/978-3-662-47507-2.![]() ![]() ![]() |
|
C. Imbert and R. Monneau, Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks, Ann. Sci. éc. Norm. Supér., (4) 50 (2017), 357-448, URL https: //hal.archives-ouvertes.fr/hal-00832545.
doi: 10.24033/asens.2323.![]() ![]() ![]() |
|
C. Imbert
, R. Monneau
and H. Zidani
, A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM Control Optim. Calc. Var., 19 (2013)
, 129-166.
doi: 10.1051/cocv/2012002.![]() ![]() ![]() |
|
S. N. Kružhkov
, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970)
, 228-255.
![]() ![]() |
|
J.-P. Lebacque
, The Godunov scheme and what it means for first order traffic flow models, in Internaional symposium on transportation and traffic theory, (1996)
, 647-677.
![]() |
|
E. Y. Panov
, On sequences of measure-valued solutions of a first-order quasilinear equation, Mat. Sb., 185 (1994)
, 87-106.
doi: 10.1070/SM1995v081n01ABEH003621.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
L. Tartar
, Nonlinear analysis and mechanics: Heriot-watt symposium, in Compensated Compactness and Applications to Partial Differential Equations, vol. IV, Pitman, Boston, (1979)
, 317-345.
![]() |
A junction consisting of