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Convergence of the Godunov scheme for a scalar conservation law with time and space discontinuities. (English) Zbl 1406.35190

MSC:
 35L65 Hyperbolic conservation laws 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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 [1] Adimurthi; Jaffré, J.; Gowda, G. D. V., Godunov-type methods for conservation laws with a flux function discontinuous in space, SIAM J. Numer. Anal., 42, 1, 179-208, (2004) · Zbl 1081.65082 [2] Adimurthi; Mishra, S.; Gowda, G. D. V., Optimal entropy solutions for conservation laws with discontinuous flux-functions, J. Hyperbolic Differ. Equ., 2, 4, 783-837, (2005) · Zbl 1093.35045 [3] Andreianov, B.; Cancès, C., The Godunov scheme for scalar conservation laws with discontinuous Bell-shaped flux functions, Appl. Math. Lett., 25, 11, 1844-1848, (2012) · Zbl 1253.65122 [4] Andreianov, B.; Cancès, C., On interface transmission conditions for conservation laws with discontinuous flux of general shape, J. Hyperbolic Differ. Equ., 12, 2, 343-384, (2015) · Zbl 1336.35230 [5] Andreianov, B.; Karlsen, K. H.; Risebro, N. H., On vanishing viscosity approximation of conservation laws with discontinuous flux, Netw. Heterogeneous Media, 5, 3, 617-633, (2010) · Zbl 1270.35305 [6] Andreianov, B.; Karlsen, K. H.; Risebro, N. H., A theory of $$L^1$$-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., 201, 1, 27-86, (2011) · Zbl 1261.35088 [7] Bretti, G.; Natalini, R.; Piccoli, B., Numerical approximations of a traffic flow model on networks, Netw. Heterogeneous Media, 1, 1, 57-84, (2006) · Zbl 1124.90005 [8] Bürger, R.; García, A.; Karlsen, K. H.; Towers, J. D., A family of numerical schemes for kinematic flows with discontinuous flux, J. Engrg. Math., 60, 387-425, (2008) · Zbl 1200.76126 [9] Bürger, R.; García, A.; Karlsen, K. H.; Towers, J. D., Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model, Netw. Heterogeneous Media, 3, 1-41, (2008) · Zbl 1173.35586 [10] Bürger, R.; Karlsen, K. H.; Klingenberg, C.; Risebro, N. H., A front tracking approach to a model of continuous sedimentation in ideal clarifier-thickener units, Nonlinear Anal. Real World Appl., 4, 3, 457-48, (2003) · Zbl 1013.35052 [11] Bürger, R.; Karlsen, K. H.; Risebro, N. H.; Towers, J. D., Well-posedness in $$B V_t$$ and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units, Numer. Math., 97, 1, 25-65, (2004) · Zbl 1053.76047 [12] Bürger, R.; Karlsen, K. H.; Towers, J. D., An engquist-osher-type scheme for conservation laws with discontinuous flux adapted to flux connections, SIAM J. Numer. Anal., 47, 3, 1684-1712, (2009) · Zbl 1201.35022 [13] Coclite, G. M.; Risebro, N. H., Conservation laws with time dependent discontinuous coefficients, SIAM J. Math. Anal., 36, 1293-1309, (2005) · Zbl 1078.35071 [14] Daganzo, C. F., The cell transmission model, part II: network traffic, Transp. Res. B, 29, 79-93, (1995) [15] Diehl, S., A regulator for continuous sedimentation in ideal clarifier-thickener units, J. Eng. Math., 60, 265-291, (2008) · Zbl 1133.76045 [16] Diehl, S., A uniqueness condition for nonlinear convection-diffusion equations with discontinuous coefficients, J. Hyperbolic Differ. Equ., 6, 1, 127-159, (2009) · Zbl 1180.35305 [17] Garavello, M.; Piccoli, B., Traffic Flow on Networks, (2006), American Institute of Mathematical Sciences, Springfield, MO, USA · Zbl 1136.90012 [18] Gimse, T.; Risebro, N. H., Solution of the Cauchy problem for a conservation law with a discontinuous flux function, SIAM J. Math. Anal., 23, 3, 635-648, (1992) · Zbl 0776.35034 [19] Goatin, P. G.; Göttlich, S.; Kolb, O., Speed limit and ramp meter control for traffic flow networks, Eng. Optim., 48, 1121-1144, (2016) [20] Holden, H.; Risebro, N. H., Front Tracking for Hyperbolic Conservation Laws, 152, (2002), Springer-Verlag, New York · Zbl 1006.35002 [21] Jin, W.; Zhang, H., The inhomogeneous kinematic wave traffic flow model as a resonant nonlinear system, Transp. Sci., 37, 294-311, (2003) [22] Karlsen, K. H.; Risebro, N. H.; Towers, J. D., On a nonlinear degenerate parabolic transport-diffusion equation with a discontinuous coefficient, Electron. J. Differ. Equ., 93, 23, (2002) · Zbl 1015.35049 [23] Karlsen, K. H.; Risebro, N. H.; Towers, J. D., Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient, IMA J. Numer. Anal., 22, 4, 623-664, (2002) · Zbl 1014.65073 [24] Karlsen, K. H.; Risebro, N. H.; Towers, J. D., $$L^1$$ stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients, Skr. K. Nor. Vidensk. Selsk., 3, 1-49, (2003) · Zbl 1036.35104 [25] Karlsen, K. H.; Towers, J. D., Convergence of the Lax-Friedrichs scheme and stability for conservation laws with a discontinuous space-time dependent flux, Chinese Ann. Math., 25B, 287-318, (2004) · Zbl 1112.65085 [26] Klingenberg, C.; Risebro, N. H., Convex conservation laws with discontinuous coefficients. existence, uniqueness and asymptotic behavior, Comm. Partial Differential Equations, 20, 11-12, 1959-1990, (1995) · Zbl 0836.35090 [27] Lebacque, J. P.; Lesort, J. B., The Godunov scheme and what it means for first order traffic flow models, Proc. 13th ISTTT Transportation and Traffic Theory, 647-677, (1996), Pergamon Press, Oxford, UK [28] LeVeque, R. J., Numerical Methods for Conservation Laws, (1992), Birkhäuser Verlag, Basel · Zbl 0847.65053 [29] Liu, H.; Zhang, L.; Sun, D.; Wang, D., Optimize the settings of variable speed limit system to improve the performance of freeway traffic, IEEE Trans. Intell. Transp. Syst., 16, 6, 3249-3257, (2015) [30] Monache, M. L. D.; Piccoli, B.; Rossi, F., Traffic regulation via controlled speed limit, SIAM J. Control Optim., 55, 5, 2936-2958, (2017) · Zbl 1377.90016 [31] Muralidharan, A.; Horowitz, R., Optimal control of freeway networks based on the link node cell transmission model, Proc. Amer. Control Conf., 5769-5774, (2012), IEEE, Montreal, Canada [32] Seguin, N.; Vovelle, J., Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients, Math. Models Methods Appl. Sci., 13, 2, 221-257, (2003) · Zbl 1078.35011 [33] Towers, J. D., Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal., 38, 2, 681-698, (2000) · Zbl 0972.65060 [34] Zhang, M.; Shu, C. W.; Wong, G. C. K.; Wong, S. C., A weighted essentially non-oscillatory numerical scheme for a multi-class Lighthill-Whitham-Richards traffic flow model, J. Comput. Phys., 191, 639-659, (2003) · Zbl 1041.90008
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