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Nonlocal systems of conservation laws in several space dimensions. (English) Zbl 1318.65046


MSC:

65M06 Finite difference 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
35L65 Hyperbolic conservation laws
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[1] D. Amadori and W. Shen, An integro-differential conservation law arising in a model of granular flow, J. Hyperbolic Differ. Equ., 9 (2012), pp. 105–131. · Zbl 1248.35118
[2] L. Ambrosio, F. Bouchut, and C. De Lellis, Well-posedness for a class of hyperbolic systems of conservation laws in several space dimensions, Comm. Partial Differential Equations, 29 (2004), pp. 1635–1651. · Zbl 1072.35116
[3] P. Amorim, R. M. Colombo, and A. Teixeira, On the numerical integration of scalar nonlocal conservation laws, ESAIM M2AN, 49 (2015), pp. 19–37. · Zbl 1317.65165
[4] F. Betancourt, R. Bürger, K. H. Karlsen, and E. M. Tory, On nonlocal conservation laws modelling sedimentation, Nonlinearity, 24 (2011), pp. 855–885. · Zbl 1381.76368
[5] S. Blandin and P. Goatin, Well-posedness of a conservation law with non-local flux arising in traffic flow modeling, Numer. Math., to appear. · Zbl 1336.65130
[6] R. Borsche, R. M. Colombo, M. Garavello, and A. Meurer, Differential equations modeling crowd interactions, J. Nonlinear Sci., to appear. · Zbl 1327.35244
[7] A. Bressan, Hyperbolic Systems of Conservation Laws, Oxford Lecture Ser. Math. Appl. 20, Oxford University Press, Oxford, UK, 2000.
[8] C. Chainais-Hillairet, Finite volume schemes for a hyperbolic equation. Convergence towards the entropy solution and error estimate, ESAIM M2AN, 33 (1999), pp. 129–156. · Zbl 0921.65071
[9] C. Chainais-Hillairet and S. Champier, Finite volume schemes for nonhomogeneous scalar conservation laws: Error estimate, Numer. Math., 88 (2001), pp. 607–639. · Zbl 1002.65105
[10] R. M. Colombo, M. Garavello, and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., 22 (2012), 1150023. · Zbl 1248.35213
[11] R. M. Colombo, M. Herty, and M. Mercier, Control of the continuity equation with a non local flow, ESAIM Control Optim. Calc. Var., 17 (2011), pp. 353–379. · Zbl 1232.35176
[12] R. M. Colombo and M. Lécureux-Mercier, Nonlocal crowd dynamics models for several populations, Acta Math. Sci. Ser. B Engl. Ed., 32 (2012), pp. 177–196. · Zbl 1265.35214
[13] M. G. Crandall and A. Majda, The method of fractional steps for conservation laws, Numer. Math., 34 (1980), pp. 285–314. · Zbl 0438.65076
[14] M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp., 34 (1980), pp. 1–21. · Zbl 0423.65052
[15] S. Göttlich, S. Hoher, P. Schindler, V. Schleper, and A. Verl, Modeling, simulation and validation of material flow on conveyor belts, Appl. Math. Model., 38 (2014), pp. 3295–3313. · Zbl 06991582
[16] K. H. Karlsen and N. H. Risebro, Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients, ESAIM M2AN, 35 (2001), pp. 239–269. · Zbl 1032.76048
[17] B. L. Keyfitz and H. C. Kranzer, A system of nonstrictly hyperbolic conservation laws arising in elasticity theory, Arch. Ration. Mech. Anal., 72 (1979/80), pp. 219–241. · Zbl 0434.73019
[18] U. Koley and N. H. Risebro, Finite difference schemes for the symmetric Keyfitz–Kranzer system, Z. Angew. Math. Phys., 64 (2013), pp. 1057–1085. · Zbl 1277.65070
[19] S. N. Kružhkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (123) (1970), pp. 228–255.
[20] N. N. Kuznecov, The accuracy of certain approximate methods for the computation of weak solutions of a first order quasilinear equation, Ž. Vyčisl. Mat. i Mat. Fiz., 16 (1976), pp. 1489–1502, 1627 (in Russian).
[21] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts Appl. Math., Cambridge University Press, Cambridge, UK, 2002.
[22] B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007. · Zbl 1185.92006
[23] R. Sanders, On convergence of monotone finite difference schemes with variable spatial differencing, Math. Comp., 40 (1983), pp. 91–106. · Zbl 0533.65061
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