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Finite volume schemes for locally constrained conservation laws. (English) Zbl 1196.65151
The paper is concerned with the study of the finite volume schemes for scalar conservation laws with local unilateral constraint of the form \(\partial_t u+\partial_x f(u)=0,\) \(t>0,x\in\mathbb R\), \(u(0,x)=u_0(x)\) and \(f(u(t,0))\leq F(t)\), \(t>0\). This type of problem models road obstacles in traffic. The authors characterize and approximate entropy solutions of the above problem in the \(L^\infty\) setting.

MSC:
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
90B20 Traffic problems in operations research
35L65 Hyperbolic conservation laws
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[1] Adimurthi S.M., Veerappa Gowda G.D.: Optimal entropy solutions for conservation laws with discontinuous flux-functions. J. Hyperbolic Diff. Equ. 2(4), 783–837 (2005) · Zbl 1093.35045 · doi:10.1142/S0219891605000622
[2] Andreianov, B., Karlsen, K.H., Risebro, N.H.: A theory of L 1-dissipative solvers for scalar conservation laws with discontinuous flux (in preparation) · Zbl 1261.35088
[3] Audusse E., Perthame B.: Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies. Proc. R. Soc. Edinb. Sect. A 135(2), 253–265 (2005) · Zbl 1071.35079 · doi:10.1017/S0308210500003863
[4] Bachmann F., Vovelle J.: Existence and uniqueness of entropy solution of scalar conservation laws with a flux function involving discontinuous coefficients. Comm. Partial Differ. Equ. 31, 371–395 (2006) · Zbl 1102.35064 · doi:10.1080/03605300500358095
[5] Baiti P., Jenssen H.K.: Well-posedness for a class of 2 \(\times\) 2 conservation laws with L data. J. Differ. Equ. 140(1), 161–185 (1997) · Zbl 0892.35097 · doi:10.1006/jdeq.1997.3308
[6] Bürger R., García A., Karlsen K.H., Towers J.D.: A family of numerical schemes for kinematic flows with discontinuous flux. J. Eng. Math. 60(3-4), 387–425 (2008) · Zbl 1200.76126 · doi:10.1007/s10665-007-9148-4
[7] 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 · doi:10.1137/07069314X
[8] Chen G.-Q., Frid H.: Divergence-measure fields and hyperbolic conservation laws. Arch. Ration. Mech. Anal. 147(2), 89–118 (1999) · Zbl 0942.35111 · doi:10.1007/s002050050146
[9] Colombo R.M., Goatin P.: A well posed conservation law with a variable unilateral constraint. J. Differ. Equ. 234(2), 654–675 (2007) · Zbl 1116.35087 · doi:10.1016/j.jde.2006.10.014
[10] DiPerna R.J.: Measure-valued solutions to conservation laws. Arch. Ration. Mech. Anal. 88(3), 223–270 (1985) · Zbl 0616.35055 · doi:10.1007/BF00752112
[11] Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Handbook of Numerical Analysis, vol. VII, pp. 713–1020. North-Holland, Amsterdam (2000) · Zbl 0981.65095
[12] 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. Vid. Selsk. 49 pp. (2003) · Zbl 1036.35104
[13] Kružkov, S.N.: First order quasilinear equations with several independent variables. (Russian) Math. Sb., 81(2), 228–255 (1970) [English transl. in Math. USSR Sb., 10 (1970)]
[14] Panov E.Yu.: Strong measure-valued solutions of the Cauchy problem for a first-order quasilinear equation with a bounded measure-valued initial function. Mosc. Univ. Math. Bull. 48(1), 18–21 (1993) · Zbl 0830.35023
[15] Panov E.Yu.: Existence of strong traces for quasi-solutions of multidimensional conservation laws. J. Hyperbolic Differ. Equ. 4(4), 729–770 (2007) · Zbl 1144.35037 · doi:10.1142/S0219891607001343
[16] Rusanov V.V.: The calculation of the interaction of non-stationary shock waves with barriers. Ž Vyčisl. Mat. i Mat. Fiz. 1, 267–279 (1961)
[17] Seguin N., Vovelle J.: Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients. Math. Model. Meth. Appl. Sci. 13(2), 221–257 (2003) · Zbl 1078.35011 · doi:10.1142/S0218202503002477
[18] Szepessy A.: Convergence of a shock-capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions. Math. Comp. 53(188), 527–545 (1989) · Zbl 0679.65072 · doi:10.1090/S0025-5718-1989-0979941-6
[19] Towers, J.D.: Convergence of a difference scheme for conservation laws with a discontinuous flux. SIAM J. Numer. Anal. 38(2), 681–698 (electronic) · Zbl 0972.65060
[20] Vasseur A.: Strong traces of multidimensional scalar conservation laws. Arch. Ration. Mech. Anal. 160(3), 181–193 (2001) · Zbl 0999.35018 · doi:10.1007/s002050100157
[21] Vol’pert A.I.: Spaces BV and quasilinear equations. Mat. Sb. (N.S.) 73(115), 255–302 (1967)
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