# zbMATH — the first resource for mathematics

Convergence rates of monotone schemes for conservation laws with discontinuous flux. (English) Zbl 1440.65107
##### MSC:
 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 35R05 PDEs with low regular coefficients and/or low regular data 35L65 Hyperbolic conservation laws 35L45 Initial value problems for first-order hyperbolic systems
Full Text:
##### References:
 [1] Adimurthi, S. Mishra, and G. V. Gowda, Conservation law with the flux function discontinuous in the space variable-II: Convex-concave type fluxes and generalized entropy solutions, J. Comput. Appl. Math., 203 (2007), pp. 310-344. · Zbl 1131.65071 [2] Adimurthi, S. Misra, and G. V. Gowda, Optimal entropy solutions for conservation laws with discontinuous flux-functions, J. Hyperbolic Differ. Equ., 2 (2005), pp. 783-837. · Zbl 1093.35045 [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), pp. 27-86. · Zbl 1261.35088 [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), pp. 253-265. · Zbl 1071.35079 [5] J. Badwaik, N. H. Risebro, and C. Klingenberg, Multilevel Monte Carlo Finite Volume Methods for Random Conservation Laws with Discontinuous Flux, preprint, https://arxiv.org/abs/1906.08991, 2019. [6] P. Baiti and H. K. Jenssen, Well-posedness for a class of 2x2 conservation laws with L^infty data, J. Differential Equations, 140 (1997), pp. 161-185. · Zbl 0892.35097 [7] R. Bürger, K. Karlsen, C. Klingenberg, and N. Risebro, A front tracking approach to a model of continuous sedimentation in ideal clarifier-thickener units, Nonlinear Anal. Real World Appl., 4 (2003), pp. 457-481. · Zbl 1013.35052 [8] G. Coclite, J. Ridder, and N. Risebro, A convergent finite difference scheme for the Ostrovsky-Hunter equation on a bounded domain, BIT, 57 (2017), pp. 93-122. · Zbl 1368.65133 [9] G. M. Coclite and N. H. Risebro, Conservation laws with time dependent discontinuous coefficients, SIAM J. Math. Anal., 36 (2005), pp. 1293-1309. · Zbl 1078.35071 [10] M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp., 34 (1980), pp. 1-21. · Zbl 0423.65052 [11] S. Diehl, A conservation law with point source and discontinuous flux function modelling continuous sedimentation, SIAM J. Appl. Math., 56 (1996), pp. 388-419. · Zbl 0849.35142 [12] T. Gimse and N. H. Risebro, Riemann problems with a discontinuous flux function, in Proceedings of the Third International Conference on Hyperbolic Problems, vol. 1, 1991, pp. 488-502. · Zbl 0789.35102 [13] T. Gimse and N. H. Risebro, Solution of the cauchy problem for a conservation law with a discontinuous flux function, SIAM J. Math. Anal., 23 (1992), pp. 635-648. · Zbl 0776.35034 [14] J. Greenberg, A. Leroux, R. Baraille, and A. Noussair, Analysis and approximation of conservation laws with source terms, SIAM J. Numer. Anal., 34 (1997), pp. 1980-2007. · Zbl 0888.65100 [15] H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer-Verlag, Berlin, 2015. · Zbl 1346.35004 [16] K. Karlsen, N. Risebro, and J. Towers, Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient, IMA J. Numer. Anal., 22 (2002), pp. 623-664. · Zbl 1014.65073 [17] K. H. Karlsen, N. H. Risebro, and J. D. Towers, On a nonlinear degenerate parabolic transport-diffusion equation with a discontinuous coefficient, Electron. J. Differential Equations, 2002 (2002), 93. · Zbl 1015.35049 [18] K. H. Karlsen and J. D. Towers, Convergence of the Lax-Friedrichs scheme and stability for conservation laws with a discontinuous space-time dependent flux, Chin. Ann. Math., 25 (2004), pp. 287-318. · Zbl 1112.65085 [19] R. A. Klausen and N. H. Risebro, Stability of conservation laws with discontinuous coefficients, J. Differential Equations, 157 (1999), pp. 41-60. · Zbl 0935.35097 [20] C. Klingenberg and N. H. Risebro, Convex conservation laws with discontinuous coefficients. Existence, uniqueness and asymptotic behavior, Comm. Partial Differential Equations, 20 (1995), pp. 1959-1990. · Zbl 0836.35090 [21] C. Klingenberg and N. H. Risebro, Stability of a resonant system of conservation laws modeling polymer flow with gravitation, J. Differential Equations, 170 (2001), pp. 344-380. · Zbl 0977.35083 [22] S. N. Kružkov, First order quasilinear equations in several independent variables, Sb. Math., 10 (1970), pp. 217-243. [23] N. Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation, USSR Comput. Math. Math. Phys., 16 (1976), pp. 105-119. · Zbl 0381.35015 [24] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts Appl. Math., Cambridge University Press, Cambridge, UK, 2002. · Zbl 1010.65040 [25] M. J. Lighthill and G. B. Whitham, On kinematic waves II. A theory of traffic flow on long crowded roads, Proc. A, 229 (1955), pp. 317-345. · Zbl 0064.20906 [26] B. J. Lucier, A moving mesh numerical method for hyperbolic conservation laws, Math. Comp., 46 (1986), pp. 59-69. · Zbl 0592.65062 [27] S. Mishra, Convergence of upwind finite difference schemes for a scalar conservation law with indefinite discontinuities in the flux function, SIAM J. Numer. Anal., 43 (2005), pp. 559-577. · Zbl 1096.35085 [28] H. Nessyahu and E. Tadmor, The convergence rate of approximate solutions for nonlinear scalar conservation laws, SIAM J. Numer. Anal., 29 (1992), pp. 1505-1519. · Zbl 0765.65092 [29] H. Nessyahu, E. Tadmor, and T. Tassa, The convergence rate of Godunov type schemes, SIAM J. Numer. Anal., 31 (1994), pp. 1-16. · Zbl 0799.65096 [30] M. Ohlberger and J. Vovelle, Error estimate for the approximation of nonlinear conservation laws on bounded domains by the finite volume method, Math. Comp., 75 (2006), pp. 113-150. · Zbl 1082.65112 [31] J. Ridder and A. M. Ruf, A convergent finite difference scheme for the Ostrovsky-Hunter equation with Dirichlet boundary conditions, BIT, 2019. · Zbl 1433.65165 [32] N. H. Risebro and A. Tveito, Front tracking applied to a nonstrictly hyperbolic system of conservation laws, SIAM J. Sci. Statist. Comput., 12 (1991), pp. 1401-1419. · Zbl 0736.65075 [33] A. M. Ruf, E. Sande, and S. Solem, The optimal convergence rate of monotone schemes for conservation laws in the Wasserstein distance, J. Sci. Comput., 2019. · Zbl 1428.65022 [34] F. Şabac, The optimal convergence rate of monotone finite difference methods for hyperbolic conservation laws, SIAM J. Numer. Anal., 34 (1997), pp. 2306-2318. · Zbl 0992.65099 [35] S. Solem, Convergence rates of the front tracking method for conservation laws in the Wasserstein distances, SIAM J. Numer. Anal., 56 (2018), pp. 3648-3666. · Zbl 06995704 [36] Z.-H. Teng and P. Zhang, Optimal L^1-rate of convergence for the viscosity method and monotone scheme to piecewise constant solutions with shocks, SIAM J. Numer. Anal., 34 (1997), pp. 959-978. · Zbl 0873.65089 [37] J. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal., 38 (2000), pp. 681-698. · Zbl 0972.65060 [38] J. Towers, A difference scheme for conservation laws with a discontinuous flux: The nonconvex case, SIAM J. Numer. Anal., 39 (2001), pp. 1197-1218. · Zbl 1055.65104 [39] D. A. Venditti and D. L. Darmofal, Adjoint error estimation and grid adaptation for functional outputs: Application to quasi-one-dimensional flow, J. Comput. Phys., 164 (2000), pp. 204-227. · Zbl 0995.76057 [40] X. Wen and S. Jin, Convergence of an immersed interface upwind scheme for linear advection equations with piecewise constant coefficients I: L^1-error estimates, J. Comput. Math., 26 (2008), pp. 1-22. · Zbl 1174.65031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.