×

zbMATH — the first resource for mathematics

Convergence of a Godunov scheme to an Audusse-Perthame adapted entropy solution for conservation laws with BV spatial flux. (English) Zbl 1451.65107
Summary: In this article we consider the initial value problem for a scalar conservation law in one space dimension with a spatially discontinuous flux. There may be infinitely many flux discontinuities, and the set of discontinuities may have accumulation points. Thus the existence of traces cannot be assumed. In [E. Audusse and B. Perthame, Proc. R. Soc. Edinb., Sect. A, Math. 135, No. 2, 253–266 (2005; Zbl 1071.35079)] proved a uniqueness result that does not require the existence of traces, using adapted entropies. We generalize the Godunov-type scheme of Adimurthi et al. [SIAM J. Numer. Anal. 42, No. 1, 179–208 (2004; Zbl 1081.65082)] for this problem with the following assumptions on the flux function, (i) the flux is BV in the spatial variable and (ii) the critical point of the flux is BV as a function of the space variable. We prove that the Godunov approximations converge to an adapted entropy solution, thus providing an existence result, and extending the convergence result of Adimurthi, Jaffré and Gowda.
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
35B44 Blow-up in context of PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
76M20 Finite difference methods applied to problems in fluid mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adimurthi; Ghoshal, SS; Veerappa Gowda, GD, Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux, Commun. Pure Appl. Math., 64, 1, 84-115 (2011) · Zbl 1223.35222
[2] Adimurthi; Jaffré, J.; Veerappa Gowda, GD, 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
[3] Adimurthi; Mishra, S.; Veerappa Gowda, GD, Optimal entropy solutions for conservation laws with discontinuous flux functions, J. Hyperbolic Differ. Equ., 2, 783-837 (2005) · Zbl 1093.35045
[4] Andreianov, B.; Cancès, C., Vanishing capillarity solutions of Buckley-Leverett equation with gravity in two-rocks’ medium, Comput. Geosci., 17, 551-572 (2013) · Zbl 1392.76033
[5] Andreianov, B.; Karlsen, KH; Risebro, NH, A theory of \(L^1\)-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., 201, 27-86 (2011) · Zbl 1261.35088
[6] Audusse, E.; Perthame, B., Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies, Proc. R. Soc. Edinb. Sect. A, 135, 253-265 (2005) · Zbl 1071.35079
[7] Baiti, P.; Jenssen, HK, Well-posedness for a class of \(2\times 2\) conservation laws with \(L^{\infty }\) data, J. Differ. Equ., 140, 161-185 (1997) · Zbl 0892.35097
[8] Bürger, R.; Karlsen, KH; Risebro, NH; Towers, JD, Well-posedness in \(BV_t\) and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units, Numer. Math., 97, 25-65 (2004) · Zbl 1053.76047
[9] Crandall, MG; Majda, A., Monotone difference approximations for scalar conservation laws, Math. Comput., 34, 1-21 (1980) · Zbl 0423.65052
[10] Diehl, S., A conservation law with point source and discontinuous flux function modelling continuous sedimentation, SIAM J. Appl. Math., 56, 388-419 (1996) · Zbl 0849.35142
[11] Garavello, M.; Natalini, R.; Piccoli, B.; Terracina, A., Conservation laws with discontinuous flux, Netw. Heterog. Media, 2, 159-179 (2007) · Zbl 1142.35511
[12] Ghoshal, SS, Optimal results on TV bounds for scalar conservation laws with discontinuous flux, J. Differ. Equ., 3, 980-1014 (2015) · Zbl 1312.35032
[13] Ghoshal, SS, BV regularity near the interface for nonuniform convex discontinuous flux, Netw. Heterog. Media, 11, 331-348 (2016) · Zbl 1343.35050
[14] Holden, H.; Risebro, NH, Front Tracking for Hyperbolic Conservation Laws (2002), New York: Springer, New York · Zbl 1006.35002
[15] Kružkov, SN, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81, 123, 228-255 (1970)
[16] May, L.; Shearer, M.; Davis, K., Scalar conservation laws with nonconstant coefficients with application to particle size segregation in granular flow, J. Nonlinear Sci., 20, 689-707 (2010) · Zbl 1209.35080
[17] Panov, EY, Existence of strong traces for quasi-solutions of multidimensional scalar conservation laws, J. Hyperbolic Differ. Equ., 4, 729-770 (2007) · Zbl 1144.35037
[18] Panov, EY, On existence and uniqueness of entropy solutions to the Cauchy problem for a conservation law with discontinuous flux, J. Hyperbolic Differ. Equ., 06, 525-548 (2009) · Zbl 1181.35145
[19] Piccoli, B.; Tournus, M., A general BV existence result for conservation laws with spatial heterogeneities, SIAM J. Math. Anal., 50, 2901-2927 (2018) · Zbl 1402.35171
[20] Shen, W., On the uniqueness of vanishing viscosity solutions for Riemann problems for polymer flooding, Nonlinear Differ. Equ. Appl., 24, 37 (2017) · Zbl 1379.35003
[21] Towers, JD, Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal., 38, 681-698 (2000) · Zbl 0972.65060
[22] Towers, JD, An existence result for conservation laws having BV spatial flux heterogeneities—without concavity, J. Differ. Equ., 269, 5754-5764 (2020) · Zbl 1440.35213
[23] Vasseur, A., Strong traces of multidimensional scalar conservation laws, Arch. Ration. Mech. Anal., 160, 181-193 (2001) · Zbl 0999.35018
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.