Free Access
Issue
ESAIM: M2AN
Volume 55, Number 3, May-June 2021
Page(s) 1039 - 1065
DOI https://doi.org/10.1051/m2an/2021011
Published online 31 May 2021
  1. R. Aae Klausen and N.H. Risebro, Stability of conservation laws with discontinuous coefficients. J. Differ. Equ. 157 (1999) 41–60. [Google Scholar]
  2. R. Abgrall, A simple, flexible and generic deterministic approach to uncertainty quantifications in non linear problems: application to fluid flow problems (2008). [Google Scholar]
  3. S.M. Adimurthi 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) 310–344. [CrossRef] [Google Scholar]
  4. S.M. Adimurthi and G.V. Gowda, Optimal entropy solutions for conservation laws with discontinuous flux-functions. J. Hyperbolic Differ. Equ. 2 (2005) 783–837. [CrossRef] [MathSciNet] [Google Scholar]
  5. B. Andreianov, K.H. Karlsen and N.H. Risebro, A theory of L1-dissipative solvers for scalar conservation laws with discontinuous flux. Arch. Ration. Mech. Anal. 201 (2011) 27–86. [Google Scholar]
  6. E. Audusse and B. Perthame, Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies. Proc. R. Soc. Edinburgh Sect. A: Math. 135 (2005) 253–265. [CrossRef] [MathSciNet] [Google Scholar]
  7. J. Badwaik and A.M. Ruf, Convergence rates of monotone schemes for conservation laws with discontinuous flux. SIAM J. Numer. Anal. 58 (2020) 607–629. [CrossRef] [Google Scholar]
  8. P. Baiti and H.K. Jenssen, Well-posedness for a class of 2×2 L data. J. Differ. Equ. 140 (1997) 161–185. [CrossRef] [Google Scholar]
  9. 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 App. 4 (2003) 457–481. [CrossRef] [Google Scholar]
  10. R. Bürger, K.H. Karlsen and J.D. Towers, An Engquist–Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections. SIAM J. Numer. Anal. 47 (2009) 1684–1712. [CrossRef] [Google Scholar]
  11. Q.-Y. Chen, D. Gottlieb and J.S. Hesthaven, Uncertainty analysis for the steady-state flows in a dual throat nozzle. J. Comput. Phys. 204 (2005) 378–398. [CrossRef] [Google Scholar]
  12. G.M. Coclite and N.H. Risebro, Conservation laws with time dependent discontinuous coefficients. SIAM J. Math. Anal. 36 (2005) 1293–1309. [CrossRef] [Google Scholar]
  13. S. Cox, M. Hutzenthaler, A. Jentzen, J. van Neerven and T. Welti, Convergence in Hölder norms with applications to Monte Carlo methods in infinite dimensions. IMA J. Numer. Anal. 41 (2021) 493–548. [CrossRef] [Google Scholar]
  14. S. Diehl, A conservation law with point source and discontinuous flux function modelling continuous sedimentation. SIAM J. Appl. Math. 56 (1996) 388–419. [Google Scholar]
  15. S.S. Ghoshal, A. Jana and J.D. Towers, Convergence of a Godunov scheme to an Audusse-Perthame adapted entropy solution for conservation laws with BV spatial flux. Numer. Math. 146 (2020) 629–659. [CrossRef] [Google Scholar]
  16. S.S. Ghoshal, J.D. Towers and G. Vaidya, Convergence of a Godunov scheme for degenerate conservation laws with BV spatial flux and a study of Panov type fluxes. Preprint: arXiv:2011.10946 (2020). [Google Scholar]
  17. S.S. Ghoshal, J.D. Towers and G. Vaidya, Well-posedness for conservation laws with spatial heterogeneities and a study of BV regularity. Preprint: arXiv:2010.13695 (2020). [Google Scholar]
  18. M. Giles, Improved multilevel Monte Carlo convergence using the Milstein scheme. In: Monte Carlo and Quasi-Monte Carlo Methods 2006. Springer (2008) 343–358. [CrossRef] [Google Scholar]
  19. M.B. Giles, Multilevel Monte Carlo path simulation. Oper. Res. 56 (2008) 607–617. [CrossRef] [Google Scholar]
  20. T. Gimse, Conservation laws with discontinuous flux functions. SIAM J. Math. Anal. 24 (1993) 279–289. [CrossRef] [Google Scholar]
  21. T. Gimse and N.H. Risebro, Riemann problems with a discontinuous flux function. In: Vol. 1 of Proceedings of Third International Conference on Hyperbolic Problems (1991) 488–502. [Google Scholar]
  22. 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) 635–648. [CrossRef] [MathSciNet] [Google Scholar]
  23. T. Gimse and N.H. Risebro, A note on reservoir simulation for heterogeneous porous media. Transp. Porous Media 10 (1993) 257–270. [CrossRef] [Google Scholar]
  24. S. Heinrich, Multilevel Monte Carlo methods. In: International Conference on Large-Scale Scientific Computing. Springer (2001) 58–67. [Google Scholar]
  25. H. Holden and N.H. Risebro, Front Tracking for Hyperbolic Conservation Laws. Springer 152 (2015). [CrossRef] [Google Scholar]
  26. 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) 287–318. [CrossRef] [MathSciNet] [Google Scholar]
  27. K.H. Karlsen and J.D. Towers, Convergence of a Godunov scheme for conservation laws with a discontinuous flux lacking the crossing condition. J. Hyperbolic Differ. Equ. 14 (2017) 671–701. [CrossRef] [Google Scholar]
  28. 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) 623–664. [CrossRef] [MathSciNet] [Google Scholar]
  29. K.H. Karlsen, N.H. Risebro and J.D. Towers, L1 stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients. Preprint Series. Pure Mathematics http://urn.nb.no/URN:NBN:no-8076 (2003). [Google Scholar]
  30. C. Klingenberg and N.H. Risebro, Convex conservation laws with discontinuous coefficients. Existence, uniqueness and asymptotic behavior. Commun. Part. Differ. Equ. 20 (1995) 1959–1990. [CrossRef] [MathSciNet] [Google Scholar]
  31. C. Klingenberg and N.H. Risebro, Stability of a resonant system of conservation laws modeling polymer flow with gravitation. J. Differ. Equ. 170 (2001) 344–380. [CrossRef] [Google Scholar]
  32. U. Koley, N.H. Risebro, C. Schwab and F. Weber, A multilevel Monte Carlo finite difference method for random scalar degenerate convection–diffusion equations. J. Hyperbolic Differ. Equ. 14 (2017) 415–454. [CrossRef] [Google Scholar]
  33. S.N. Kružkov, First order quasilinear equations in several independent variables. Math. USSR-Sbornik 10 (1970) 217–243. [CrossRef] [Google Scholar]
  34. M. Ledoux and M. Talagrand, Probability in Banach Spaces: Isoperimetry and Processes. Springer Science & Business Media (2013). [Google Scholar]
  35. M.J. Lighthill and G.B. Whitham, On kinematic waves II. A theory of traffic flow on long crowded roads. Proc. R. Soc. London. Ser. A. Math. Phys. Sci. 229 (1955) 317–345. [Google Scholar]
  36. G. Lin, C. Su and G. Karniadakis, The stochastic piston problem. Proc. Nat. Acad. Sci. USA 101 (2004) 15840–15845. [CrossRef] [Google Scholar]
  37. 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) 559–577. [CrossRef] [Google Scholar]
  38. S. Mishra and C. Schwab, Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random initial data. Math. Comput. 81 (2012) 1979–2018. [CrossRef] [Google Scholar]
  39. S. Mishra, C. Schwab and J. Šukys, Multi-level Monte Carlo finite volume methods for uncertainty quantification in nonlinear systems of balance laws. In: Uncertainty Quantification in Computational Fluid Dynamics. Springer (2013) 225–294. [CrossRef] [Google Scholar]
  40. S. Mishra, D. Ochsner, A.M. Ruf and F. Weber, Bayesian inverse problems in the Wasserstein distance and application to conservation laws. in preparation (2021). [Google Scholar]
  41. S. Mishra, N.H. Risebro, C. Schwab and S. Tokareva, Numerical solution of scalar conservation laws with random flux functions. SIAM/ASA J. Uncertainty Quant. 4 (2016) 552–591. [CrossRef] [Google Scholar]
  42. B. Piccoli and M. Tournus, A general BV existence result for conservation laws with spatial heterogeneities. SIAM J. Math. Anal. 50 (2018) 2901–2927. [CrossRef] [Google Scholar]
  43. G. Poëtte, B. Després and D. Lucor, Uncertainty quantification for systems of conservation laws. J. Comput. Phys. 228 (2009) 2443–2467. [CrossRef] [Google Scholar]
  44. N.H. Risebro and A. Tveito, Front tracking applied to a nonstrictly hyperbolic system of conservation laws. SIAM J. Sci. Stat. Comput. 12 (1991) 1401–1419. [CrossRef] [Google Scholar]
  45. N.H. Risebro, C. Schwab and F. Weber, Correction to: Multilevel Monte Carlo front-tracking for random scalar conservation laws. BIT Numer. Math. 58 (2018) 247–255. [Google Scholar]
  46. A.M. Ruf, Flux-stability for conservation laws with discontinuous flux and convergence rates of the front tracking method. IMA J. Numer. Anal. 101 (2021) draa101. [Google Scholar]
  47. 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. 80 (2019) 1764–1776. [Google Scholar]
  48. W. Shen, On the uniqueness of vanishing viscosity solutions for riemann problems for polymer flooding. Nonlinear Differ. Equ. App. NoDEA 24 (2017) 37. [Google Scholar]
  49. J. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux. SIAM J. Numer. Anal. 38 (2000) 681–698. [Google Scholar]
  50. J.D. Towers, A difference scheme for conservation laws with a discontinuous flux: the nonconvex case. SIAM J. Numer. Anal. 39 (2001) 1197–1218. [Google Scholar]
  51. J.D. Towers, An existence result for conservation laws having BV spatial flux heterogeneities – without concavity. J. Differ. Equ. 269 (2020) 5754–5764. [Google Scholar]
  52. J. Tryoen, O. Le Maitre, M. Ndjinga and A. Ern, Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems. J. Comput. Phys. 229 (2010) 6485–6511. [CrossRef] [Google Scholar]
  53. J. Van Neerven, Stochastic evolution equations. ISEM Lecture Notes (2008). [Google Scholar]
  54. X. Wan and G.E. Karniadakis, Long-term behavior of polynomial chaos in stochastic flow simulations. Comput. Methods Appl. Mech. Eng. 195 (2006) 5582–5596. [CrossRef] [Google Scholar]
  55. X. Wen and S. Jin, Convergence of an immersed interface upwind scheme for linear advection equations with piecewise constant coefficients I: L1-error estimates. J. Comput. Math. 26 (2008) 1–22. [Google Scholar]

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