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 |
- R. Aae Klausen and N.H. Risebro, Stability of conservation laws with discontinuous coefficients. J. Differ. Equ. 157 (1999) 41–60. [Google Scholar]
- R. Abgrall, A simple, flexible and generic deterministic approach to uncertainty quantifications in non linear problems: application to fluid flow problems (2008). [Google Scholar]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- M.B. Giles, Multilevel Monte Carlo path simulation. Oper. Res. 56 (2008) 607–617. [CrossRef] [Google Scholar]
- T. Gimse, Conservation laws with discontinuous flux functions. SIAM J. Math. Anal. 24 (1993) 279–289. [CrossRef] [Google Scholar]
- 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]
- 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]
- 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]
- S. Heinrich, Multilevel Monte Carlo methods. In: International Conference on Large-Scale Scientific Computing. Springer (2001) 58–67. [Google Scholar]
- H. Holden and N.H. Risebro, Front Tracking for Hyperbolic Conservation Laws. Springer 152 (2015). [CrossRef] [Google Scholar]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- S.N. Kružkov, First order quasilinear equations in several independent variables. Math. USSR-Sbornik 10 (1970) 217–243. [CrossRef] [Google Scholar]
- M. Ledoux and M. Talagrand, Probability in Banach Spaces: Isoperimetry and Processes. Springer Science & Business Media (2013). [Google Scholar]
- 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]
- G. Lin, C. Su and G. Karniadakis, The stochastic piston problem. Proc. Nat. Acad. Sci. USA 101 (2004) 15840–15845. [CrossRef] [Google Scholar]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- G. Poëtte, B. Després and D. Lucor, Uncertainty quantification for systems of conservation laws. J. Comput. Phys. 228 (2009) 2443–2467. [CrossRef] [Google Scholar]
- 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]
- 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]
- 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]
- 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]
- 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]
- J. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux. SIAM J. Numer. Anal. 38 (2000) 681–698. [Google Scholar]
- 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]
- 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]
- 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]
- J. Van Neerven, Stochastic evolution equations. ISEM Lecture Notes (2008). [Google Scholar]
- 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]
- 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]
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.