Free Access
Issue
ESAIM: M2AN
Volume 46, Number 1, January-February 2012
Page(s) 187 - 206
DOI https://doi.org/10.1051/m2an/2011044
Published online 03 October 2011
  1. R. Abgrall and S. Karni, Two-layer shallow water system: a relaxation approach. SIAM. J. Sci. Comput. 31 (2009) 1603–1627. [CrossRef] [MathSciNet] [Google Scholar]
  2. R. Abgrall and S. Karni, A comment on the computation of non-conservative products. J. Comput. Phys. 229 (2010) 2759–2763. [CrossRef] [MathSciNet] [Google Scholar]
  3. E. Audusse, F. Bouchut, M.O. Bristeau, R. Klien and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM. J. Sci. Comput. 25 (2004) 2050–2065. [CrossRef] [MathSciNet] [Google Scholar]
  4. E. Audusse and M.O. Bristeau, Finite volume solvers for multi-layer Saint-Venant system. Int. J. Appl. Math. Comput. Sci. 17 (2007) 311–319. [CrossRef] [MathSciNet] [Google Scholar]
  5. M.J. Castro, P. LeFloch, M.L. Munoz Ruiz and C. Pares, Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes. J. Comput. Phys. 227 (2008) 8107–8129. [CrossRef] [MathSciNet] [Google Scholar]
  6. G.-Q. Chen, C. Christoforou and Y. Zhang, Continuous dependence of entropy solutions to the euler equations on the adiabatic exponent and mach number. Arch. Ration. Mech. Anal. 189 (2008) 97–130. [CrossRef] [MathSciNet] [Google Scholar]
  7. G. Dal Maso, P. LeFloch and F. Murat, Definition and weak stability of nonconservative products. J. Math. Pures. Appl. 74 (1995) 483–548. [MathSciNet] [Google Scholar]
  8. U.S. Fjordholm, S. Mishra and E. Tadmor, Energy preserving and energy stable schemes for the shallow water equations,Foundations of Computational Mathematics, Proc. FoCM held in Hong Kong 2008, London Math. Soc. Lecture Notes Ser. 363, edited by F. Cucker, A. Pinkus and M. Todd (2009) 93–139. [Google Scholar]
  9. U.S. Fjordholm, S. Mishra and E. Tadmor, Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography. J. Comput. Phys. 230 (2011) 5587–5609. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
  10. E. Godlewski and P.-A. Raviart, Hyperbolic systems of conservation laws. Ellipses (1991). [Google Scholar]
  11. S. Gottlieb, C.-W. Shu and E. Tadmor, High order time discretization methods with the strong stability property, SIAM Rev. 43 (2001) 89–112. [Google Scholar]
  12. S. Karni, Viscous shock profiles and primitive formulations. SIAM J. Numer. Anal. 29 (1992) 1592–1609. [CrossRef] [MathSciNet] [Google Scholar]
  13. P.G. LeFloch, Entropy weak solutions to nonlinear hyperbolic systems in nonconservative form. Comm. Partial Differential Equations 13 (1988) 669–727. [CrossRef] [MathSciNet] [Google Scholar]
  14. T.Y. Hou and P.G. LeFloch, Why nonconservative schemes converge to wrong solutions. Error analysis. Math. Comput. 62 (1994) 497–530. [CrossRef] [MathSciNet] [Google Scholar]
  15. P.G. LeFloch and M.D. Thanh, The Riemann problem for fluid flows in a nozzle with discontinuous cross-section. Commun. Math. Sci. 1 (2003) 763–797. [CrossRef] [MathSciNet] [Google Scholar]
  16. P.G. LeFloch, J.M. Mercier and C. Rohde, Fully discrete entropy conservative schemes of arbitrary order. SIAM J. Numer. Anal. 40 (2002) 1968–1992. [CrossRef] [MathSciNet] [Google Scholar]
  17. R.J. LeVeque, Finite volume methods for hyperbolic problems.Cambridge university press, Cambridge (2002). [Google Scholar]
  18. T.P. Liu, Shock waves for compressible Navier–Stokes equations are stable. Comm. Pure Appl. Math. 39 (1986) 565–594. [CrossRef] [MathSciNet] [Google Scholar]
  19. M.L. Munoz Ruiz and C. Pares, Godunov method for non-conservative hyperbolic systems. Math. Model. Num. Anal. 41 (2007) 169–185. [CrossRef] [EDP Sciences] [Google Scholar]
  20. C. Pares and M.J. Castro, On the well-balance property of Roe’s method for nonconservative hyperbolic systems. Applications to shallow water equations. Math. Model. Num. Anal. 38 (2004) 821–852. [CrossRef] [EDP Sciences] [Google Scholar]
  21. C. Pares, Numerical methods for non-conservative hyperbolic systems: a theoretical framework. SIAM. J. Num. Anal. 44 (2006) 300–321. [CrossRef] [MathSciNet] [Google Scholar]
  22. E. Tadmor, The numerical viscosity of entropy stable schemes for systems of conservation laws, I. Math. Comp. 49 (1987) 91–103. [CrossRef] [MathSciNet] [Google Scholar]
  23. E. Tadmor, Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer. 12 (2003) 451–512. [CrossRef] [MathSciNet] [Google Scholar]
  24. E. Tadmor and W. Zhong, Entropy stable approximations of Navier–Stokes equations with no artificial numerical viscosity. J. Hyperbolic Differ. Equ. 3 (2006) 529–559. [CrossRef] [MathSciNet] [Google Scholar]
  25. E. Tadmor and W. Zhong, Energy preserving and stable approximations for the two-dimensional shallow water equations,in Mathematics and computation: A contemporary view, Proc. of the third Abel symposium. Ålesund, Norway, Springer (2008) 67–94. [Google Scholar]
  26. E. Romenski, D. Drikakis and E. Toro, Conservative models and numerical methods for compressible two-phase flow. J. Sci. Comput. 42 (2010) 68–95. [CrossRef] [MathSciNet] [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.

Recommended for you