Free access
Issue
ESAIM: M2AN
Volume 38, Number 5, September-October 2004
Page(s) 821 - 852
DOI http://dx.doi.org/10.1051/m2an:2004041
Published online 15 October 2004
  1. N. Andronov and G. Warnecke, On the solution to the Riemann problem for the compressible duct flow. SIAM J. Appl. Math. 64 (2004) 878–901. [CrossRef] [MathSciNet]
  2. F. Bouchut, An introduction to finite volume methods for hyperbolic systems of conservation laws with source, in Free surface geophysical flows. Tutorial Notes. INRIA, Rocquencourt (2002).
  3. A. Bermúdez and M.E. Vázquez, Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23 (1994) 1049–1071. [CrossRef] [MathSciNet]
  4. M.J. Castro, J. Macías and C. Parés, A Q-Scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system. ESAIM: M2AN 35 (2001) 107–127. [CrossRef] [EDP Sciences]
  5. M.J. Castro, J.A. García-Rodríguez, J.M. González-Vida, J. Macías, C. Parés and M.E. Vázquez-Cendón, Numerical simulation of two-layer Shallow Water flows through channels with irregular geometry. J. Comp. Phys. 195 (2004) 202–235. [CrossRef] [MathSciNet]
  6. T. Chacón, A. Domínguez and E.D. Fernández, A family of stable numerical solvers for Shallow Water equations with source terms. Comp. Meth. Appl. Mech. Eng. 192 (2003) 203–225. [CrossRef] [MathSciNet]
  7. T. Chacón, A. Domínguez and E.D. Fernández, An entropy-correction free solver for non-homogeneous shallow water equations. ESAIM: M2AN 37 (2003) 755–772. [CrossRef] [EDP Sciences]
  8. T. Chacón, E.D. Fernández and M. Gómez Mármol, A flux-splitting solver for shallow water equations with source terms. Int. Jour. Num. Meth. Fluids 42 (2003) 23–55. [CrossRef]
  9. T. Chacón, A. Domínguez and E.D. Fernández, Asymptotically balanced schemes for non-homogeneous hyperbolic systems – application to the Shallow Water equations. C.R. Acad. Sci. Paris, Ser. I 338 (2004) 85–90.
  10. J.F. Colombeau, A.Y. Le Roux, A. Noussair and B. Perrot, Microscopic profiles of shock waves and ambiguities in multiplications of distributions. SIAM J. Num. Anal. 26 (1989) 871–883. [CrossRef] [MathSciNet]
  11. G. Dal Masso, P.G. LeFloch and F. Murat, Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74 (1995) 483–548. [MathSciNet]
  12. E.D. Fernández Nieto, Aproximación Numérica de Leyes de Conservación Hiperbólicas No Homogéneas. Aplicación a las Ecuaciones de Aguas Someras. Ph.D. Thesis, Universidad de Sevilla (2003).
  13. A.C. Fowler, Mathematical Model in the Applied Sciences. Cambridge (1997).
  14. P. García-Navarro and M.E. Vázquez-Cendón, On numerical treatment of the source terms in the shallow water equations. Comput. Fluids 29 (2000) 17–45.
  15. P. Goatin and P.G. LeFloch, The Riemann problem for a class of resonant hyperbolic systems of balance laws, preprint (2003).
  16. E. Godlewski and P.A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer-Verlag, New York (1996).
  17. L. Gosse, A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms. Comp. Math. Appl. 39 (2000) 135–159. [CrossRef] [MathSciNet]
  18. L. Gosse, A well-balanced scheme using non-conservative products designed for hyperbolic system of conservation laws with source terms. Mat. Mod. Meth. Appl. Sc. 11 (2001) 339–365. [CrossRef] [MathSciNet]
  19. J.M. Greenberg and A.Y. LeRoux, A well balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 1–16. [CrossRef] [MathSciNet]
  20. J.M. Greenberg, A.Y. LeRoux, R. Baraille and A. Noussair, Analysis and approximation of conservation laws with source terms. SIAM J. Numer. Anal. 34 (1997) 1980–2007. [CrossRef] [MathSciNet]
  21. A. Harten and J.M. Hyman, Self-adjusting grid methods for one-dimensional hyperbolic conservation laws. J. Comp. Phys. 50 (1983) 235–269. [CrossRef] [MathSciNet]
  22. P.G. LeFloch, Propagating phase boundaries; formulation of the problem and existence via Glimm scheme. Arch. Rat. Mech. Anal. 123 (1993) 153–197. [CrossRef]
  23. R. LeVeque, Numerical Methods for Conservation Laws. Birkhäuser (1990).
  24. R. LeVeque, Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comp. Phys. 146 (1998) 346–365. [NASA ADS] [CrossRef] [MathSciNet]
  25. R. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press (2002).
  26. B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38 (2001) 201–231. [CrossRef] [MathSciNet]
  27. B. Perthame and C. Simeoni, Convergence of the upwind interface source method for hyperbolic conservation laws, in Proc. of Hyp 2002, Thou and Tadmor Eds., Springer (2003).
  28. P.A. Raviart and L. Sainsaulieu, A nonconservative hyperbolic system modeling spray dynamics. I. Solution of the Riemann problem. Math. Mod. Meth. Appl. Sci. 5 (1995) 297–333. [CrossRef] [MathSciNet]
  29. P.L. Roe, Approximate Riemann solvers, parameter vectors and difference schemes. J. Comp. Phys. 43 (1981) 357–371. [NASA ADS] [CrossRef] [MathSciNet]
  30. P.L. Roe, Upwinding difference schemes for hyperbolic conservation laws with source terms, in Proc. of the Conference on Hyperbolic Problems, Carasso, Raviart and Serre Eds., Springer (1986) 41–51.
  31. J.J. Stoker, Water Waves. Interscience, New York (1957).
  32. E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction. Springer-Verlag (1997).
  33. E.F. Toro, Shock-Capturing Methods for Free-Surface Shallow Flows. Wiley (2001).
  34. E.F. Toro and M.E. Vázquez-Cendón, Model hyperbolic systems with source terms: exact and numerical solutions, in Proc. of Godunov methods: Theory and Applications (2000).
  35. I. Toumi, A weak formulation of Roe's approximate Riemann Solver. J. Comp. Phys. 102 (1992) 360–373. [CrossRef] [MathSciNet]
  36. M.E. Vázquez-Cendón, Estudio de Esquemas Descentrados para su Aplicación a las Leyes de Conservación Hiperbólicas con Términos Fuente. Ph.D. Thesis, Universidad de Santiago de Compostela (1994).
  37. M.E. Vázquez-Cendón, Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. J. Comp. Phys. 148 (1999) 497–526. [CrossRef] [MathSciNet]
  38. A.I. Volpert, The space BV and quasilinear equations. Math. USSR Sbornik 73 (1967) 225–267. [CrossRef]

Recommended for you