Free Access
Issue
ESAIM: M2AN
Volume 41, Number 6, November-December 2007
Page(s) 1089 - 1123
DOI https://doi.org/10.1051/m2an:2007048
Published online 15 December 2007
  1. R. Abeyaratne and J. Knowles, Kinetic relations and the propagation of phase boundaries in solids. Arch. Ration. Mech. Anal. 114 (1991) 119–154. [CrossRef] [Google Scholar]
  2. R. Abgrall and S. Karni, Compressible multifluid flows. J. Comput. Phys. 169 (2001) 594–623. [CrossRef] [MathSciNet] [Google Scholar]
  3. R. Abgrall and R. Saurel, Discrete equations for physical and numerical compressible multiphase mixtures. J. Comput. Phys. 186 (2003) 361–396. [CrossRef] [MathSciNet] [Google Scholar]
  4. T.D. Aslam, A level set algorithm for tracking discontinuities in hyperbolic conservation laws II: Systems of equations. J. Sci. Comput. 19 (2003) 37–62. [CrossRef] [MathSciNet] [Google Scholar]
  5. N. Bedjaoui and P.G. LeFloch, Diffusive-dispersive travelling waves and kinetic relations. II. A hyperbolic-elliptic model of phase-transition dynamics. Proc. Roy. Soc. Edinburgh Sect. A 132A (2002) 1–21. [Google Scholar]
  6. S. Benzoni-Gavage, Stability of multi-dimensional phase transitions in a van der Waals fluid. Nonlinear Anal., Theory Methods Appl. 31 (1998) 243–263. [Google Scholar]
  7. C. Chalons, Transport-Equilibrium Schemes for Computing Nonclassical Shocks. I. Scalar Conservation Laws. Preprint, Laboratoire Jacques-Louis Lions (2005). [Google Scholar]
  8. C. Chalons and P.G. LeFloch, Computing undercompressive waves with the random choice scheme. Interfaces Free Bound. 5 (2003) 129–158. [CrossRef] [MathSciNet] [Google Scholar]
  9. R.M. Colombo and A. Corli, Continuous dependence in conservation laws with phase transitions. SIAM J. Math. Anal. 31 (1999) 34–62. [CrossRef] [MathSciNet] [Google Scholar]
  10. R.M. Colombo and A. Corli, Stability of the Riemann semigroup with respect to the kinetic condition. Quart. Appl. Math. 62 (2004) 541–551. [MathSciNet] [Google Scholar]
  11. C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics. Grundlehren der mathematischen Wisenschaften 325. Springer (2000). [Google Scholar]
  12. H. Fan and M. Slemrod, Dynamic flows with liquid/vapor phase transitions, in Handbook of mathematical fluid dynamics, Vol. I, North-Holland, Amsterdam (2002) 373–420. [Google Scholar]
  13. R.P. Fedkiw, T. Aslam, B. Merriman and S. Osher, A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. 152 (1999) 457–492. [CrossRef] [MathSciNet] [Google Scholar]
  14. R.P. Fedkiw, T. Aslam and S. Xu, The ghost fluid method for deflagration and detonation discontinuities. J. Comput. Phys. 154 (1999) 393–427. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
  15. E. Godlewski and P.-A. Raviart, Numerical approximation of hyperbolic systems of conservation laws. Appl. Math. Sci. 118 Springer (1996). [Google Scholar]
  16. E. Godlewski and N. Seguin, The Riemann problem for a simple model of phase transition. Commun. Math. Sci. 4 (2006) 227–247. [MathSciNet] [Google Scholar]
  17. B. Hayes and P.G. LeFloch, Nonclassical shocks and kinetic relations: strictly hyperbolic systems. SIAM J. Math. Anal. 31 (2000) 941–991. [CrossRef] [MathSciNet] [Google Scholar]
  18. T.Y. Hou, P. Rosakis and P.G. LeFloch, A level-set approach to the computation of twinning and phase-transition dynamics. J. Comput. Phys. 150 (1999) 302–331. [CrossRef] [MathSciNet] [Google Scholar]
  19. M. Kac, G.E. Uhlenbeck and P.C. Hemmer, On the van der Waals theory of the vapor-liquid equilibrium. I. Discussion of a one-dimensional model. J. Math. Phys. 4 (1963) 216–228. [NASA ADS] [CrossRef] [Google Scholar]
  20. S. Karni and R. Abgrall, Ghost-Fluids for the Poor: A Single Fluid Algorithm for Multifluids, in Lecture Notes in Mathematics, Proceedings of the 10th International Conference on Hyperbolic problems, theory and numerics, Springer (2001) 293–302. [Google Scholar]
  21. O. Le Métayer, J. Massoni and R. Saurel, Modelling evaporation fronts with reactive Riemann solvers. J. Comput. Phys. 205 (2005) 567–610. [CrossRef] [MathSciNet] [Google Scholar]
  22. P.G. LeFloch, Propagating phase boundaries: Formulation of the problem and existence via the Glimm method. Arch. Ration. Mech. Anal. 123 (1993) 153–197. [CrossRef] [Google Scholar]
  23. P.G. LeFloch, Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves. Lectures in Mathematics. ETH Zürich, Birkhäuser (2002). [Google Scholar]
  24. P.G. LeFloch and M.D. Thanh, Nonclassical Riemann solvers and kinetic relations. II. An hyperbolic-elliptic model of phase transitions. Proc. Royal Soc. Edinburgh A 131A (2001) 1–39. [Google Scholar]
  25. 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]
  26. T.G. Liu, B.C. Khoo and K.S. Yeo, Ghost fluid method for strong impacting on material interfaces. J. Comput. Phys. 190 (2003) 651–681. [CrossRef] [Google Scholar]
  27. C. Merkle, Dynamical Phase Transitions in Compressible Media. Doctoral dissertation, Albert-Ludwigs-Universität Freiburg (2006) http://www.freidok.uni-freiburg.de/volltexte/2674/. [Google Scholar]
  28. C. Merkle and C. Rohde, Computation of dynamical phase transitions in solids. Appl. Numer. Math. 56 (2006) 1450–1463. [CrossRef] [MathSciNet] [Google Scholar]
  29. W. Mulder, S. Osher and J. Sethian, Computing interface motion in compressible gas dynamics. J. Comput. Phys. 100 (1992) 209–228. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
  30. S. Müller and A. Voß, The Riemann problem for the Euler equations with nonconvex and nonsmooth equation of state: construction of wave curves. SIAM J. Sci. Comput. 28 (1992) 651–681. [Google Scholar]
  31. S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces. Appl. Math. Sci. 153. Springer (2003). [Google Scholar]
  32. S. Osher and J. Sethian, Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79 (1988) 12–49. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
  33. S. Osher and C.-W. Shu, High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal. 28 (1991) 907–922. [CrossRef] [MathSciNet] [Google Scholar]
  34. D. Peng, B. Merriman, S. Osher, H.K. Zhao and M. Kang, A PDE-based fast local level set method. J. Comput. Phys. 155 (1999) 410–438. [CrossRef] [MathSciNet] [Google Scholar]
  35. G. Russo and P. Smereka, A remark on computing distance functions. J. Comput. Phys. 163 (2000) 51–67. [CrossRef] [MathSciNet] [Google Scholar]
  36. D. Serre, Systems of Conservation Laws 1. Cambridge University Press (1999). [Google Scholar]
  37. M. Sussman, P. Smereka and S. Osher, A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114 (1994) 146–154. [NASA ADS] [CrossRef] [Google Scholar]
  38. E.F. Toro, Multi-Stage Predictor-Corrector Fluxes for Hyperbolic Equations. Technical Report NI03037-NPA Isaac Newton Institute for Mathematical Sciences (2003). [Google Scholar]
  39. L. Truskinovsky, Kinks versus Shocks, in Shock induced transitions and phase structures in general media, Springer, New York (1993) 185–229. [Google Scholar]
  40. L. Truskinovsky and A. Vainchtein, Explicit kinetic relation from “first principles”, in Mechanics of material forces 11, Advances in Mechanics and Mathematics, P. Steinmann and G.A. Maugin (Eds.), Springer (2005) 43–50. [Google Scholar]
  41. X. Zhong, T.Y. Hou and P.G. LeFloch, Computational method for propagating phase boundaries. J. Comput. Phys. 124 (1996) 192–216. [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