The sharp-interface approach for fluids with phase change: Riemann problems and ghost fluid techniques

Mathematical Modelling and Numerical Analysis

Volumn 41, Issue 6, 2007, Pages 1089-1123

  Merkle, Christian ^a   Rohde, Christian ^b  

^a UNIVERSITY OF FREIBURG   (Germany)

^b UNIVERSITY OF STUTTGART   (Germany)

Author keywords

Dynamical phase transitions in compressible media; Ghost fluid approach; Kinetic relations; Riemann solver; Van der Waals pressure

Indexed keywords

EID: 37449014907     PISSN: 0764583X     EISSN: 12903841     Source Type: Journal    
DOI: 10.1051/m2an:2007048     Document Type: Article

References (41)

1. R. Abeyaratne and J. Knowles, Kinetic relations and the propagation of phase boundaries in solids. Arch. Ration. Mech. Anal. 114 (1991) 119-154.
2. R. Abgrall and S. Karni, Compressible multifluid flows. J. Comput. Phys. 169 (2001) 594-623.
3. R. Abgrall and R. Saurel, Discrete equations for physical and numerical compressible multiphase mixtures. J. Comput. Phys. 186 (2003) 361-396.
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.
5. N. Bedjaoui and P.G. LeFloch, Diffusive-dispersive travelling waves and kinetic relations. IL A hyperbolic-elliptic model of phase-transition dynamics. Proc. Roy. Soc. Edinburgh Sect. A 132A (2002) 1-21.
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.
7. C. Chalons, Transport-Equilibrium Schemes for Computing Nonclassical Shocks. I. Scalar Conservation Laws. Preprint, Laboratoire Jacques-Louis Lions (2005).
8. C. Chalons and P.G. LeFloch, Computing undercompressive waves with the random choice scheme. Interfaces Free Bound. 5 (2003) 129-158.
9. R.M. Colombo and A. Corli, Continuous dependence in conservation laws with phase transitions. SIAM J. Math. Anal. 31 (1999) 34-62.
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.
11. C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics. Grundlehren der mathematischen Wisenschaften 325. Springer (2000).
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.
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.
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.
15. E. Godlewski and P.-A. Raviart, Numerical approximation of hyperbolic systems of conservation laws. Appl. Math. Sci. 118 Springer (1996).
16. E. Godlewski and N. Seguin, The Riemann problem for a simple model of phase transition. Commun. Math. Sci. 4 (2006) 227-247.
17. B. Hayes and P.G. LeFloch, Nonclassical shocks and kinetic relations: strictly hyperbolic systems. SIAM J. Math. Anal. 31 (2000) 941-991.
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.
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.
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.
21. O. Le Métayer, J. Massoni and R. Saurel, Modelling evaporation fronts with reactive Riemann solvers. J. Comput. Phys. 205 (2005) 567-610.
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.
23. P.G. LeFloch, Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves. Lectures in Mathematics. ETH Zürich, Birkhäuser (2002).
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.
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.
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.
27. C. Merkle, Dynamical Phase Transitions in Compressible Media. Doctoral dissertation, Albert-Ludwigs-Universität Freiburg (2006) http ://www.freidok.uni-freiburg.de/volltexte/2674/.
28. C. Merkle and C. Rohde, Computation of dynamical phase transitions in solids. Appl. Numer. Math. 56 (2006) 1450-1463.
29. W. Mulder, S. Osher and J. Sethian, Computing interface motion in compressible gas dynamics. J. Comput. Phys. 100 (1992) 209-228.
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.
31. S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces. Appl. Math. Sci. 153. Springer (2003).
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.
33. S. Osher and C.-W. Shu, High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal. 28 (1991) 907-922.
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.
35. G. Russo and P. Smereka, A remark on computing distance functions. J. Comput. Phys. 163 (2000) 51-67.
36. D. Serre, Systems of Conservation Laws 1. Cambridge University Press (1999).
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.
38. E.F. Toro, Multi-Stage Predictor-Corrector Fluxes for Hyperbolic Equations. Technical Report NI03037-NPA Isaac Newton Institute for Mathematical Sciences (2003).
39. L. Truskinovsky, Kinks versus Shocks, in Shock induced transitions and phase structures in general media, Springer, New York (1993) 185-229.
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.
41. X. Zhong, T.Y. Hou and P.G. LeFloch, Computational method for propagating phase boundaries. J. Comput. Phys. 124 (1996) 192-216.