Systems of mixed hyperbolic-elliptic conservation laws can serve as models for the evolution of a liquid-vapor fluid with possible sharp dynamical phase changes. We focus on the equations of ideal hydrodynamics in the isothermal case and introduce a thermodynamically consistent solution of the Riemann problem in one space dimension. This result is the basis for an algorithm of ghost fluid type to solve the sharp-interface model numerically. In particular the approach allows to resolve phase transitions sharply, i.e., without artificial smearing in the physically irrelevant elliptic region. Numerical experiments demonstrate the reliability of the method.
Mots clés : dynamical phase transitions in compressible media, van-der-Waals pressure, kinetic relations, Riemann solver, ghost fluid approach
@article{M2AN_2007__41_6_1089_0,
author = {Merkle, Christian and Rohde, Christian},
title = {The sharp-interface approach for fluids with phase change : {Riemann} problems and ghost fluid techniques},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {1089--1123},
publisher = {EDP-Sciences},
volume = {41},
number = {6},
year = {2007},
doi = {10.1051/m2an:2007048},
mrnumber = {2377108},
zbl = {1134.35074},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2007048/}
}
TY - JOUR AU - Merkle, Christian AU - Rohde, Christian TI - The sharp-interface approach for fluids with phase change : Riemann problems and ghost fluid techniques JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2007 SP - 1089 EP - 1123 VL - 41 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2007048/ DO - 10.1051/m2an:2007048 LA - en ID - M2AN_2007__41_6_1089_0 ER -
%0 Journal Article %A Merkle, Christian %A Rohde, Christian %T The sharp-interface approach for fluids with phase change : Riemann problems and ghost fluid techniques %J ESAIM: Modélisation mathématique et analyse numérique %D 2007 %P 1089-1123 %V 41 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2007048/ %R 10.1051/m2an:2007048 %G en %F M2AN_2007__41_6_1089_0
Merkle, Christian; Rohde, Christian. The sharp-interface approach for fluids with phase change : Riemann problems and ghost fluid techniques. ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 6, pp. 1089-1123. doi : 10.1051/m2an:2007048. http://www.numdam.org/articles/10.1051/m2an:2007048/
[1] and, Kinetic relations and the propagation of phase boundaries in solids. Arch. Ration. Mech. Anal. 114 (1991) 119-154. | Zbl
[2] and, Compressible multifluid flows. J. Comput. Phys. 169 (2001) 594-623. | Zbl
[3] and, Discrete equations for physical and numerical compressible multiphase mixtures. J. Comput. Phys. 186 (2003) 361-396. | Zbl
[4] , A level set algorithm for tracking discontinuities in hyperbolic conservation laws II: Systems of equations. J. Sci. Comput. 19 (2003) 37-62. | Zbl
[5] and, 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. | Zbl
[6] , Stability of multi-dimensional phase transitions in a van der Waals fluid. Nonlinear Anal., Theory Methods Appl. 31 (1998) 243-263. | Zbl
[7] , Transport-Equilibrium Schemes for Computing Nonclassical Shocks. I. Scalar Conservation Laws. Preprint, Laboratoire Jacques-Louis Lions (2005). | MR
[8] and, Computing undercompressive waves with the random choice scheme. Interfaces Free Bound. 5 (2003) 129-158. | Zbl
[9] and, Continuous dependence in conservation laws with phase transitions. SIAM J. Math. Anal. 31 (1999) 34-62. | Zbl
[10] and, Stability of the Riemann semigroup with respect to the kinetic condition. Quart. Appl. Math. 62 (2004) 541-551. | Zbl
[11] , Hyperbolic Conservation Laws in Continuum Physics. Grundlehren der mathematischen Wisenschaften 325. Springer (2000). | MR | Zbl
[12] and, Dynamic flows with liquid/vapor phase transitions, in Handbook of mathematical fluid dynamics, Vol. I, North-Holland, Amsterdam (2002) 373-420. | Zbl
[13] ,, and, A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. 152 (1999) 457-492. | Zbl
[14] , and, The ghost fluid method for deflagration and detonation discontinuities. J. Comput. Phys. 154 (1999) 393-427. | Zbl
[15] and, Numerical approximation of hyperbolic systems of conservation laws. Appl. Math. Sci. 118 Springer (1996). | MR | Zbl
[16] and, The Riemann problem for a simple model of phase transition. Commun. Math. Sci. 4 (2006) 227-247. | Zbl
[17] and, Nonclassical shocks and kinetic relations: strictly hyperbolic systems. SIAM J. Math. Anal. 31 (2000) 941-991. | Zbl
[18] , and, A level-set approach to the computation of twinning and phase-transition dynamics. J. Comput. Phys. 150 (1999) 302-331. | Zbl
[19] , and, 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. | Zbl
[20] and, 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] , and, Modelling evaporation fronts with reactive Riemann solvers. J. Comput. Phys. 205 (2005) 567-610. | Zbl
[22] , Propagating phase boundaries: Formulation of the problem and existence via the Glimm method. Arch. Ration. Mech. Anal. 123 (1993) 153-197. | Zbl
[23] , Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves. Lectures in Mathematics. ETH Zürich, Birkhäuser (2002). | MR | Zbl
[24] and, Nonclassical Riemann solvers and kinetic relations. II. An hyperbolic-elliptic model of phase transitions. Proc. Royal Soc. Edinburgh A 131A (2001) 1-39. | Zbl
[25] , and, Fully discrete, entropy conservative schemes of arbitrary order. SIAM J. Numer. Anal. 40 (2002) 1968-1992. | Zbl
[26] , and, Ghost fluid method for strong impacting on material interfaces. J. Comput. Phys. 190 (2003) 651-681. | Zbl
[27] , Dynamical Phase Transitions in Compressible Media. Doctoral dissertation, Albert-Ludwigs-Universität Freiburg (2006) http://www.freidok.uni-freiburg.de/volltexte/2674/. | Zbl
[28] and, Computation of dynamical phase transitions in solids. Appl. Numer. Math. 56 (2006) 1450-1463. | Zbl
[29] , and, Computing interface motion in compressible gas dynamics. J. Comput. Phys. 100 (1992) 209-228. | Zbl
[30] and, 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. | Zbl
[31] and, Level Set Methods and Dynamic Implicit Surfaces. Appl. Math. Sci. 153. Springer (2003). | MR | Zbl
[32] and, Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79 (1988) 12-49. | Zbl
[33] and, High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal. 28 (1991) 907-922. | Zbl
[34] ,,, and, A PDE-based fast local level set method. J. Comput. Phys. 155 (1999) 410-438. | Zbl
[35] and, A remark on computing distance functions. J. Comput. Phys. 163 (2000) 51-67. | Zbl
[36] , Systems of Conservation Laws 1. Cambridge University Press (1999). | MR | Zbl
[37] , and, A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114 (1994) 146-154. | Zbl
[38] , Multi-Stage Predictor-Corrector Fluxes for Hyperbolic Equations. Technical Report NI03037-NPA Isaac Newton Institute for Mathematical Sciences (2003).
[39] , Kinks versus Shocks, in Shock induced transitions and phase structures in general media, Springer, New York (1993) 185-229. | Zbl
[40] and, 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] , and, Computational method for propagating phase boundaries. J. Comput. Phys. 124 (1996) 192-216. | Zbl
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