Communications in Mathematical Sciences

Volume 14 (2016)

Number 8

A relaxation model for liquid-vapor phase change with metastability

Pages: 2179 – 2214



François James (Mathématiques, Analyse, Probabilités, Modélisation, Orléans (MAPMO), Université d’Orléans, France)

Hélène Mathis (Laboratoire Jean Leray, Universit´e de Nantes, France)


We propose a model that describes phase transition including metastable states present in the van der Waals equation of state. From a convex optimization problem on the Helmholtz free energy of a mixture, we deduce a dynamical system that is able to depict the mass transfer between two phases, for which equilibrium states are either metastable states, stable states or a coexistent state. The dynamical system is then used as a relaxation source term in an isothermal $4 \times 4$ two-phase model. We use a finite volume scheme that treats the convective part and the source term in a fractional step way. Numerical results illustrate the ability of the model to capture phase transition and metastable states.


thermodynamics of phase transition, metastable states, nonlinear hyperbolic system with relaxation, van der Waals equation of state

2010 Mathematics Subject Classification

35L40, 35Q79, 37N10, 76T10, 80A10

Published 26 October 2016