A Positive and Entropy-Satisfying Finite Volume Scheme for the Baer-Nunziato Model

TL;DR

This paper introduces a new finite volume relaxation scheme for the Baer-Nunziato two-phase flow model that guarantees positivity, entropy dissipation, and stability, with improved accuracy and efficiency over existing methods.

Contribution

The paper presents the first relaxation scheme for the Baer-Nunziato model that ensures positivity, entropy dissipation, and stability without restrictive conditions, and demonstrates superior performance compared to existing schemes.

Findings

The scheme maintains positivity of phase fractions, densities, and energies.

It preserves real sound speeds for ideal and stiffened gases.

Numerical results show higher accuracy and lower computational cost than standard methods.

Abstract

We present a relaxation scheme for approximating the entropy dissipating weak solutions of the Baer-Nunziato two-phase flow model. This relaxation scheme is straightforwardly obtained as an extension of the relaxation scheme designed in [16] for the isentropic Baer-Nunziato model and consequently inherits its main properties. To our knowledge, this is the only existing scheme for which the approximated phase fractions, phase densities and phase internal energies are proven to remain positive without any restrictive condition other than a classical fully computable CFL condition. For ideal gas and stiffened gas equations of state, real values of the phasic speeds of sound are also proven to be maintained by the numerical scheme. It is also the only scheme for which a discrete entropy inequality is proven, under a CFL condition derived from the natural sub-characteristic condition associated with the relaxation approximation. This last property, which ensures the non-linear stability of the numerical method, is satisfied for any admissible equation of state. We provide a numerical study for the convergence of the approximate solutions towards some exact Riemann solutions. The numerical simulations show that the relaxation scheme compares well with two of the most popular existing schemes available for the Baer-Nunziato model, namely Schwendeman-Wahle-Kapila's Godunov-type scheme [39] and Toro-Tokareva's HLLC scheme [42]. The relaxation scheme also shows a higher precision and a lower computational cost (for comparable accuracy) than a standard numerical scheme used in the nuclear industry, namely Rusanov's scheme. Finally, we assess the good behavior of the scheme when approximating vanishing phase solutions.