An entropy preserving finite-element/finite-volume pressure correction scheme for the drift-flux model

TL;DR

This paper introduces a stable, entropy-preserving pressure correction scheme for the drift-flux model that combines finite element and finite volume methods, ensuring physical bounds and dissipation properties.

Contribution

The novel scheme integrates finite element and finite volume discretizations with a pressure correction step, guaranteeing stability, physical bounds, and entropy decrease in the drift-flux model.

Findings

The scheme is conservative and maintains physical bounds.

It exhibits near-first-order convergence in numerical tests.

The algorithm ensures stability and dissipation properties.

Abstract

We present in this paper a pressure correction scheme for the drift-flux model combining finite element and finite volume discretizations, which is shown to enjoy essential stability features of the continuous problem: the scheme is conservative, the unknowns are kept within their physical bounds and, in the homogeneous case (i.e. when the drift velocity vanishes), the discrete entropy of the system decreases; in addition, when using for the drift velocity a closure law which takes the form of a Darcy-like relation, the drift term becomes dissipative. Finally, the present algorithm preserves a constant pressure and a constant velocity through moving interfaces between phases. To ensure the stability as well as to obtain this latter property, a key ingredient is to couple the mass balance and the transport equation for the dispersed phase in an original pressure correction step. The existence of a solution to each step of the algorithm is proven; in particular, the existence of a solution to the pressure correction step is derived as a consequence of a more general existence result for discrete problems associated to the drift-flux model. Numerical tests show a near-first-order convergence rate for the scheme, both in time and space, and confirm its stability.