Energy consistent discontinuous Galerkin methods for a quasi-incompressible diffuse two phase flow model

TL;DR

This paper introduces energy consistent discontinuous Galerkin schemes for a complex two-phase flow model, ensuring mass conservation and energy dissipation without artificial numerical effects, thus accurately reflecting the continuous system's physics.

Contribution

The paper develops novel discontinuous Galerkin methods that are energy consistent and mass conservative for a quasi-incompressible two-phase flow model with phase transitions.

Findings

Scheme is mass conservative.

Scheme is monotonically energy dissipative.

No artificial numerical dissipation is introduced.

Abstract

We design consistent discontinuous Galerkin finite element schemes for the approximation of a quasi-incompressible two phase flow model of Allen-Cahn/Cahn-Hilliard/Navier-Stokes-Korteweg type which allows for phase transitions. We show that the scheme is mass conservative and monotonically energy dissipative. In this case the dissipation is isolated to discrete equivalents of those effects already causing dissipation on the continuous level, that is, there is no artificial numerical dissipation added into the scheme. In this sense the methods are consistent with the energy dissipation of the continuous PDE system.