Energy consistent DG methods for the Navier-Stokes-Korteweg system

TL;DR

This paper develops energy consistent discontinuous Galerkin schemes for the Euler-Korteweg and Navier-Stokes-Korteweg systems, ensuring conservation and dissipation properties that align with the continuous models.

Contribution

It introduces novel DG schemes that are energy and mass conservative or dissipative, matching the physical properties of the PDE systems.

Findings

Euler-Korteweg scheme is energy and mass conservative.

Navier-Stokes-Korteweg scheme is mass conservative and energy dissipative.

Methods are consistent with the energy behavior of the continuous systems.

Abstract

We design consistent discontinuous Galerkin finite element schemes for the approximation of the Euler-Korteweg and the Navier-Stokes-Korteweg systems. We show that the scheme for the Euler-Korteweg system is energy and mass conservative and that the scheme for the Navier-Stokes-Korteweg system is mass conservative and monotonically energy dissipative. In this case the dissipation is isolated to viscous effects, that is, there is no numerical dissipation. In this sense the methods is consistent with the energy dissipation of the continuous PDE systems.