A Structure Preserving Finite Volume Scheme for the Navier-Stokes-Korteweg Equations

TL;DR

This paper introduces a finite volume scheme for Navier-Stokes-Korteweg equations that preserves physical properties like mass, momentum, and energy, and demonstrates convergence in numerical experiments.

Contribution

It develops a direct, structure-preserving finite volume scheme for Navier-Stokes-Korteweg systems on Cartesian meshes, avoiding auxiliary variables.

Findings

Conserves mass and momentum

Proves energy stability

Achieves first-order convergence

Abstract

We present a semi-discrete finite volume scheme for the local NavierStokes-Korteweg and Euler-Korteweg systems. Our scheme is applicable for equidistant Cartesian meshes in one and two space dimensions. In contrast to other works, which employ, for example, hyperbolic approximations of the equations or auxiliary-variable approaches leading to extended systems, our scheme operates directly on the original system. We prove that it conserves mass and momentum and is energy stable. Numerical experiments complement our theoretical findings, showing that the scheme is convergent of order one if employed with explicit or implicit time discretisation.