Convergence analysis for a finite-volume scheme for the Euler- and Navier-Stokes-Korteweg system via energy-variational solutions

TL;DR

This paper proves that a structure-preserving finite-volume scheme for the Euler- and Navier-Stokes-Korteweg systems converges to energy-variational solutions, a new solution concept for these hyperbolic conservation laws.

Contribution

It introduces a convergence proof for a finite-volume scheme to energy-variational solutions, extending this concept to the NSK model.

Findings

Numerical solutions converge to energy-variational solutions under mesh refinement.

The scheme preserves structure and yields uniform estimates.

Energy-variational solutions are stable under weak convergence.

Abstract

We consider a structure-preserving finite-volume scheme for the Euler-Korteweg (EK) and Navier-Stokes-Korteweg (NSK) equations. We prove that its numerical solutions converge to energy-variational solutions of EK or NSK under mesh refinement. Energy-variational solutions constitute a novel solution concept that has recently been introduced for hyperbolic conservation laws, including the EK system, and which we extend to the NSK model. Our proof is based on establishing uniform estimates following from the properties of the structure-preserving scheme, and using the stability of the energy-variational formulation under weak convergence in the natural energy spaces.