Convergence of a Finite Volume Scheme for the Navier-Stokes-Korteweg Model via Dissipative Solutions

TL;DR

This paper introduces dissipative weak solutions for the Navier-Stokes-Korteweg system and proves that a finite volume scheme converges to such solutions, extending the framework of the Lax theorem to nonlinear fluid models.

Contribution

It establishes the convergence of a structure-preserving finite volume scheme to dissipative weak solutions for the NSK system, building on recent advances in related fluid models.

Findings

Proves convergence of the FV scheme to DW solutions of NSK.

Leverages conservation and dissipation properties of the scheme.

Extends the concept of dissipative solutions to the NSK system.

Abstract

We propose a concept of dissipative weak (DW) solutions for the Navier-Stokes-Korteweg (NSK) system and prove conditional convergence of a structure-preserving finite volume scheme towards such a solution. DW solutions provide a generalized solution concept in computational fluid dynamics and have recently attracted significant attention. They provide an extension of the famous Lax Equivalence Theorem to nonlinear problems, i.e. consistency and stability of a numerical scheme imply convergence. Our work builds on recent advances where convergence towards DW solutions of structure-preserving schemes has been established for the Euler and Navier-Stokes equations. Indeed, we prove convergence of a recently proposed FV scheme by leveraging its conservation and dissipation properties as well as its consistency.