Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Jan Giesselmann, Philipp Öffner, Robert Sauerborn

TL;DR

The paper tackles structure-preserving discretization of the local Navier-Stokes-Korteweg and Euler-Korteweg equations on equidistant Cartesian grids. It introduces a semi-discrete finite-volume scheme that discretizes the Korteweg stress directly on the original variables, and proves mass and momentum conservation along with energy dissipation in 1D (with 2D results in the appendix). The numerical tests, including method-of-manufactured-solutions MMS, a Riemann problem, and a thin-film experiment, confirm first-order convergence and energy stability, and demonstrate robust behavior as the capillarity parameter tends to zero. The approach provides a practical, physically faithful baseline for structure-preserving NSK/EK simulations and offers a pathway toward convergence proofs within the DMV-solution framework.

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.

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

TL;DR

The paper tackles structure-preserving discretization of the local Navier-Stokes-Korteweg and Euler-Korteweg equations on equidistant Cartesian grids. It introduces a semi-discrete finite-volume scheme that discretizes the Korteweg stress directly on the original variables, and proves mass and momentum conservation along with energy dissipation in 1D (with 2D results in the appendix). The numerical tests, including method-of-manufactured-solutions MMS, a Riemann problem, and a thin-film experiment, confirm first-order convergence and energy stability, and demonstrate robust behavior as the capillarity parameter tends to zero. The approach provides a practical, physically faithful baseline for structure-preserving NSK/EK simulations and offers a pathway toward convergence proofs within the DMV-solution framework.

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.
Paper Structure (14 sections, 4 theorems, 37 equations, 3 figures, 3 tables)

This paper contains 14 sections, 4 theorems, 37 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Semidiscrete Methods
  3. Semidiscretization of the 1D NSK system
  4. Semidiscretization of the 2D NSK system
  5. Properties of proposed schemes
  6. Properties of the 1D method
  7. Properties of the 2D method
  8. Numerical Tests
  9. Convergence Studies
  10. 1D case
  11. 2D case
  12. Riemann Problem
  13. Thin Film Experiment
  14. Proofs of structure preserving properties for the 2D method

Key Result

Theorem 2

The method (semi1d) conserves the total mass and momentum given by $\frac{1}{N} \sum_{i=1}^{N} \rho_{i}$ and $\frac{1}{N} \sum_{i=1}^{N} \rho_{i}u_i$ respectively.

Figures (3)

  • Figure 1: Riemann problem with contact discontinuity, densities at $t = 0.1$
  • Figure 2: Riemann problem with contact discontinuity
  • Figure 3: Profile of fluid surface at t = 1

Theorems & Definitions (9)

  • Remark 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • Theorem 5
  • proof : Proof of Theorem \ref{['2Dconservation']}
  • proof : Proof of Theorem \ref{['2Ddissipation']}