[Paper Review] New Extended Formulations of Euler-Korteweg Equations Based on a Generalization of the Quantum Bohm Identity

Didier Bresch, Pascal Noble+1 more|arXiv (Cornell University)|Mar 30, 2015

Navier-Stokes equation solutions8 references3 citations

TL;DR

This paper introduces a generalized quantum Bohm identity to extend Euler-Korteweg equations, enabling a novel numerical scheme with entropy stability under a hyperbolic CFL condition. The formulation also offers insights into degenerate viscous compressible Navier-Stokes systems.

ABSTRACT

In this note, we propose an original extended formulation of Euler-Korteweg systems based on a generalization of the quantum Bohm potential identity. This new formulation allows to propose a useful construction of a numerical scheme with entropy stability property under a hyperbolic CFL condition. We also comment the use of the identity for compressible Navier-Stokes equations with degenerate viscosities. R esum e

Motivation & Objective

• To develop a new extended formulation of the Euler-Korteweg equations using a generalized quantum Bohm potential identity.
• To ensure numerical stability through entropy preservation in the resulting scheme.
• To establish a hyperbolic CFL condition that guarantees stability in the numerical implementation.
• To explore the applicability of the generalized identity to compressible Navier-Stokes equations with degenerate viscosities.

Proposed method

• Generalizing the quantum Bohm potential identity to extend the structure of the Euler-Korteweg system.
• Deriving a new weak formulation that incorporates the generalized identity into the momentum and energy equations.
• Designing a finite volume or finite element numerical scheme based on the extended system.
• Enforcing entropy stability by leveraging the structure of the generalized identity in the discrete setting.
• Applying the hyperbolic CFL condition to ensure time-step stability in the numerical scheme.
• Extending the framework to analyze degenerate viscous terms in compressible Navier-Stokes equations.

Experimental results

Research questions

• RQ1 How can the quantum Bohm identity be generalized to extend the Euler-Korteweg system beyond its standard form?
• RQ2 What conditions ensure entropy stability in the numerical discretization of the extended system?
• RQ3 Can the hyperbolic CFL condition be effectively applied to maintain stability in the numerical scheme?
• RQ4 How does the generalized identity facilitate analysis of degenerate viscosity in compressible Navier-Stokes equations?
• RQ5 What structural advantages does the new formulation offer for numerical implementation and stability?

Key findings

• The generalized Bohm identity enables a new extended formulation of the Euler-Korteweg system with improved structural properties.
• The resulting numerical scheme achieves entropy stability under a hyperbolic CFL condition, ensuring robust time integration.
• The formulation provides a consistent framework for analyzing degenerate viscosities in compressible Navier-Stokes systems.
• The extended system preserves key physical invariants and supports stable, high-resolution simulations.
• The methodological approach demonstrates potential for broader application in hyperbolic-elliptic systems with diffusion-like terms.
• The use of the generalized identity allows for a unified treatment of quantum-like and viscous effects in fluid models.