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

Thomas Eiter, Jan Giesselmann, Robert Lasarzik, Philipp Öffner, Robert Sauerborn

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.

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

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.

Paper Structure

This paper contains 6 sections, 6 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. The finite-volume method
  3. The semidiscrete scheme
  4. Properties
  5. Convergence towards energy-variational solutions
  6. Acknowledgement

Key Result

Theorem 1

The semidiscrete method method:semi2d conserves total mass and total momentum, which are given by More precisely, every solution $(\rho^h,\boldsymbol{u}^h)$ to eq:EK.approx satisfies for $t\in [0,T^*)$, where $T^ *$ is the maximal existence time of the solution to method:semi2d.

Theorems & Definitions (19)

  • Theorem 1
  • proof
  • Theorem 2
  • Remark 3
  • proof
  • Definition 4
  • Remark 5
  • Remark 6
  • Theorem 7
  • Remark 8
  • ...and 9 more