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Convergence of a finite volume method to weak solutions for the compressible Navier-Stokes-Fourier system

Eduard Feireisl, Maria Lukacova-Medvidova, Bangwei She, Yuhuan Yuan

Abstract

We prove strong convergence of an upwind-type finite volume method to a weak solution of the Navier-Stokes-Fourier system with the Dirichlet boundary conditions. The limit solution satisfies a weak form of the mass and momentum equations, together with a weak form of the entropy and ballistic energy inequalities, and complies with the weak-strong uniqueness principle. The finite volume method uses piecewise-constant spatial approximations. The convergence proof is based on a combination of delicate consistency estimates with a careful analysis of the oscillations of numerical densities via renormalisation of the continuity equation.

Convergence of a finite volume method to weak solutions for the compressible Navier-Stokes-Fourier system

Abstract

We prove strong convergence of an upwind-type finite volume method to a weak solution of the Navier-Stokes-Fourier system with the Dirichlet boundary conditions. The limit solution satisfies a weak form of the mass and momentum equations, together with a weak form of the entropy and ballistic energy inequalities, and complies with the weak-strong uniqueness principle. The finite volume method uses piecewise-constant spatial approximations. The convergence proof is based on a combination of delicate consistency estimates with a careful analysis of the oscillations of numerical densities via renormalisation of the continuity equation.
Paper Structure (28 sections, 26 theorems, 228 equations, 1 figure)

This paper contains 28 sections, 26 theorems, 228 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Weak formulation and the numerical method
  3. Notation
  4. Numerical scheme
  5. Discrete initial data.
  6. Uniform bounds - stability
  7. Discretization error - compatibility and consistency
  8. Equation of continuity
  9. Momentum equation
  10. Internal energy equation and entropy balance
  11. Ballistic energy balance
  12. Compatibility of discrete differential operators
  13. Main result - convergence of the numerical scheme
  14. Weak convergence
  15. Strong convergence of the approximate velocities
  16. ...and 13 more sections

Key Result

Lemma 2.4

Let $\{ \varrho_h ,{\bf u}_h, \vartheta_h \}_{h \searrow 0}$ be a family of numerical solutions obtained by the FV method VFV, VFV_bound with $h \approx \Delta t$, and $\alpha \in (-1,1)$. Suppose the numerical solutions are uniformly bounded, specifically, Then we have The constant $C$ depends on $T$, $\|\vartheta_{B}\|_{W^{1,\infty}((0,T) \times\Omega)}$ and $\underline{\varrho}, \overline{\va

Figures (1)

  • Figure 1: Boundary cell $K$.

Theorems & Definitions (45)

  • Definition 2.1
  • Definition 2.2: Numerical method
  • Remark 2.3
  • Lemma 2.4: Stability estimates
  • Corollary 2.5
  • Lemma 2.6: Consistency error in the continuity equation
  • Lemma 2.7: Consistency error in the momentum equation
  • Lemma 2.8: Consistency error in the internal energy/entropy equation
  • Lemma 2.9: Consistency error in the ballistic inequality
  • Lemma 2.10: Compatibility
  • ...and 35 more