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Convergence of numerical methods for the Navier-Stokes-Fourier system driven by uncertain initial/boundary data

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

TL;DR

This work develops a rigorous convergence framework for numerical approximations of the compressible Navier–Stokes–Fourier system under random data, proving convergence of both stochastic collocation and Monte Carlo methods under a bounded-in-probability hypothesis. Central to the analysis is the ballistic energy structure, which yields stability in the absence of conventional energy estimates, together with a probabilistic toolkit (Skorokhod representation and Gyöngy–Krylov) to pass to limits and identify the limit with a strong solution. The authors establish that bounded consistent approximations converge to the unique strong solution and extend these results to random data, obtaining convergence in law and in probability for the numerical schemes. The abstract results are corroborated by Rayleigh–Bénard convection simulations, where stochastic methods yield correct convergence behavior for statistically meaningful quantities such as expectations and deviations. Overall, the paper provides a robust pathway for uncertainty quantification in compressible, thermally conducting fluids with Dirichlet boundaries, with practical implications for reliable simulations of turbulent convection under data uncertainty.

Abstract

We consider the Navier-Stokes-Fourier system governing the motion of a general compressible, heat conducting, Newtonian fluid driven by random initial/boundary data. Convergence of the stochastic collocation and Monte Carlo numerical methods is shown under the hypothesis that approximate solutions are bounded in probability. Abstract results are illustrated by numerical experiments for the Rayleigh-Benard convection problem.

Convergence of numerical methods for the Navier-Stokes-Fourier system driven by uncertain initial/boundary data

TL;DR

This work develops a rigorous convergence framework for numerical approximations of the compressible Navier–Stokes–Fourier system under random data, proving convergence of both stochastic collocation and Monte Carlo methods under a bounded-in-probability hypothesis. Central to the analysis is the ballistic energy structure, which yields stability in the absence of conventional energy estimates, together with a probabilistic toolkit (Skorokhod representation and Gyöngy–Krylov) to pass to limits and identify the limit with a strong solution. The authors establish that bounded consistent approximations converge to the unique strong solution and extend these results to random data, obtaining convergence in law and in probability for the numerical schemes. The abstract results are corroborated by Rayleigh–Bénard convection simulations, where stochastic methods yield correct convergence behavior for statistically meaningful quantities such as expectations and deviations. Overall, the paper provides a robust pathway for uncertainty quantification in compressible, thermally conducting fluids with Dirichlet boundaries, with practical implications for reliable simulations of turbulent convection under data uncertainty.

Abstract

We consider the Navier-Stokes-Fourier system governing the motion of a general compressible, heat conducting, Newtonian fluid driven by random initial/boundary data. Convergence of the stochastic collocation and Monte Carlo numerical methods is shown under the hypothesis that approximate solutions are bounded in probability. Abstract results are illustrated by numerical experiments for the Rayleigh-Benard convection problem.
Paper Structure (35 sections, 16 theorems, 211 equations, 10 figures)

This paper contains 35 sections, 16 theorems, 211 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Field equations -- Navier--Stokes--Fourier system
  3. Boundary conditions
  4. Problems with uncertain data
  5. Strategy of the convergence analysis
  6. Strong solutions
  7. Local existence of strong solutions
  8. Conditional regularity
  9. Stability with respect to data
  10. Approximate solutions
  11. Consistent approximation
  12. Convergent approximations
  13. Solutions with random data
  14. Convergence for random data
  15. Convergence in law
  16. ...and 20 more sections

Key Result

Theorem 2.1

Let $Q \subset \mathbb{R}^d$, $d=2,3$ be a bounded domain of class $C^4$. Suppose the data $(\varrho_D, \vartheta_D, {\bm u}_D, {\bm g})$ belong to the class s1 and comply with the compatibility conditions s2. Then there exists $T_{\rm max}$, $0 < T_{\rm max} \leq \infty$ such that the Navier--Stoke for any $0 < T < T_{\rm max}$.

Figures (10)

  • Figure 1: Deterministic Rayleigh--Bénard experiment: FV solutions obtained at time $T = 8$ (left) and $T=25$ (right) on a mesh with $640\times 320$ cells. From top to bottom: density (top); temperature (middle); streamline (bottom).
  • Figure 2: Rayleigh--Bénard experiment: the deterministic FV solution (left) and the SCFV solutions (right) at time $T = 8$ on a mesh with $640\times 320$ cells. From top to bottom: density (top); temperature (middle); streamline (bottom).
  • Figure 3: SCFV solutions $\varrho$ obtained with $80$ stochastic collocation points on a mesh with $640 \times 320$ cells. Top: expectation (left) and variance (right) of $\varrho$. Bottom: from left to right: $\varrho$ at lines $x_2 = -3/4$ and $x_2 =3/4$.
  • Figure 4: SCFV solutions $\vartheta$ obtained with $80$ stochastic collocation points on a mesh with $640 \times 320$ cells. Top: expectation (left) and variance (right) of $\vartheta$. Bottom: from left to right: $\vartheta$ at lines $x_2 = -3/4$ and $x_2 =3/4$.
  • Figure 5: SCFV total errors of the expectation $E_1^{SC}(U,h,n(h))$ and the deviation $E_2^{SC}(U,h,n(h))$ with $U\in \{ \varrho, {\bm m}, S, {\bm u}, \vartheta \}$ and $(h,1/n(h)) =(h,\mathcal{O}(h)) = (2/(20\cdot 2^l), \, 1/(5\cdot 2^l)), \, l =0,\dots,3.$ The black solid line without any marker denotes the reference line $h$.
  • ...and 5 more figures

Theorems & Definitions (37)

  • Remark 1.1
  • Theorem 2.1: Local existence
  • Remark 2.2
  • Theorem 2.3: Regularity estimates
  • Corollary 2.4: Conditional regularity
  • Theorem 2.5: Stability with respect to data
  • Definition 3.1: Bounded consistent approximate method
  • Theorem 3.2: Convergence of bounded approximations
  • Remark 3.3
  • proof
  • ...and 27 more