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Random compressible Euler flows

Maria Lukacova-Medvidova, Simon Schneider

TL;DR

The work addresses uncertainty in the compressible Euler equations by employing a nonintrusive stochastic collocation method built on a viscous finite volume discretization. By coupling deterministic convergence theory for the VFV scheme with stochastic compactness arguments, the authors prove convergence in probability of the stochastic solutions to a strong, unique Euler solution as the discretization in both physical and random space is refined. Key contributions include a rigorous a priori bounds framework, construction of uniformly Lipschitz approximations, and a probabilistic convergence result under bounded discrete gradients, enabling reliable uncertainty quantification for hyperbolic flows. The approach mitigates Gibbs phenomena by using piecewise-continuous representations in both deterministic and random spaces, with implications for robust simulations of random compressible flows in engineering and geophysical contexts.

Abstract

We propose a finite volume stochastic collocation method for the random Euler system. We rigorously prove the convergence of random finite volume solutions under the assumption that the discrete differential quotients remain bounded in probability. Convergence analysis combines results on the convergence of a deterministic FV method with stochastic compactness arguments due to Skorokhod and Gyöngy-Krylov.

Random compressible Euler flows

TL;DR

The work addresses uncertainty in the compressible Euler equations by employing a nonintrusive stochastic collocation method built on a viscous finite volume discretization. By coupling deterministic convergence theory for the VFV scheme with stochastic compactness arguments, the authors prove convergence in probability of the stochastic solutions to a strong, unique Euler solution as the discretization in both physical and random space is refined. Key contributions include a rigorous a priori bounds framework, construction of uniformly Lipschitz approximations, and a probabilistic convergence result under bounded discrete gradients, enabling reliable uncertainty quantification for hyperbolic flows. The approach mitigates Gibbs phenomena by using piecewise-continuous representations in both deterministic and random spaces, with implications for robust simulations of random compressible flows in engineering and geophysical contexts.

Abstract

We propose a finite volume stochastic collocation method for the random Euler system. We rigorously prove the convergence of random finite volume solutions under the assumption that the discrete differential quotients remain bounded in probability. Convergence analysis combines results on the convergence of a deterministic FV method with stochastic compactness arguments due to Skorokhod and Gyöngy-Krylov.
Paper Structure (9 sections, 13 theorems, 29 equations)

This paper contains 9 sections, 13 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Discretization of the deterministic problem
  3. The viscous finite volume method
  4. Conditional regularity of the viscous finite volume method
  5. A priori bounds
  6. Lipschitz continuous approximation of numerical solutions
  7. Proof of Theorem \ref{['thm:deterministic_convergence']}
  8. The stochastic collocation viscous finite volume scheme
  9. Main result: convergence

Key Result

Theorem 1.1

If $(\varrho_1, \, \mathbf{m}_1, \,E_1)$ and $(\varrho_2, \, \mathbf{m}_2, \,E_2)$ are strong solutions of (eq:EulerSystem) with the same initial data, then $(\varrho_1, \, \mathbf{m}_1, \,E_1)=(\varrho_2, \, \mathbf{m}_2, \,E_2)$.

Theorems & Definitions (18)

  • Definition 1.1: Weak solution
  • Theorem 1.1: Uniqueness of strong solutions
  • Definition 2.1: Admissible deterministic discrete initial data
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.1
  • Lemma 2.3
  • Lemma 2.4
  • ...and 8 more