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Discontinuous Galerkin methods for the complete stochastic Euler equations

Dominic Breit, Thamsanqa Castern Moyo, Philipp Öffner

TL;DR

This work addresses solving the complete stochastic Euler equations with momentum forcing using entropy-dissipative discontinuous Galerkin schemes (including a finite-volume variant). It proves convergence of the numerical method to dissipative measure-valued martingale solutions under stopping-time bounds and demonstrates, for strong pathwise solutions, a convergence rate of $1/2$ in the expected relative energy. The approach relies on a relative-entropy framework to connect discrete and continuous solutions and to obtain consistency and convergence results, complemented by a suite of one- and two-dimensional numerical experiments that illustrate robustness and the influence of stochastic forcing. The results extend deterministic structure-preserving convergence theory to the stochastic setting and lay groundwork for further error analysis, limiter design, and turbulence modeling under uncertainty.

Abstract

In recent years, stochastic effects have become increasingly relevant for describing fluid behaviour, particularly in the context of turbulence. The most important model for inviscid fluids in computational fluid dynamics are the Euler equations of gas dynamics which we focus on in this paper. To take stochastic effects into account, we incorporate a stochastic forcing term in the momentum equation of the Euler system. To solve the extended system, we apply an entropy dissipative discontinuous Galerkin spectral element method including the Finite Volume setting, adjust it to the stochastic Euler equations and analyze its convergence properties. Our analysis is grounded in the concept of dissipative martingale solutions, as recently introduced by Moyo (J. Diff. Equ. 365, 408-464, 2023). Assuming no vacuum formation and bounded total energy, we proof that our scheme converges in law to a dissipative martingale solution. During the lifespan of a pathwise strong solution, we achieve convergence of at least order 1/2, measured by the expected $L^1$ norm of the relative energy. The results built a counterpart of corresponding results in the deterministic case. In numerical simulations, we show the robustness of our scheme, visualise different stochastic realizations and analyze our theoretical findings.

Discontinuous Galerkin methods for the complete stochastic Euler equations

TL;DR

This work addresses solving the complete stochastic Euler equations with momentum forcing using entropy-dissipative discontinuous Galerkin schemes (including a finite-volume variant). It proves convergence of the numerical method to dissipative measure-valued martingale solutions under stopping-time bounds and demonstrates, for strong pathwise solutions, a convergence rate of in the expected relative energy. The approach relies on a relative-entropy framework to connect discrete and continuous solutions and to obtain consistency and convergence results, complemented by a suite of one- and two-dimensional numerical experiments that illustrate robustness and the influence of stochastic forcing. The results extend deterministic structure-preserving convergence theory to the stochastic setting and lay groundwork for further error analysis, limiter design, and turbulence modeling under uncertainty.

Abstract

In recent years, stochastic effects have become increasingly relevant for describing fluid behaviour, particularly in the context of turbulence. The most important model for inviscid fluids in computational fluid dynamics are the Euler equations of gas dynamics which we focus on in this paper. To take stochastic effects into account, we incorporate a stochastic forcing term in the momentum equation of the Euler system. To solve the extended system, we apply an entropy dissipative discontinuous Galerkin spectral element method including the Finite Volume setting, adjust it to the stochastic Euler equations and analyze its convergence properties. Our analysis is grounded in the concept of dissipative martingale solutions, as recently introduced by Moyo (J. Diff. Equ. 365, 408-464, 2023). Assuming no vacuum formation and bounded total energy, we proof that our scheme converges in law to a dissipative martingale solution. During the lifespan of a pathwise strong solution, we achieve convergence of at least order 1/2, measured by the expected norm of the relative energy. The results built a counterpart of corresponding results in the deterministic case. In numerical simulations, we show the robustness of our scheme, visualise different stochastic realizations and analyze our theoretical findings.

Paper Structure

This paper contains 25 sections, 6 theorems, 120 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Mathematical framework
  3. Constitutive relations
  4. Probability framework
  5. Discontinuous Galerkin schemes
  6. Concept of solutions and main results
  7. Measure-valued solutions
  8. Strong solutions
  9. Consistency and convergence
  10. Consistency
  11. The consistency formulation
  12. Compactness Argument
  13. Passage to the limit
  14. Convergence
  15. The relative entropy
  16. ...and 10 more sections

Key Result

Theorem 3.2

Assume (eq:HS) holds. Let $\Lambda$ be a Borel probability measure on $L^{\gamma}(\mathcal{O})\times L^{\gamma}(\mathcal{O})\times L^{\frac{2\gamma}{\gamma +1}}(\mathcal{O})$ such that where $\underline{\vartheta}, \overline{\vartheta},\underline{\varrho}, \overline{\varrho}$ are deterministic constants. Moreover, the moment estimate holds for all $p\geq 1$. Then there exists a dissipative measu

Figures (2)

  • Figure 1: Different noise (left) and different noise strength(right)
  • Figure 2: Kelvin-Helmholtz and influence of the noise

Theorems & Definitions (17)

  • Example 2.1
  • Remark 2.2: Interpretation and solvability
  • Definition 3.1: Dissipative measure-valued martingale solution
  • Theorem 3.2
  • Definition 3.3: Strong Solution
  • Theorem 3.4
  • Remark 3.5
  • Theorem 3.6
  • Remark 3.7
  • Theorem 3.8
  • ...and 7 more