Discontinuous Galerkin methods for the complete stochastic Euler equations

TL;DR

This paper develops and analyzes a stochastic discontinuous Galerkin method for the Euler equations with stochastic forcing, proving convergence to martingale solutions and demonstrating robustness through numerical simulations.

Contribution

It introduces a stochastic DG spectral element scheme for the Euler equations, proving convergence to dissipative martingale solutions under certain conditions.

Findings

Scheme converges in law to a dissipative martingale solution.

Achieves at least 1/2 order convergence during the lifespan of a strong solution.

Numerical simulations confirm robustness and theoretical predictions.

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.