Discontinuous Galerkin methods for the complete stochastic Euler equations

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 analyse its convergence properties. Our analysis is based on 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 prove that the 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 entropy. The results build a counterpart of those obtained in the deterministic case. In numerical simulations, we show the robustness of our scheme, visualise different stochastic realisations and support/extend the theoretical findings.