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.

翻译：近年来，随机效应在描述流体行为（特别是在湍流背景下）中日益重要。计算流体动力学中无粘性流体的最重要模型是气体动力学欧拉方程，这也是本文的研究重点。为考虑随机效应，我们在欧拉系统的动量方程中引入了随机强迫项。为求解该扩展系统，我们采用包含有限体积框架的熵耗散不连续伽辽金谱元法，将其调整适用于随机欧拉方程，并分析其收敛特性。我们的分析基于Moyo（J. Diff. Equ. 365, 408-464, 2023）近期提出的耗散鞅解概念。在假设无真空形成且总能量有界的条件下，我们证明该格式依分布收敛于耗散鞅解。在路径强解的存在区间内，通过相对能量的期望$L^1$范数度量，我们获得至少1/2阶的收敛速度。这些结果构成了确定性情形对应结论的随机版本。在数值模拟中，我们展示了该格式的鲁棒性，可视化了不同的随机实现，并对理论结果进行了验证分析。