标题： 不连续伽辽金方法用于完整的随机欧拉方程 显示英文标题

标题： Discontinuous Galerkin methods for the complete stochastic Euler equations

摘要： 近年来，随机效应在描述流体行为方面变得越来越相关，尤其是在湍流的背景下。计算流体力学中无粘流体最重要的模型是气体动力学的欧拉方程，本文我们专注于这些方程。为了考虑随机效应，我们在欧拉系统的动量方程中引入了一个随机强迫项。为了求解扩展后的系统，我们应用了一种熵耗散的不连续伽辽金谱元方法，包括有限体积设置，将其调整到随机欧拉方程，并分析其收敛性。我们的分析基于最近由Moyo（J. Diff. Equ. 365, 408-464, 2023）引入的耗散鞅解的概念。假设没有真空形成且总能量有界，我们证明了我们的方案在法律上收敛到一个耗散鞅解。在路径强解的生命周期内，我们实现了至少1/2阶的收敛性，通过相对能量的期望$L^1$范数来衡量。这些结果建立了确定性情况下相应结果的对应结果。在数值模拟中，我们展示了我们方案的鲁棒性，可视化了不同的随机实现，并分析了我们的理论结果。 显示英文摘要

摘要： 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.