In this paper, we analyze a semi-discrete finite volume scheme for the three-dimensional barotropic compressible Euler equations driven by a multiplicative Brownian noise. We derive necessary a priori estimates for numerical approximations, and show that the Young measure generated by the numerical approximations converge to a dissipative measure--valued martingale solution to the stochastic compressible Euler system. These solutions are probabilistically weak in the sense that the driving noise and associated filtration are integral part of the solution. Moreover, we demonstrate strong convergence of numerical solutions to the regular solution of the limit systems at least on the lifespan of the latter, thanks to the weak (measure-valued)--strong uniqueness principle for the underlying system. To the best of our knowledge, this is the first attempt to prove the convergence of numerical approximations for the underlying system.
翻译:在本文中,我们分析了由多种复制型的布朗噪音驱动的三维barotropic可压缩的 Euler 方程式的半分解量计法。我们从数字近似值中得出必要的先验估计值,并表明数字近似值产生的Young 度量法与蒸发式压缩 Euler 系统分解的量估的马丁格尔解决方案相融合。这些解决方案在概率上是薄弱的,因为驱动噪音和相关过滤是解决方案的组成部分。此外,我们展示了至少在后者寿命期间,数字解决方案与限制系统的常规解决方案的高度趋同,这要归功于基本系统的弱(量估)强强的独特性原则。据我们所知,这是首次试图证明基本系统数字近似值的趋同。