论文标题
与随机数据的可压缩流体流的收敛性和误差分析：蒙特卡洛法
Convergence and error analysis of compressible fluid flows with random data: Monte Carlo method
论文作者
论文摘要
本文的目的是研究带有随机数据的Navier-Stokes方程的Monte Carlo方法的收敛性和错误估计。为了离散空间和时间，将蒙特卡洛方法与合适的确定性离散方案（例如有限体积方法）结合使用。我们假设初始数据，力和粘度系数是随机变量，并且同时研究统计收敛速率以及近似误差。由于可压缩的Navier-Stokes方程在全局弱解决方案类别中不可唯一可解决，因此我们不能应用路径参数来分析随机的Navier-Stokes方程。相反，我们必须通过Skorokhod代表定理和Gyöngy-Krylov方法应用内在的随机紧凑性参数。假设数值解具有概率，我们证明蒙特卡洛有限体积方法会收敛到统计强溶液。也讨论了收敛速率。数值实验说明了理论结果。
The goal of this paper is to study convergence and error estimates of the Monte Carlo method for the Navier-Stokes equations with random data. To discretize in space and time, the Monte Carlo method is combined with a suitable deterministic discretization scheme, such as a finite volume method. We assume that the initial data, force and the viscosity coefficients are random variables and study both, the statistical convergence rates as well as the approximation errors. Since the compressible Navier-Stokes equations are not known to be uniquely solvable in the class of global weak solutions, we cannot apply pathwise arguments to analyze the random Navier-Stokes equations. Instead we have to apply intrinsic stochastic compactness arguments via the Skorokhod representation theorem and the Gyöngy-Krylov method. Assuming that the numerical solutions are bounded in probability, we prove that the Monte Carlo finite volume method converges to a statistical strong solution. The convergence rates are discussed as well. Numerical experiments illustrate theoretical results.
