Convergence and error analysis of compressible fluid flows with random data: Monte Carlo method

0569928 - MÚ 2023 RIV SG eng J - Journal Article
Feireisl, Eduard - Lukáčová-Medviďová, M. - She, Bangwei - Yuan, Y.
Convergence and error analysis of compressible fluid flows with random data: Monte Carlo method.
Mathematical Models and Methods in Applied Sciences. Roč. 32, č. 14 (2022), s. 2887-2925. ISSN 0218-2025. E-ISSN 1793-6314
R&D Projects: GA ČR(CZ) GA21-02411S
Institutional support: RVO:67985840
Keywords : deterministic and statistical convergence rates * finite volume method * Monte Carlo method
OECD category: Pure mathematics
Impact factor: 3.5, year: 2022
Method of publishing: Limited access
https://doi.org/10.1142/S0218202522500671

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 (FV) 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 FV method converges to a statistical strong solution. The convergence rates are discussed as well. Numerical experiments illustrate theoretical results.
Permanent Link: https://hdl.handle.net/11104/0341247