Abstract
We propose a stochastic collocation method based on the piecewise constant interpolation on the probability space combined with a finite volume method to solve the compressible Navier–Stokes system at the nodal points. We show convergence of numerical solutions to a statistical solution of the Navier–Stokes system on condition that the numerical solutions are bounded in probability. The analysis uses the stochastic compactness method based on the Skorokhod/Jakubowski representation theorem and the criterion of convergence in probability due to Gyöngy and Krylov.
Funding Statement
The work of E.F. was partially supported by the Czech Sciences Foundation (GAČR), Grant Agreement 21–02411S. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO:67985840.
M.L. has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Project number 233630050—TRR 146 as well as by TRR 165 Waves to Weather. She is grateful to the Gutenberg Research College and Mainz Institute of Multiscale Modelling for supporting her research.
Acknowledgment
We would like to thank Yuhuan Yuan (Mainz) for providing the figures of numerical simulations.
Citation
Eduard Feireisl. Mária Lukáčová-Medviďová. "Convergence of a stochastic collocation finite volume method for the compressible Navier–Stokes system." Ann. Appl. Probab. 33 (6A) 4936 - 4963, December 2023. https://doi.org/10.1214/23-AAP1937
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