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<p>We show several results on the convergence of the Monte Carlo method applied to a family of consistent approximations of the isentropic Euler system of gas dynamics with uncertain initial data. Our approach is based on a combination of several new ideas developed recently in the context of the Euler system:<UnorderedList Mark="Bullet"> <ItemContent> <p>The concept of a dissipative weak solution as a universal closure of families of consistent approximations.</p> </ItemContent> <ItemContent> <p>A set–valued version of the Strong law of large numbers for general multivalued mappings with closed range.</p> </ItemContent> <ItemContent> <p>Application of the Komlós version of the convergence of empirical averages of integrable functions.</p> </ItemContent> <ItemContent> <p>Unconditionally convergent finite volume approximation schemes.</p> </ItemContent> </UnorderedList> The theoretical results are illustrated by a series of numerical simulations obtained by a viscosity finite volume scheme combined with the Monte Carlo method.</p>

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Monte Carlo method and the random isentropic Euler system

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年4月7日

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  • Eduard Feireisl,
  • Mária Lukáčová-Medvid’ová,
  • Hana Mizerová,
  • Changsheng Yu

摘要

We show several results on the convergence of the Monte Carlo method applied to a family of consistent approximations of the isentropic Euler system of gas dynamics with uncertain initial data. Our approach is based on a combination of several new ideas developed recently in the context of the Euler system:

The concept of a dissipative weak solution as a universal closure of families of consistent approximations.

A set–valued version of the Strong law of large numbers for general multivalued mappings with closed range.

Application of the Komlós version of the convergence of empirical averages of integrable functions.

Unconditionally convergent finite volume approximation schemes.

 The theoretical results are illustrated by a series of numerical simulations obtained by a viscosity finite volume scheme combined with the Monte Carlo method.