Convergence of a finite volume scheme for a stochastic conservation law involving a Q-Brownian motion

Tadahisa Funaki, Yueyuan Gao, Danielle Hilhorst*

*この研究の対応する著者

研究成果: Article査読

2 被引用数 (Scopus)

抄録

We study a time explicit finite volume method for a first order conservation law with a multiplicative source term involving a Q-Wiener process. After having presented the definition of a measure-valued weak entropy solution of the stochastic conservation law, we apply a finite volume method together with Godunov scheme for the space discretization, and we denote by (uT ,k) its discrete solution. We present some a priori estimates including a weak BV estimate. After performing a time interpolation, we prove two entropy inequalities for the discrete solution. We show that the discrete solution (uT ,k) converges along a subsequence to a measure-valued entropy solution of the conservation law in the sense of Young measures as the maximum diameter of the volume elements and the time step tend to zero. Some numerical simulations are presented in the case of the stochastic Burgers equation. The empirical average turns out to be a regularization of the deterministic solution; moreover, the variance in the case of the Q-Brownian motion converges to a constant while that in the Brownian motion case keeps increasing as time tends to infinity.

本文言語English
ページ(範囲)1459-1502
ページ数44
ジャーナルDiscrete and Continuous Dynamical Systems - Series B
23
4
DOI
出版ステータスPublished - 2018 6月

ASJC Scopus subject areas

  • 離散数学と組合せ数学
  • 応用数学

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