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

Tadahisa Funaki, Yueyuan Gao, Danielle Hilhorst

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)1459-1502
Number of pages44
JournalDiscrete and Continuous Dynamical Systems - Series B
Volume23
Issue number4
DOIs
Publication statusPublished - 2018 Jun

Keywords

  • Convergence of numerical methods
  • Finite volume methods
  • Measure-valued entropy solution
  • Numerical simulations
  • Stochastic partial differential equations

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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