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June  2018, 23(4): 1459-1502. doi: 10.3934/dcdsb.2018159

## Convergence of a finite volume scheme for a stochastic conservation law involving a $Q$-brownian motion

 1 Department of Mathematics, School of Fundamental Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan 2 MathAM-OIL, Advanced Industrial Science and Technology Tohoku, c/o AIMR, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan 3 Laboratoire de Mathématiques, University Paris-Sud Paris-Saclay and CNRS, 91405 Orsay Cedex, France

* Corresponding author: Danielle Hilhorst

Received  January 2017 Revised  January 2018 Published  April 2018

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 $\{u_{\mathcal{T}, k}\}$ 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 $\{u_{\mathcal{T}, k}\}$ 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.

Citation: Tadahisa Funaki, Yueyuan Gao, Danielle Hilhorst. Convergence of a finite volume scheme for a stochastic conservation law involving a $Q$-brownian motion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1459-1502. doi: 10.3934/dcdsb.2018159
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Solutions in the deterministic case
The positions of the shock
Comparing the solution in the deterministic case with the empirical average (left) and one realization (right) in the Brownian motion case with $\alpha_B = 1/(2\pi)$ at $t = 0.05$
Comparing the solution in the deterministic case with the empirical average (left) and one realization (right) in the Brownian motion case with $\alpha_B = 1/(2\pi)$ at $t = 1$
Comparing the solution in the deterministic case with the empirical average (left) and one realization (right) in the Brownian motion case with $\alpha_B = 1/(2\pi)$ at $t = 20$
Comparing the solution in the deterministic case with the empirical average (left) and one realization (right) in the $Q$-Brownian motion case with $\alpha_Q = 1$ and $\zeta = 1$ at $t = 0.05$
Comparing the solution in the deterministic case with the empirical average (left) and one realization (right) in the $Q$-Brownian motion case with $\alpha_Q = 1$ and $\zeta = 1$ at $t = 1$
Comparing the solution in the deterministic case with the empirical average (left) and one realization (right) in the $Q$-Brownian motion case with $\alpha_Q = 1$ and $\zeta = 1$ at $t = 20$
Variance in the Brownian motion case (left) and in the $Q$-Brownian motion case (right) for fixed time, with $\alpha_B = 1/(2\pi)$, $\alpha_Q = 1$ and $\zeta = 1$
$L^1$ norm of the variance as a function of time in the case of Brownian motion (left) and $Q$-Brownian motion (right) with $\alpha_B = 1/(2\pi)$, $\alpha_Q = 1$ and $\zeta = 1$
$L^1$ norm of the variance as a function of time in the case of the Brownian motion with $\alpha_B = 1/(2\pi)$ (left) and $\alpha_B = 1/\pi$ (right)
Covariance in the case of Brownian motion (left) and $Q$-Brownian motion (right) as a function of time with $\alpha_Q = 1$ and $\zeta = 1$
Comparing the solution in the deterministic case with the empirical average (left) and one realization (right) in the $Q$-Brownian motion case with $\alpha_Q = 2\pi$ and $\zeta = 1$ at $t = 0.05$
Comparing the solution in the deterministic case with the empirical average (left) and one realization (right) in the $Q$-Brownian motion case with $\alpha_Q = 2\pi$ and $\zeta = 1$ at $t = 1$
Comparing the solution in the deterministic case with the empirical average (left) and one realization (right) in the $Q$-Brownian motion case with $\alpha_Q = 2\pi$ and $\zeta = 1$ at $t = 20$
The $L^1$ norm of the variance in the cases that $\alpha_Q = 1$ (left) and $\alpha_Q = 2\pi$ (right) as a function of 10:36:59
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