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

Tadahisa Funaki; Yueyuan Gao; Danielle Hilhorst

doi:10.3934/dcdsb.2018159

Article Contents

2018, Volume 23, Issue 4: 1459-1502. Doi: 10.3934/dcdsb.2018159

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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
^* Corresponding author: Danielle Hilhorst

Received: January 2017

Revised: January 2018

Early access: April 2018

Published: April 2018

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

Keywords:

Finite volume methods,
stochastic partial differential equations,
convergence of numerical methods,
measure-valued entropy solution,
numerical simulations.

Mathematics Subject Classification: Primary: 65M08, 60H15; Secondary: 65M12.

Citation:

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Figure 1. Solutions in the deterministic case

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Figure 2. The positions of the shock

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Figure 3. 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 $t = 0.05$

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Figure 4. 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 $t = 1$

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Figure 5. 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 $t = 20$

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Figure 6. 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 $t = 0.05$

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Figure 7. 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 $t = 1$

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Figure 8. 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 $t = 20$

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Figure 9. 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 $\zeta = 1$

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Figure 10. $L^1$ 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 $\zeta = 1$

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Figure 11. $L^1$ 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)

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Figure 12. Covariance in the case of Brownian motion (left) and $Q$-Brownian motion (right) as a function of time with $\alpha_Q = 1$ and $\zeta = 1$

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Figure 13. 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 $t = 0.05$

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Figure 14. 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 $t = 1$

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Figure 15. 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 $t = 20$

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Figure 16. 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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