TY - JOUR

T1 - Convergence of a finite volume scheme for a stochastic conservation law involving a Q-Brownian motion

AU - Funaki, Tadahisa

AU - Gao, Yueyuan

AU - Hilhorst, Danielle

N1 - Publisher Copyright:
© 2018 American Institute of Mathematical Sciences. All rights reserved.
Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.

PY - 2018/6

Y1 - 2018/6

N2 - 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.

AB - 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.

KW - Convergence of numerical methods

KW - Finite volume methods

KW - Measure-valued entropy solution

KW - Numerical simulations

KW - Stochastic partial differential equations

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U2 - 10.3934/dcdsb.2018159

DO - 10.3934/dcdsb.2018159

M3 - Article

AN - SCOPUS:85046826718

VL - 23

SP - 1459

EP - 1502

JO - Discrete and Continuous Dynamical Systems - Series B

JF - Discrete and Continuous Dynamical Systems - Series B

SN - 1531-3492

IS - 4

ER -