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 journalArticle

    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 1

    Fingerprint

    Finite Volume Scheme
    Brownian movement
    Conservation Laws
    Brownian motion
    Conservation
    Entropy Solution
    Finite Volume Method
    Entropy
    Finite volume method
    Tend
    Godunov Scheme
    Converge
    Young Measures
    Entropy Inequality
    Wiener Process
    Explicit Methods
    Burgers Equation
    Source Terms
    A Priori Estimates
    Subsequence

    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

    Cite this

    Convergence of a finite volume scheme for a stochastic conservation law involving a Q-Brownian motion. / Funaki, Tadahisa; Gao, Yueyuan; Hilhorst, Danielle.

    In: Discrete and Continuous Dynamical Systems - Series B, Vol. 23, No. 4, 01.06.2018, p. 1459-1502.

    Research output: Contribution to journalArticle

    @article{3d2d934f497e44398b97cdf734eacd76,
    title = "Convergence of a finite volume scheme for a stochastic conservation law involving a Q-Brownian motion",
    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.",
    keywords = "Convergence of numerical methods, Finite volume methods, Measure-valued entropy solution, Numerical simulations, Stochastic partial differential equations",
    author = "Tadahisa Funaki and Yueyuan Gao and Danielle Hilhorst",
    year = "2018",
    month = "6",
    day = "1",
    doi = "10.3934/dcdsb.2018159",
    language = "English",
    volume = "23",
    pages = "1459--1502",
    journal = "Discrete and Continuous Dynamical Systems - Series B",
    issn = "1531-3492",
    publisher = "Southwest Missouri State University",
    number = "4",

    }

    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

    PY - 2018/6/1

    Y1 - 2018/6/1

    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

    UR - http://www.scopus.com/inward/record.url?scp=85046826718&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=85046826718&partnerID=8YFLogxK

    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 -