Convergence of a finite volume scheme for a stochastic conservation law involving a &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$Q$\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-brownian motion

Funaki, Tadahisa; Gao, Yueyuan; Hilhorst, Danielle

doi:10.3934/dcdsb.2018159

Cited by 3 publications

(2 citation statements)

References 15 publications

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“…Concerning the numerical analysis of these variational solutions, it is clear that in the past the use of finite-element methods has been favored and extensively employed (we refer to [11], [12] for a thorough exposition of existing papers). But we note that recently, for first order scalar conservation laws with multiplicative noise, finite-volume discretizations have been developed in [4], [5], [6], [22], [26], [16], [3], and [17] by adapting the deterministic framework. Then, in [7] and [8], convergence of finite-volume scheme for the particular case of stochastic heat equation (with respectively linear and non-linear multiplicative noise) has been investigated.…”

Section: State Of the Artmentioning

confidence: 99%

Convergence of a finite-volume scheme for a heat equation with a multiplicative Lipschitz noise

Bauzet

Nabet²,

Schmitz³

et al. 2023

ESAIM: M2AN

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We study here the approximation by a finite-volume scheme of a heat equation forced by a Lipschitz continuous multiplicative noise in the sense of Itô. More precisely, we consider a discretization which is semi-implicit in time and a two-point flux approximation scheme (TPFA) in space. We adapt the method based on the theorem of Prokhorov to obtain a convergence in distribution result, then Skorokhod's representation theorem yields the convergence of the scheme towards a martingale solution and the Gyöngy-Krylov argument is used to prove convergence in probability of the scheme towards the unique variational solution of our parabolic problem.

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Section: State Of the Artmentioning

confidence: 99%

Convergence of a finite-volume scheme for a heat equation with a multiplicative Lipschitz noise

Bauzet

Nabet²,

Schmitz³

et al. 2023

ESAIM: M2AN

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“…• In space dimension 1, with strongly monotone fluxes, [24] proved the convergence of a semi-discrete Finite Volume Method towards the solution defined in [16] • In space dimension N ≥ 1, [3] proved the convergence of a fully discrete flux-splitting Finite Volume Method towards the solution defined in [5]. [18] proved it for stochastic source term in infinite dimensions. • In space dimension N ≥ 1, with monotone fluxes, [4] proved the convergence of a fully discrete Finite Volume Method towards the solution defined in [5].…”

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confidence: 98%

Convergence of the Finite Volume Method for Stochastic Hyperbolic Scalar Conservation Laws: A Proof by Truncation on the Sample-Time Space

Sylvain Dotti

2024

IJNAM

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We prove the almost sure convergence of the explicit-in-time Finite Volume Method with monotone fluxes towards the unique solution of the scalar hyperbolic balance law with locally Lipschitz continuous flux and additive noise driven by a cylindrical Wiener process. We use the standard CFL condition and a martingale exponential inequality on sets whose probabilities are converging towards one. Then, with the help of stopping times on those sets, we apply theorems of convergence for approximate kinetic solutions of balance laws with stochastic forcing.

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Existence and uniqueness of the entropy solution of a stochastic conservation law with a Q‐Brownian motion

Funaki

Gao

Hilhorst

2020

Math Methods in App Sciences

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In this paper, we prove the existence and uniqueness of the entropy solution for a first-order stochastic conservation law with a multiplicative source term involving a Q-Brownian motion. After having defined a measure-valued weak entropy solution of the stochastic conservation law, we present the Kato inequality, and as a corollary, we deduce the uniqueness of the measure-valued weak entropy solution, which coincides with the unique weak entropy solution of the problem. The Kato inequality is proved by a doubling of variables method; to that purpose, we prove the existence and the uniqueness of the strong solution of an associated stochastic nonlinear parabolic problem by means of an implicit time discretization scheme; we also prove its convergence to a measure-valued entropy solution of the stochastic conservation law, which proves the existence of the measure-valued entropy solution. KEYWORDS associated parabolic problem, existence and uniqueness of the entropy solution, Kato inequality, Q-Brownian motion, stochastic first-order conservation law MSC CLASSIFICATION 60H15; 35L60; 35A07 Several articles have been devoted to the study of stochastic perturbations of nonlinear first-order hyperbolic problems. Let us mention the article of Bauzet-Vallet-Wittbold 1 who prove the existence and uniqueness of a stochastic entropy Math Meth Appl

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Convergence of a finite volume scheme for a stochastic conservation law involving a <inline-formula><tex-math id="M1">\begin{document}$Q$\end{document}</tex-math></inline-formula>-brownian motion

Cited by 3 publications

References 15 publications

Convergence of a finite-volume scheme for a heat equation with a multiplicative Lipschitz noise

Convergence of a finite-volume scheme for a heat equation with a multiplicative Lipschitz noise

Convergence of the Finite Volume Method for Stochastic Hyperbolic Scalar Conservation Laws: A Proof by Truncation on the Sample-Time Space

Existence and uniqueness of the entropy solution of a stochastic conservation law with a Q‐Brownian motion

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