Finite Volume Approximations for Non-linear Parabolic Problems with Stochastic Forcing

Caroline Bauzet; Flore Nabet; Kerstin Schmitz; Aleksandra Zimmermann

doi:10.1007/978-3-031-40864-9_10

Finite Volume Approximations for Non-linear Parabolic Problems with Stochastic Forcing

Caroline Bauzet
, Flore Nabet
, Kerstin Schmitz
, Aleksandra Zimmermann

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

We propose a two-point flux approximation finite-volume scheme for a stochastic non-linear parabolic equation with a multiplicative noise. The time discretization is implicit except for the stochastic noise term in order to be compatible with stochastic integration in the sense of Itô. We show existence and uniqueness of solutions to the scheme and the appropriate measurability for stochastic integration follows from the uniqueness of approximate solutions.

Original language	English
Title of host publication	Finite Volumes for Complex Applications 10—Volume 1, Elliptic and Parabolic Problems - FVCA10, 2023, Invited Contributions
Editors	Emmanuel Franck, Victor Michel-Dansac, Laurent Navoret, Jürgen Fuhrmann
Publisher	Springer
Pages	157-166
Number of pages	10
ISBN (Print)	9783031408632
DOIs	https://doi.org/10.1007/978-3-031-40864-9_10
Publication status	Published - 1 Jan 2023
Event	10th International Symposium on Finite Volumes for Complex Applications, FVCA10 2023 - Strasbourg, France Duration: 30 Oct 2023 → 3 Nov 2023

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	432
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Conference

Conference	10th International Symposium on Finite Volumes for Complex Applications, FVCA10 2023
Country/Territory	France
City	Strasbourg
Period	30/10/23 → 3/11/23

Keywords

Diffusion-convection equation
Finite-volume method
Multiplicative Lipschitz noise
Stochastic non-linear parabolic equation
Upwind scheme
Variational approach

Access to Document

10.1007/978-3-031-40864-9_10

Cite this

Bauzet, C., Nabet, F., Schmitz, K., & Zimmermann, A. (2023). Finite Volume Approximations for Non-linear Parabolic Problems with Stochastic Forcing. In E. Franck, V. Michel-Dansac, L. Navoret, & J. Fuhrmann (Eds.), Finite Volumes for Complex Applications 10—Volume 1, Elliptic and Parabolic Problems - FVCA10, 2023, Invited Contributions (pp. 157-166). (Springer Proceedings in Mathematics and Statistics; Vol. 432). Springer. https://doi.org/10.1007/978-3-031-40864-9_10

Bauzet, Caroline ; Nabet, Flore ; Schmitz, Kerstin et al. / Finite Volume Approximations for Non-linear Parabolic Problems with Stochastic Forcing. Finite Volumes for Complex Applications 10—Volume 1, Elliptic and Parabolic Problems - FVCA10, 2023, Invited Contributions. editor / Emmanuel Franck ; Victor Michel-Dansac ; Laurent Navoret ; Jürgen Fuhrmann. Springer, 2023. pp. 157-166 (Springer Proceedings in Mathematics and Statistics).

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abstract = "We propose a two-point flux approximation finite-volume scheme for a stochastic non-linear parabolic equation with a multiplicative noise. The time discretization is implicit except for the stochastic noise term in order to be compatible with stochastic integration in the sense of It{\^o}. We show existence and uniqueness of solutions to the scheme and the appropriate measurability for stochastic integration follows from the uniqueness of approximate solutions.",

keywords = "Diffusion-convection equation, Finite-volume method, Multiplicative Lipschitz noise, Stochastic non-linear parabolic equation, Upwind scheme, Variational approach",

author = "Caroline Bauzet and Flore Nabet and Kerstin Schmitz and Aleksandra Zimmermann",

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Bauzet, C, Nabet, F, Schmitz, K & Zimmermann, A 2023, Finite Volume Approximations for Non-linear Parabolic Problems with Stochastic Forcing. in E Franck, V Michel-Dansac, L Navoret & J Fuhrmann (eds), Finite Volumes for Complex Applications 10—Volume 1, Elliptic and Parabolic Problems - FVCA10, 2023, Invited Contributions. Springer Proceedings in Mathematics and Statistics, vol. 432, Springer, pp. 157-166, 10th International Symposium on Finite Volumes for Complex Applications, FVCA10 2023, Strasbourg, France, 30/10/23. https://doi.org/10.1007/978-3-031-40864-9_10

Finite Volume Approximations for Non-linear Parabolic Problems with Stochastic Forcing. / Bauzet, Caroline; Nabet, Flore; Schmitz, Kerstin et al.
Finite Volumes for Complex Applications 10—Volume 1, Elliptic and Parabolic Problems - FVCA10, 2023, Invited Contributions. ed. / Emmanuel Franck; Victor Michel-Dansac; Laurent Navoret; Jürgen Fuhrmann. Springer, 2023. p. 157-166 (Springer Proceedings in Mathematics and Statistics; Vol. 432).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AB - We propose a two-point flux approximation finite-volume scheme for a stochastic non-linear parabolic equation with a multiplicative noise. The time discretization is implicit except for the stochastic noise term in order to be compatible with stochastic integration in the sense of Itô. We show existence and uniqueness of solutions to the scheme and the appropriate measurability for stochastic integration follows from the uniqueness of approximate solutions.

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Bauzet C, Nabet F, Schmitz K, Zimmermann A. Finite Volume Approximations for Non-linear Parabolic Problems with Stochastic Forcing. In Franck E, Michel-Dansac V, Navoret L, Fuhrmann J, editors, Finite Volumes for Complex Applications 10—Volume 1, Elliptic and Parabolic Problems - FVCA10, 2023, Invited Contributions. Springer. 2023. p. 157-166. (Springer Proceedings in Mathematics and Statistics). doi: 10.1007/978-3-031-40864-9_10