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Volume 21, Issue 1
Convergence of the Finite Volume Method for Stochastic Hyperbolic Scalar Conservation Laws: A Proof by Truncation on the Sample-Time Space

Sylvain Dotti

DOI: 10.4208/ijnam2024-1005

Int. J. Numer. Anal. Mod., 21 (2024), pp. 120-164.

Published online: 2024-01

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  • Abstract

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.

  • Keywords

Finite volume method, stochastic balance law, kinetic formulation.

  • AMS Subject Headings

65M08, 35L60, 35L65, 35R60, 60H15, 65M12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-21-120, author = {Dotti , Sylvain}, title = {Convergence of the Finite Volume Method for Stochastic Hyperbolic Scalar Conservation Laws: A Proof by Truncation on the Sample-Time Space}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2024}, volume = {21}, number = {1}, pages = {120--164}, abstract = {

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.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2024-1005}, url = {http://global-sci.org/intro/article_detail/ijnam/22331.html} }
TY - JOUR T1 - Convergence of the Finite Volume Method for Stochastic Hyperbolic Scalar Conservation Laws: A Proof by Truncation on the Sample-Time Space AU - Dotti , Sylvain JO - International Journal of Numerical Analysis and Modeling VL - 1 SP - 120 EP - 164 PY - 2024 DA - 2024/01 SN - 21 DO - http://doi.org/10.4208/ijnam2024-1005 UR - https://global-sci.org/intro/article_detail/ijnam/22331.html KW - Finite volume method, stochastic balance law, kinetic formulation. AB -

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.

Sylvain Dotti. (2024). Convergence of the Finite Volume Method for Stochastic Hyperbolic Scalar Conservation Laws: A Proof by Truncation on the Sample-Time Space. International Journal of Numerical Analysis and Modeling. 21 (1). 120-164. doi:10.4208/ijnam2024-1005
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