{"title":"Convergence of the Finite Volume Method for Stochastic Hyperbolic Scalar Conservation Laws: A Proof By Truncation on the Sample-Time Space","authors":"Sylvain Dotti","doi":"10.4208/ijnam2024-1005","DOIUrl":null,"url":null,"abstract":"We prove the almost sure convergence of the explicit-in-time Finite Volume Method\nwith monotone fluxes towards the unique solution of the scalar hyperbolic balance law with locally\nLipschitz continuous flux and additive noise driven by a cylindrical Wiener process. We use the\nstandard CFL condition and a martingale exponential inequality on sets whose probabilities are\nconverging towards one. Then, with the help of stopping times on those sets, we apply theorems\nof convergence for approximate kinetic solutions of balance laws with stochastic forcing.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"70 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal of Numerical Analysis and Modeling","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4208/ijnam2024-1005","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
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.
期刊介绍：
The journal is directed to the broad spectrum of researchers in numerical methods throughout science and engineering, and publishes high quality original papers in all fields of numerical analysis and mathematical modeling including: numerical differential equations, scientific computing, linear algebra, control, optimization, and related areas of engineering and scientific applications. The journal welcomes the contribution of original developments of numerical methods, mathematical analysis leading to better understanding of the existing algorithms, and applications of numerical techniques to real engineering and scientific problems. Rigorous studies of the convergence of algorithms, their accuracy and stability, and their computational complexity are appropriate for this journal. Papers addressing new numerical algorithms and techniques, demonstrating the potential of some novel ideas, describing experiments involving new models and simulations for practical problems are also suitable topics for the journal. The journal welcomes survey articles which summarize state of art knowledge and present open problems of particular numerical techniques and mathematical models.
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