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

IF 1.3 4区数学 Q1 MATHEMATICS

International Journal of Numerical Analysis and Modeling Pub Date : 2024-01-01 DOI:10.4208/ijnam2024-1005

Sylvain Dotti

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.

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随机双曲标量守恒定律有限体积法的收敛性：采样时间空间截断证明

我们证明了具有单调通量的显式实时有限体积法几乎肯定收敛于具有局部利普希茨连续通量和圆柱维纳过程驱动的加性噪声的标量双曲平衡定律的唯一解。我们使用标准 CFL 条件和概率趋近于 1 的集合上的马氏指数不等式。然后，借助这些集合上的停止时间，我们将收敛定理应用于具有随机强迫的平衡定律的近似动力学解。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

International Journal of Numerical Analysis and Modeling 数学-数学

CiteScore

2.10

自引率

9.10%

发文量

审稿时长

6-12 weeks

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.