A Bhatnagar-Gross-Krook Approximation to Stochastic Scalar Conservation Laws

TL;DR

This paper introduces a BGK-like approximation method for stochastic scalar conservation laws with multiplicative noise, demonstrating existence of solutions and convergence to the kinetic solution as the approximation parameter tends to zero.

Contribution

It develops a novel BGK-like approximation framework for stochastic conservation laws and proves convergence to the kinetic solution in the limit.

Findings

Existence of solutions for fixed approximation parameter.

Convergence of the approximation to the kinetic solution as the parameter tends to zero.

Application of stochastic characteristics method to analyze the problem.

Abstract

We study a BGK-like approximation to hyperbolic conservation laws forced by a multiplicative noise. First, we make use of the stochastic characteristics method and establish the existence of a solution for any fi xed parameter $\varepsilon$. In the next step, we investigate the limit as $\varepsilon$ tends to 0 and show the convergence to the kinetic solution of the limit problem.