On a time-splitting method for a scalar conservation law with a multiplicative stochastic perturbation and numerical experiments

Abstract

In this paper, we present a numerical scheme for a first-order hyperbolic equation of nonlinear type perturbed by a multiplicative noise. The problem is set in a bounded domain D of \({\mathbb{R}^{d}}\) and with homogeneous Dirichlet boundary condition. Using a time-splitting method, we are able to show the existence of an approximate solution. The result of convergence of such a sequence is based on the work of Bauzet–Vallet–Wittbold (J Funct Anal, 2013), where the authors used the concept of measure-valued solution and Kruzhkov’s entropy formulation to show the existence and uniqueness of the stochastic weak entropy solution. Then, we propose numerical experiments by applying this scheme to the stochastic Burgers’ equation in the one-dimensional case.