Convergence of flux-splitting finite volume schemes for hyperbolic scalar conservation laws with a multiplicative stochastic perturbation

Abstract

We study here explicit flux-splitting finite volume discretizations of multi-dimensional nonlinear scalar conservation laws perturbed by a multiplicative noise with a given initial data in L 2 (R d ). Under a stability condition on the time step, we prove the convergence of the finite volume approximation towards the unique stochastic entropy solution of the equation. Keywords : Stochastic PDE • first-order hyperbolic equation • Itô integral • multiplicative noise • finite volume method • flux-splitting scheme • Engquist-Osher scheme • Lax-Friedrichs scheme • upwind scheme • Young measures • Kruzhkov smooth entropy. Mathematics Subject Classification (2000) : 35L60 • 60H15 • 65M08 • 65M12

Convergence of monotone finite volume schemes for hyperbolic scalar conservation laws with multiplicative noise

“…In a recent submitted work, [3] proposed a time and space discretization of the problem and showed the convergence of a class of flux-splitting finite volume schemes towards the unique stochastic entropy solution of the problem by using the theoretical framework of [4]. For a thorough exposition of all these papers, we refer the reader to the introduction of [3]. Note that to the best of our knowledge, in the case of a space and time dependent flux-function, stochastic equations of type (1) have not been studied yet from a theoretical (respectively numerical) point of view, neither by means of entropy formulation (respectively finite volume) framework nor by any other approachs.…”

“…Let us also mention the paper of [18] where a space-discretization of the equation is investigated by considering monotone numerical fluxes. In a recent submitted work, [3] proposed a time and space discretization of the problem and showed the convergence of a class of flux-splitting finite volume schemes towards the unique stochastic entropy solution of the problem by using the theoretical framework of [4]. For a thorough exposition of all these papers, we refer the reader to the introduction of [3].…”

“…For technical reasons, as in [3] and [4], we also need to consider a generalized notion of entropy solution. In fact, in a first step, we will only prove the convergence of the finite volume approximate solution u T ,k to a measure-valued entropy solution.…”

“…In order to treat the first sum we just have to apply the same argument as in the proof of the point I.1, whereas the second term can be dealt by using similar argument to the one used in the proof of Proposition 4 of [3], since we recall that the modified Lax-Friedrichs scheme belongs to the family of flux-splitting schemes.…”

Numerical simulations of the inviscid burgers equation with periodic boundary conditions and stochastic forcing

“…During the completion of the present work, we remarked that approximations by a flux-splitting scheme (i.e. a special choice of the numerical flux) have been recently shown to converge to the stochastic entropic solution of a scalar conservation law under a fixed CFL condition on the time step [3], in the case of a multiplicative noise which vanishes when u = 0.…”