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

Abstract

We study here the discretization by monotone finite volume schemes of multi-dimensional nonlinear scalar conservation laws forced by a multiplicative noise with a time and space dependent flux-function and a given initial data in $L^{2}(\R^d)$. After establishing the well-posedness theory for solutions of such kind of stochastic problems, we prove under a stability condition on the time step the convergence of the finite volume approximation towards the unique stochastic entropy solution of the equation

Convergence of Approximations to Stochastic Scalar Conservation Laws

“…Here, Ω 2 ⊂ Ω is of full measure. Next, we use the estimate (2.22): there exists a set of full measure Ω 3 in Ω such that, for every ω ∈ Ω 3 ,…”

Convergence of the Finite Volume method for scalar conservation laws with multiplicative noise: an approach by kinetic formulation

“…Our final result, cf. Theorem 7.4, should be compared to [3,Theorem 2]. This latter gives the convergence of the Finite Volume method with monotone fluxes in a very similar context, under the slightly stronger hypothesis that the ratio of the time step ∆t with the spatial characteristic size h of the mesh tends to 0 when h tends to 0.…”

Numerical methods for conservation laws with rough flux

“…where f, σ are nonlinear functions and W (t) is a (finite or infinite dimensional) Wiener process. Numerical methods, based on operator splitting [5,48,54] or finite volume discretizations [7,6,23,24,58], have been proposed and successfully analyzed for (1.1) and similar equations. In another direction, several works [2,31,67,69] have explored linear transport equations with low-regularity velocity coefficient b(x) and "transportation noise", (1.2) du + b(x) · ∇u dt + ∇u • dW (t) = 0, where • refers to the Stratonovich differential (integral).…”