Semi-discretization for stochastic scalar conservation laws with multiple rough fluxes

TL;DR

This paper introduces a semi-discretization method for stochastic scalar conservation laws with multiple rough fluxes, utilizing rough path theory and Brenier's algorithm, with proven convergence results.

Contribution

It develops a novel semi-discretization scheme for rough flux conservation laws, leveraging pathwise entropy solutions and kinetic formulation for convergence analysis.

Findings

Strong $L^1$-convergence for inhomogeneous fluxes

Convergence rate established for homogeneous fluxes

Method based on rough path theory and transport-collapse algorithm

Abstract

We develop a semi-discretization approximation for scalar conservation laws with multiple rough time dependence in inhomogeneous fluxes. The method is based on Brenier's transport-collapse algorithm and uses characteristics defined in the setting of rough paths. We prove strong $L^1$-convergence for inhomogeneous fluxes and provide a rate of convergence for homogeneous one's. The approximation scheme as well as the proofs are based on the recently developed theory of pathwise entropy solutions and uses the kinetic formulation which allows to define globally the (rough) characteristics.