A finite difference scheme for conservation laws driven by Levy noise

TL;DR

This paper develops and analyzes a finite difference scheme for conservation laws influenced by Levy noise, proving convergence to the entropy solution with a specific rate, advancing numerical methods for stochastic PDEs.

Contribution

It introduces a semi-discrete finite difference scheme for Levy-driven conservation laws and proves its convergence with a quantifiable rate, which is a novel contribution.

Findings

Convergence of the scheme to the entropy solution as mesh size decreases

Expected L^1 difference converges at rate O(√Dx)

BV estimates ensure compactness and convergence

Abstract

In this paper, we analyze a semi-discrete finite difference scheme for a conservation laws driven by a homogeneous multiplicative Levy noise. Thanks to BV estimates, we show a compact sequence of approximate solutions, generated by the finite difference scheme, converges to the unique entropy solution of the underlying problem, as the spatial mesh size \Dx-->0. Moreover, we show that the expected value of the L^1-difference between the approximate solution and the unique entropy solution converges at a rate O(\sqrt{\Dx}).