On the rate of convergence of a numerical scheme for fractional conservation laws with noise

TL;DR

This paper analyzes the convergence rate of a finite volume numerical scheme for degenerate fractional conservation laws with noise, employing BV estimates, Young measures, and adapted Kruzkov theory.

Contribution

It introduces a novel approach to estimate convergence rates for fractional conservation laws with noise, addressing the challenges posed by degenerate fractional operators.

Findings

Established convergence rate estimates for the numerical scheme.

Demonstrated numerical convergence rates through simulations.

Extended classical Kruzkov theory to fractional, degenerate, stochastic settings.

Abstract

We consider a semi-discrete finite volume scheme for a degenerate fractional conservation laws driven by a cylindrical Wiener process. Making use of the bounded variation (BV) estimates, Young measure theory, and a clever adaptation of classical Kruzkov theory, we provide estimates on the rate of convergence for approximate solutions to fractional problems. The main difficulty stems from the degenerate fractional operator, and requires a significant departure from the existing strategy to establish Kato's type of inequality. Finally, as an application of this theory, we demonstrate numerical convergence rates.