Convergence of a stochastic particle approximation for fractional scalar conservation laws

TL;DR

This paper introduces a probabilistic numerical method using particle systems driven by Lévy alpha-stable processes to solve fractional conservation laws with nonlinear drift, demonstrating convergence of the Euler scheme.

Contribution

It presents a novel particle-based approach for fractional PDEs with nonlinear drift and proves convergence of the numerical scheme.

Findings

Convergence of the particle approximation to the PDE solution.

Effective handling of fractional diffusion via Lévy stable processes.

Validation of the Euler scheme's convergence.

Abstract

We give a probabilistic numerical method for solving a partial differential equation with fractional diffusion and nonlinear drift. The probabilistic interpretation of this equation uses a system of particles driven by L\'evy alpha-stable processes and interacting with their drift through their empirical cumulative distribution function. We show convergence to the solution for the associated Euler scheme.