On rate of convergence of finite difference scheme for degenerate parabolic-hyperbolic PDE with Levy noise

TL;DR

This paper analyzes the convergence rate of a finite difference scheme for a degenerate parabolic-hyperbolic PDE with Levy noise, establishing a specific rate of convergence in the expected $L^1$-difference.

Contribution

It introduces a convergence analysis for a semi-discrete finite difference scheme applied to PDEs with Levy noise, providing a quantitative rate of convergence.

Findings

Expected $L^1$-difference converges at rate $( riangle x)^{1/7}$.

Uses bounded variation estimates and Kruzhkov's doubling of variables.

First such rate established for this class of stochastic PDEs.

Abstract

In this article, we consider a semi discrete finite difference scheme for a degenerate parabolic-hyperbolic PDE driven by L\'evy noise in one space dimension. Using bounded variation estimations and a variant of classical Kru\v{z}kov's doubling of variable approach, we prove that expected value of the $L^1$-difference between the unique entropy solution and approximate solution converges at a rate of $(\Delta x)^\frac{1}{7}$, where $\Delta x$ is the spatial mesh size.