Convergence rates for a finite volume scheme of a stochastic non-linear parabolic equation

TL;DR

This paper establishes convergence rates for a finite volume numerical scheme applied to a stochastic non-linear parabolic PDE, demonstrating its accuracy under certain regularity conditions.

Contribution

It provides the first error estimates for a finite volume scheme solving a stochastic non-linear parabolic equation with multiplicative noise.

Findings

Error bounds in the L2-norm for the discretization

Convergence rates depend on initial data regularity

Scheme performs well under Lipschitz noise conditions

Abstract

In this contribution, we provide convergence rates for a finite volume scheme of a stochastic non-linear parabolic equation with multiplicative Lipschitz noise and homogeneous Neumann boundary conditions. More precisely, we give an error estimate for the $L^2$-norm of the space-time discretization by a semi-implicit Euler scheme with respect to time and a two-point flux approximation finite volume scheme with respect to space and the variational solution. The only regularity assumptions additionally needed is spatial regularity of the initial datum and smoothness of the diffusive term.