Study of a TPFA scheme for the stochastic Allen-Cahn problem with constraint through numerical experiments

TL;DR

This paper investigates a finite volume scheme with a splitting method for the stochastic Allen-Cahn equation, demonstrating its accuracy and error behavior through numerical experiments on bounded domains.

Contribution

It introduces a splitting method for a finite volume scheme applied to the stochastic Allen-Cahn equation with constraints, analyzing its accuracy and error estimates.

Findings

The splitting method is accurate for the scheme.

The squared L2-error w.r.t. time is of order 1 for small noise.

Error order may deteriorate with larger noise.

Abstract

This contribution provides numerical experiments for a finite volume scheme for an approximation of the stochastic Allen-Cahn equation with homogeneous Neumann boundary conditions. The approximation is done by a Yosida approximation of the subdifferential operator. The problem is set on a polygonal bounded domain in two or three dimensions. The non-linear character of the projection term induces challenges to implement the scheme. To this end, we provide a splitting method for the finite volume scheme. We show that the splitting method is accurate. The computational error estimates induce that the squared $L^2$-error w.r.t. time is of order $1$ as long as the noise term is small enough. For larger noise terms the order of convergence w.r.t. time might become worse.