A numerical study of a PDE-ODE system with a stochastic dynamical boundary condition: a nonlinear model for sulphation phenomena

TL;DR

This paper presents a numerical analysis of a stochastic PDE-ODE system modeling sulphation in cultural heritage, highlighting the effects of boundary noise on solution behavior through innovative schemes.

Contribution

It introduces a novel numerical scheme combining Lamperti transformation and splitting strategies for a stochastic boundary value problem in sulphation modeling.

Findings

Boundary noise significantly affects solution behavior.

The scheme preserves positivity and stability.

Qualitative analysis of slow and fast regimes.

Abstract

We investigate the qualitative behaviour of the solutions of a stochastic boundary value problem on the half-line for a nonlinear system of parabolic reaction-diffusion equations, from a numerical point of view. The model describes the chemical aggression of calcium carbonate stones under the attack of sulphur dioxide. The dynamical boundary condition is given by a Pearson diffusion, which is original in the context of the degradation of cultural heritage. We first discuss a scheme based on the Lamperti transformation for the stochastic differential equation to preserve the boundary and a splitting strategy for the partial differential equation based on recent theoretical results. Positiveness, boundedness, and stability are stated. The impact of boundary noise on the solution and its qualitative behaviour both in the slow and fast regimes is discussed in several numerical experiments.