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

TL;DR

The paper addresses a nonlinear PDE-ODE system on the half-line coupled to a stochastic boundary condition driven by a Pearson diffusion dΨ_t = α(γ−Ψ_t) dt + σ sqrt(Ψ_t(η−Ψ_t)) dW_t, modeling sulphation of CaCO3 by SO2. It introduces a Lamperti transform to obtain constant diffusion, and a splitting-based numerical scheme (Lamperti Sloping Smooth Truncation, LSST) to discretize the coupled system while preserving positivity and stability. The authors establish well-posedness of the boundary process, develop a fully discrete, pathwise-stable scheme, and perform extensive pathwise and statistical numerical experiments in slow and fast reaction regimes to reveal boundary-noise effects, front formation, and invariant-boundary behavior. The results provide a robust computational framework for stochastic PDE-ODE systems with bounded boundary noise and yield insights into degradation patterns in cultural heritage materials, with potential applications in prediction and conservation planning.

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.

Figures (12)

• Figure 1: Drift $f(x)$ of equation \ref{['eq:Lamperti_equation']} (dashed line) and the LSST drift $f_{\Delta}(x)$, (solid line) for $\alpha=7, \gamma=1,\eta=1.5, \Delta=10^{-4}, k=0.22$. (a) $\sigma_1=0.25$, $x^*=1.912$ , $C_0=0.48$; (b) $\sigma_3=1$, $x^*=1.938$, $C_0=0.25$.
• Figure 2: Log-log of the estimations of the $L^2$-errors at time $T$$\left(\Delta,\hat{e}_{{2,T,\Delta}}\right)_\Delta$ (solid line-first raw), and the uniform one $\left(\Delta,\hat{\epsilon}_{{2,\Delta}}\right)_\Delta$ (solid line-second row). Parameters are: $\alpha=7,\gamma=1,\eta=1.5$ and different diffusion coefficients: (a)-(c) $\sigma_1=0.25$; (b)-(d) $\sigma_3=1$. Dashed black line: slope reference 1; dashed blue line: slope reference 0.8; dashed green line: slop reference 0.6.
• Figure 3: Log-log representation of the estimations of the $L^2$-errors at time $T$$\left(\Delta,\hat{e}_{{2,T,\Delta}}\right)_\Delta$ (a), and the uniform one $\left(\Delta,\hat{\epsilon}_{{2,\Delta}}\right)_\Delta$ (b). Parameters are: $\alpha=3.9,\gamma=0.9,\eta=1.5$ and $\sigma_3=1$. Dashed black line: slope reference 1; dashed green line: slop reference 0.5.
• Figure 4: Two random trajectories (filled line) kept within the max and the min of all the trajectories (dotted lines); the sampled mean process (dashed centered line) and boundary of the domain (lower and higher dashed lines) $[0,\eta]$ for $\alpha=7,\gamma=1,\eta=1.5$, $\Psi_0=0$ and different diffusion coefficients: (a) $\sigma_1=0.25$; (b) $\sigma_3=1$.
• Figure 5: Random heat solution in the case $\varphi_1=0.2$, for different values of $\sigma$. Top-left: $\sigma_0=0$; top-right: $\sigma_1=0.25$; bottom-left : $\sigma_2=0.7$; bottom-right : $\sigma_3=1$.

Theorems & Definitions (27)

• Proposition 2.1
• proof
• Definition 2.2
• Definition 2.3
• Proposition 2.4
• proof
• Remark 1
• Proposition 2.5
• proof
• Proposition 2.6