MODELLING AND NUMERICAL STUDY OF MARBLE SULPHATION IN CULTURALE HERITAGE: A RANDOM PDE-ODE SYSTEM

The final scope of this thesis is to investigate from a numerical point of view a new approach to the mathematical modelling of the degradation process in the context of Cultural Heritage, focusing on the description of the marble sulphation chemical reaction. Our contribution is a numerical analysis of a novel random model of the sulphation reaction on the half-line. Given a system of non-linear reaction-diffusion partial and ordinary differential equations (PDE-ODE) system, already present in the literature and describing the penetration of sulphur dioxide through the pores of a calcium carbonate-based surface which reacts with a consequent formation of gypsum, a source of randomness has been introduced as a Dirichlet dynamical boundary condition, with the aim to better model the sulphur dioxide evolution in the air. A direct observation of some recent sulphur dioxide time series recorded in Milan suggests that an appropriate model needs to incorporate a significant stochastic fluctuation component, boundedness and a mean-reverting behaviour. The novelty relies on the choice of a class of stochastic processes with such properties, known as Pearson diffusions, and which can be introduced as solutions of an It\^o's stochastic differential equations (SDE). For completeness, positiveness, boundedness, and stability for the Pearson SDE, as well as its numerical scheme, known as Lamperti Sloping Smooth Truncation, based on the well-known Euler-Maruyama method, are discussed and implemented, respectively. The entirely original results are presented in the second part of the thesis. The numerical approximation of the overall system, combination of a non-linear reaction-diffusion PDE-ODE system with a stochastic boundary condition given by a specific Pearson process is investigated. Following recent theoretical results about the existence and uniqueness of a mild solution of this random system, a new numerical splitting scheme is proposed: an autonomous heat equation which inherits the stochastic boundary condition, and a non-linear and non local random system, coupled with the first, but with deterministic boundary conditions. We present an FTCS approximation and we provide the stability conditions for the discrete scheme. Furthermore, the order of spatial accuracy is numerically estimated to be one. Numerical quantitative and qualitative analyses of the impact of the randomness at the boundary are illustrated, both in the framework of single realizations of the process, i.e. in a pathwise sense, and also by considering distribution, first and second moments of large samples. We analyse both slow and accelerated regimes, by considering increasing values of the reaction rates. In the fast reaction case, a formation of a moving stochastic front can be observed. The final part of the thesis presents a theoretical convergence result. We consider a semi-discrete approximation of the system and prove the convergence of the discrete solution to the unique mild solution of the original random system, through a space-discrete approach. The same splitting strategy is adopted to derive some a priori estimates for the splitted discrete variables. Relevant results on compact embedding for time-space Besov spaces on the lattice as well some recent estimate results for the discrete heat kernel are exploited to prove a useful estimate for the solution to the discrete random heat equation. Furthermore, an $L^2$ a priori estimate for the second splitted variable together with its discrete derivative is provided. The final convergence result is obtained by a compactness argument, by first introducing piecewise constant interpolation and discretization operators on the lattice and by exploiting the uniform a priori estimate for the total system variable.