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.

翻译：我们从数值角度研究了一个在半直线上具有非线性抛物型反应-扩散方程组的随机边值问题解的定性行为。该模型描述了碳酸钙石材在二氧化硫侵蚀下的化学腐蚀过程。其动态边界条件由皮尔逊扩散给出，这在文化遗产劣化研究背景下具有原创性。我们首先讨论了一种基于Lamperti变换的随机微分方程数值格式以保持边界特性，并结合基于最新理论结果的偏微分方程分裂策略。研究阐明了解的正性、有界性和稳定性。通过多组数值实验，探讨了边界噪声对解的影响及其在慢速与快速两种状态下的定性行为。