Problem oriented discretizations for reaction-diffusion models in sustainability

Sustainability represents a current trend in the field of interdisciplinary research. In mathematics, it is interesting to study the modelling of sustainability problems through functional equations, then solving them by means of appropriate numerical methods. In this talk, we focus on the accurate and efficient numerical solution of sustainability models consisting of systems of reaction-diffusion Partial Differential Equations (PDEs). They have been introduced, e.g., to model environmental problems, vegetation phenomena (Eigentler et al. 2019), problems of deterioration and corrosion of materials (Mai et al. 2016, Waschinsky et al. 2021), issues related to production of solar cells (Gagliardi et al. 2017, Maldon et al. 2020). Very often, reaction-diffusion sustainability models of PDEs are characterized by high stiffness and particular properties to be preserved in the discrete setting, forcing the use of specific non-trivial numerical techniques in order to compute the solution accurately and efficiently. In this talk, we show techniques for the construction of efficient and strongly problem-oriented numerical methods, which are stable, i.e. able to handle stiffness preserving the main properties of the solution (e.g. long term behavior, any positivity or oscillation frequency) even for large discretization steps [2, 3]. In particular, we show how the use of Time-Accurate and highly-Stable Explicit (TASE) operators (Calvo et al. 2021) can lead to a class of efficient and parallelizable numerical schemes [2]. Such methods, called TASE-peer, have an explicit structure, and involve only the solution of a small and fixed number of linear systems per step depending on the Jacobian of the problem (or an approximation thereof [4]). Numerical results testify that TASE-peer methods are competitive in solving reaction-diffusion sustainability models. Finally, we propose a conservative numerical method [1] for the solution of time fractional PDEs, which are used, e.g., to model some phenomena that are responsible of corrosion, such as the diffusion of chloride ions in reinforced concrete (Chen et al. 2020). REFERENCES [1] Cardone, A. and Frasca-Caccia, G. Numerical conservation laws of time fractional diffusion PDEs. Fract. Calc. Appl. Anal. (2022) 25(4): 1459–1483. [2] Conte, D., Pagano, G. and Paternoster, B. Time-accurate and highly-stable explicit peer methods for stiff differential problems. Commun. Nonlinear Sci. Numer. Simul. (2023) 119:107136. [3] Conte, D. and Frasca-Caccia, G. A MATLAB code for the computational solution of a phase field model for pitting corrosion. Dolomites Res. Notes Approx. (2022) 15(2):47–65. [4] Conte, D., Mart´ın-Vaquero, J., Pagano, G. and Paternoster, B. Stability of TASE Runge-Kutta methods with inexact Jacobian. In preparation.