Enhanced Numerical Solution Of Application-Oriented Reaction-Diffusion Equations

In this talk, we focus on the accurate and efficient numerical solution of mathematical models deriving from applications consisting of systems of reaction-diffusion Partial Differential Equations (PDEs). In particular, we will consider models of vegetation phenomena (Eigentler et al. 2019), problems of deterioration and corrosion of metallic materials (Mai et al. 2016, Waschinsky et al. 2021) and of architectural works, issues related to production of solar cells (Gagliardi et al. 2017, Maldon et al. 2020). The mentioned problems 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. Here, 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. 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. Such methods, called TASE-peer, have nice stability properties and involve the solution of a small and fixed number of linear systems per step depending on the Jacobian of the problem (or an approximation thereof). Numerical results testify that TASE methods are competitive in solving the reaction-diffusion models mentioned above.