Adapted Numerical Methods for Reaction-Diffusion Problems

Several areas of applied sciences require the use of reaction-diffusion Partial Differential Equations (PDEs) to model phenomena of interest, such as vegetation growth (Eigentler et al., 2019), material corrosion processes (Mai et al., 2016), solar cells production (Maldon et al., 2020). Such models are usually multiscale and characterized by large spatial domains and high stiffness. The knowledge of these characteristics, together with a-priori known properties of the model, e.g. positivity or presence of oscillations, is very useful when constructing a numerical method for the related solution. In this talk, we show numerical techniques for the construction of efficient and strongly problem-oriented methods, which are stable, i.e. able to handle stiffness preserving the main properties of the solution even for large discretization steps. In particular, we focus on the use of TASE (Time-Accurate and highly-Stable Explicit) operators (Calvo et al., 2021) to stabilize parallelizable explicit peer methods. Numerical experiments confirming the efficiency of the derived methods in solving some reaction-diffusion systems of PDEs from applications are carried out.