Adapted numerical treatment of stiff PDEs models from applications

Mathematical models expressed through Partial Differential Equations (PDEs) represent powerful a tool for the description and prediction of real phenomena in time and space. In the scientific literature, there are several numerical methods that have been constructed to solve these problems. However, PDEs describing a specific model may be endowed with a-priori known characteristics of fundamental importance, which are not always captured by the used numerical discretization, if this is not carefully chosen. In this talk, starting from a numerical approach recently introduced in the scientific literature for solving stiff ordinary differential equations (Calvo et al. J. Comput. Phys., 436, 110316, 2021), which is based on an appropriate preconditioning of the problem to be solved, we derive new efficient, stable and accurate methods, called TASE (Time Accurate and highly-Stable Explicit) methods. We will describe the derivation and consistency and stability properties of TASE methods, showing their efficiency and competitiveness with other well-known numerical schemes of the scientific literature for stiff initial value problems. Finally, we will consider three models of PDEs of interest related to the development of vegetation in arid environments (Eigentler et al. Bull. Math. Biol., 81, 2290–2322, 2019), to the corrosion of metallic materials (Mai et al. Corros. Sci., 110, 157–166, 2016), and to the movement of electrons in solar cell components (Maldon et al. Entropy, 22(2), 248, 2020). These models are characterized by a-priori known properties, such as positivity, oscillations in space, severe stiffness mainly induced by the diffusion term, which require the use of ad hoc temporal and spatial discretizations. With this in mind, we will adapt the TASE methods for the related numerical solution, showing that they are able to efficiently deal with stiffness and to preserve any positivity and oscillations.