The use of functional equations represents the most common strategy for modeling real phenomena. In particular, we are interested in the numerical solution of models of Partial Differential Equations (PDEs) coming from applications in real contexts, such as corrosion [3, 7], sustainability , vegetation . The numerical treatment of these problems is not trivial, since they are often characterized by high stiffness, which requires the use of very dense spatial and temporal discretizations. This obviously leads to unacceptable computing times. Furthermore, a numerical method is not always able to exploit a-priori known properties of the problem, such as any positivity or oscillating trend of the solution, the asymptotic stability, and so on. In this talk, we focus on a reaction-diffusion vegetation model that has been introduced to investigate the coexistence of two different plant species in arid environments, characterized by scarce presence of water . The considered system of PDEs is an extension of the well-known Klausmeier model. The latter investigates the growth of a single type of plant with varying water availability. The first, on the other hand, constitutes a generalization of the Klausmeier model, as it considers the competition of two different species of plants, which must somehow try to survive by sharing the same limiting resource. The model under investigation has high stiffness, and is characterized by positivity and oscillating behavior in space. We therefore show numerical techniques capable of dealing with the stiffness of the problem, and also of preserving the a-priori known properties of the exact solution for each choice of the spatial and temporal discretization steps . In particular, to preserve positivity, we extend non-standard finite differences using exponential integrators and TASE operators, which have been recently introduced to stabilize explicit Runge-Kutta methods . This also helps to deal with the stiffness of the problem. Furthermore, to preserve the spatial oscillations of the solution, we integrate the exponential fitting framework within the non-standard discretizations. To further improve the efficiency of the proposed approaches, we show the use of adapted parallel peer methods for the considered vegetation problem [2, 5]. Finally, numerical tests are shown to confirm the effectiveness of the proposed techniques. References  M. Bassenne, L. Fu, and A. Mani. Time-accurate and highly-stable explicit operators for stiff differential equations. , J. Comput. Phys., 424:Paper No. 109847, 24, 2021.  D. Conte, P. De Luca, A. Galletti, G. Giunta, L. Marcellino, G. Pagano and B. Paternoster. First Experiences on Parallelizing Peer Methods for Numerical Solution of a Vegetation Model., Lect. Notes Comput. Sci., 13376, 384–394, 2022.  D. Conte and G. Frasca-Caccia. A Matlab code for the computational solution of a phase field model for pitting corrosion., Dolomites Res. Notes Approx., 15:47–65, 2022.  D. Conte, G. Pagano, and B. Paternoster. Nonstandard finite differences numerical methods for a vegetation reaction-diffusion model., J. Comput. Appl. Math., 419:Paper No. 114790, 17, 2023.  D. Conte, G. Pagano, and B. Paternoster. Time-accurate and highly-stable explicit peer methods for stiff differential problems. , Commun. Nonlinear Sci. Numer. Simul., 119:Paper No. 107136, 20, 2023.  L. Eigentler and J. A. Sherratt. Metastability as a coexistence mechanism in a model for dryland vegetation patterns., Bull. Math. Biol., 81:2290–2322, 2019.  H. Gao, L. Ju, R. Duddu, and H. Li. An efficient second-order linear scheme for the phase field model of corrosive dissolution., J. Comput. Appl. Math., 367:112472, 16 pp., 2020.  B. Maldon and N. Thamwattana. Review of diffusion models for charge-carrier densities in dye-sensitized solar cells., J. Phys. Commun., 4:Paper No. 082001, 2020.
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