Nonstandard finite differences numerical methods for a vegetation reaction–diffusion model

In this work we derive NonStandard Finite Differences (NSFDs) (Anguelov and Lubuma, 2001; Mickens, 2020) numerical schemes to solve a model consisting of reaction-diffusion Partial Differential Equations (PDEs) that describes the coexistence of plant species in arid environments (Eigentler and Sherratt, 2019). The new methods are constructed by exploiting a-priori known properties of the exact solution, such as positivity and oscillating behavior in space. Furthermore, we extend the definition of NSFDs inspired by the Time-Accurate and High-Stable Explicit (TASE) (Bassenne et al., 2021) methodology, also exploring the existing connections between nonstandard methods and the Exponential-Fitting (EF) (Ixaru, 1997; Ixaru and Berghe, 2010) technique. Several numerical tests are performed to highlight the best properties of the new NSFDs methods compared to the related standard ones. In fact, at the same cost, the former are much more stable than the latter, and unlike them preserve all the most important features of the model.