On the advantages of nonstandard finite differences discretizations for differential problems

The goal of this work is to highlight the advantages of using NonStandard Finite Differences (NSFD) numerical schemes for the resolution of Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs) of which some properties of the exact solution are a-priori known, such as positivity. The main reference considered is Mickens' work [14], in which the author derives NSFD schemes for ODEs and PDEs that describe real phenomena, and therefore widely used in applications. We rigorously demonstrate that NSFD methods can have a higher order of convergence than the related classical ones, deriving also the conditions that guarantee the stability of the analyzed schemes. Furthermore, we carry out in-depth numerical tests comparing the classical methods with the NSFD ones proposed by Mickens, evaluating when the latter are decidedly advantageous.