On the Advantages of Nonstandard Finite Difference Discretizations for Differential Problems
Dajana Conte, Nicolina Guarino, Giovanni Pagano, Beatrice Paternoster
Abstract
The goal of this work is to highlight the advantages of using NonStandard Finite Difference (NSFD) numerical schemes for the solution of ordinary differential equations (ODEs) and Partial Differential Equations (PDEs) of which some properties of the exact solution, such as positivity, are a priori known. 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 conditions that guarantee the stability of the analyzed schemes. Furthermore, we carry out in-depth numerical tests comparing classical methods with the NSFD ones proposed by Mickens, evaluating when the latter are decidedly advantageous.