Nonstandard finite difference methods: recent trends and further developments

Abstract

In this paper, we review many recent developments and further applications of nonstandard finite difference (NSFD) methods encountered in the past decade. In particular, it is a follow up article of the one published in 2005 [K.C. Patidar, On the use of non-standard finite difference methods, J. Differ. Equ. Appl. 11 (2005), pp. 735–758]. It also includes those research contributions in this field that are very significant and published prior to the above article but were not included in the above paper simply because we did not have access to them when we wrote the above article. We also give a detailed account on various definitions/notions of NSFD methods appeared in the literature in past two decades. All contributions are listed chronologically except that in some instances we have grouped certain works to show connectivity in those fields. While categorizing these research contributions, we considered a number of different application areas. Moreover, due to space limitations, firstly, we have not included all works that used NSFD methodology but certainly important contributions are given due consideration, and secondly, we have only included the salient features of the proposed numerical schemes for many of these contributions and ignored other contents where the involved approaches sound fairly standard thus implying why these methods are known robust. Of course, the cases when there is a specific variation, in the theoretical analysis of these NSFD schemes, are highlighted with more details.