Nonstandard finite difference schemes

Nonstandard finite difference schemes are a generalization of the usual discrete models of differential equations. Their most important property is, in many cases, the complete absence of the elementary numerical instabilities which plague the usual finite difference schemes. While the general criterion for constructing such schemes are not known at the present time, several important rules have been discovered. The main purpose of this chapter is to introduce the concept of exact schemes, use them to formulate a number of nonstandard modeling rules, explain the significance of these rules, and illustrate their use in the construction of nonstandard finite difference schemes for a variety of model ordinary and partial differential equations. Several unresolved problems and related issues are also discussed, along with possible future directions for research in the area of nonstandard schemes.