Contributions to the mathematics of the nonstandard finite difference method and applications

We formalize the transfer of essential properties of the solution of a diff erential equation to the solution of a discrete scheme as qualitative stabi lity with respect to the properties. This permits us to motivate some rules (viz. on the order of the difference equation, on the renormalization of t he denominator of the discrete derivative, and on nonlocal approximation of nonlinear terms) used in the design of nonstandard finite difference schem es. Extensions of some models are considered, and numerical examples confir ming the efficiency of the nonstandard approach are provided. (C) 2001 John Wiley & Sons, Inc.