Contributions to the Mathematics of the Nonstandard Finite Difference Method and Applications Roumen Anguelov, 1 Jean M.-S. Lubuma 2 1 Department of Mathematics and Statistics, Vista University, Private Bag X1311, Silverton, Pretoria 0127, South Africa 2 Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa Received 1 August 2000; accepted 1 August 2000 We formalize the transfer of essential properties of the solution of a differential equation to the solution of a discrete scheme as qualitative stability with respect to the properties. This permits us to motivate some rules (viz. on the order of the difference equation, on the renormalization of the denominator of the discrete derivative, and on nonlocal approximation of nonlinear terms) used in the design of nonstandard finite difference schemes. Extensions of some models are considered, and numerical examples confirming the efficiency of the nonstandard approach are provided. c  2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 518–543, 2001 Keywords: nonstandard finite differences; elementary stability; schemes preserving physical properties I. INTRODUCTION The finite difference method has been extensively used for the numerical treatment of ordinary and partial differential equations. (See, for example, [1-3] and more recently [4, 5]). Traditionally, two requirements of contemporary numerical analysis form an integral part of the study of finite difference schemes. These are: the consistency of the discrete scheme with the original differential equation and the stability, i.e., zero-stability in the terminology of [6], of the discrete method. That these requirements are important cannot be doubted, because they guarantee convergence with, in many cases, optimal rates of convergence of the discrete solution to the exact one. One shortcoming of these standard requirements is that essential qualitative properties of the exact solution are not transferred to the numerical solution. In practice, the limit 0 of the step- Correspondence to: Jean M-S Lubuma, University of Pretoria, Department of Mathematics and Applied Mathematics, Mathematics Building 2-18, Pretoria 0002, South Africa (e-mail: jlubuma@scientia.up.ac.za) c  2001 John Wiley & Sons, Inc.