The main purpose of this book is to provide a concise introduction to the methods and philosophy of constructing nonstandard finite difference schemes and illustrate how such techniques can be applied to several important problems. Chapter 1 gives an overview of the subject and summarizes previous work. Chapters 2 and 3 consider in detail the construction and numerical implementation of schemes for physical problems involving convection-diffusion-reaction equations that arise in groundwater pollution and scattering of electromagnetic waves using Maxwell's equations. Chapter 4 examines certain mathematical issues related to the nonstandard discretization of competitive and cooperative models for ecology. Chapter 5 discusses exactness, stability properties, and the symplecticity of various schemes including the conditions for which Runge-Kutta methods are exact. The application chapters illustrate well the power of nonstandard methods. In particular, for the same accuracy as obtained by standard techniques, larger step sizes can be used.This volume will satisfy the needs of scientists, engineers, and mathematicians who wish to know how to construct nonstandard schemes and see how these are applied to obtain numerical solutions of the differential equations which arise in the study of nonlinear dynamical systems modeling important physical phenomena.
Nonstandard finite difference schemes, R.E. Mickens; nonstandard methods for advection-diffusion-reaction equations, H.V. Kojouharov and B.M. Chen; application of nonstandard finite differences to solve the wave equation and Maxwell's equations, J.B. Cole; nonstandard discretization methods for some biological models, H. Al-Kahby et al; an introduction to numerical integrators preserving physical properties, M.J. Gander and R. Meyer-Spasche.