Applications Of Nonstandard Finite Difference Schemes

An introduction to the methods and philosophy of constructing nonstandard finite difference schemes. It illustrates how such techniques can be applied to several important problems. Chapter One gives an overview of the subject and summarizes previous work. Chapters Two and Three consider in detail the construction and numerical implementation of schemes for physical problems involving convection-diffusion-reaction equations that rise in groundwater pollution and scattering of electromagnetic waves using Maxwell's equations. Chapter Four examines certain mathematical issues related to the nonstandard discretization of competitive and co-operative models for ecology. Chapter Five discusses exactness, stability properties, and the symplecticity of various schemes including the conditions for which Runge-Kutta methods are exact. The application chapters illustrate the power of nonstandard methods. In particular, for the same accuracy as obtained by standard techniques, larger step sizes can be used.