Nonstandard Finite Difference Models of Differential Equations

This book provides a summary of the work of the author on the construction of non-standard finite difference schemes for the numerical integration of differential equations. The major thrust of the book is to show that discrete models of differential equations exist such that the elementary types of numerical instabilities do not occur. A consequence of this result is that in general, bigger step-sizes can often be used in actual calculations and/or finite difference schemes can be constructed that are conditionally stable in many instances where using standard techniques no such schemes exist. The theoretical basis of this work is centered on the concepts of "exact" and "best" finite difference schemes. In addition, a set of rules are given for the discrete modelling of derivatives and nonlinear expressions that occur in differential equations. These rules often lead to a unique non-standard finite difference model for a given differential equation.