Nonstandard Finite Difference Schemes

This second edition of Nonstandard Finite Difference Models of Differential Equations provides an update on the progress made in both the theory and application of the NSFD methodology during the past two and a half decades. In addition to discussing details related to the determination of the denominator functions and the nonlocal discrete representations of functions of dependent variables, we include many examples illustrating just how this should be done.Of real value to the reader is the inclusion of a chapter listing many exact difference schemes, and a chapter giving NSFD schemes from the research literature. The book emphasizes the critical roles played by the 'principle of dynamic consistency' and the use of sub-equations for the construction of valid NSFD discretizations of differential equations.Contents: A Second Edition ... Why?IntroductionNumerical InstabilitiesNonstandard Finite Difference SchemesFirst-Order ODE'sSecond-Order, Nonlinear Oscillator EquationsTwo First-Order, Coupled Ordinary Differential EquationsPartial Differential EquationsSchrödinger Differential EquationsThe NSFD MethodologySome Exact Finite Difference SchemesApplications and Related TopicsAppendices:Difference EquationsLinear Stability AnalysisDiscrete WKB Method
Readership: This book is critical reading for anyone interested in the numerical integration of differential equations by use of finite differences. It is for both advanced undergraduate and graduate students, as well as researchers and practitioners in the field of numercial analysis. Selected sections of this book could provide supplementary materials for courses, both elementary and advanced, in difference and/or differential equations. Such courses exist in all of the natural and engineering sciences. NSFD Schemes;NSFD Methodology;Non-Local Discretization;Numerical Analysis of Differential Equations;Denominator Functions;Sub-equations for Differential Equations;Dynamic Consistency;Exact Finite Difference Schemes;Discrete Conservation Laws;Discrete Modelling;Exponential Numerical Integration;Geometric Numerical Integration;Discrete Hamiltonian Systems;Discrete Lagrangian Systems0Key Features:No other book currently exists on the most recent advances and applications of the NSFD methodologyThis book explains directly and clearly, key concepts which are essential for understanding and constructing valid NSFD discretizations of differential equationsSeveral value-added features include a chapter on explicit representations of exact finite difference schemes for a number of important differential equations and a chapter on applications to a broad range of phenomena in the natural sciencesAlso provided is an extensive listing of references to the original research literature