1st Edition
Applications of Lie Groups to Difference Equations
Intended for researchers, numerical analysts, and graduate students in various fields of applied mathematics, physics, mechanics, and engineering sciences, Applications of Lie Groups to Difference Equations is the first book to provide a systematic construction of invariant difference schemes for nonlinear differential equations. A guide to methods and results in a new area of application of Lie groups to difference equations, difference meshes (lattices), and difference functionals, this book focuses on the preservation of complete symmetry of original differential equations in numerical schemes. This symmetry preservation results in symmetry reduction of the difference model along with that of the original partial differential equations and in order reduction for ordinary difference equations.
A substantial part of the book is concerned with conservation laws and first integrals for difference models. The variational approach and Noether type theorems for difference equations are presented in the framework of the Lagrangian and Hamiltonian formalism for difference equations.
In addition, the book develops difference mesh geometry based on a symmetry group, because different symmetries are shown to require different geometric mesh structures. The method of finite-difference invariants provides the mesh generating equation, any special case of which guarantees the mesh invariance. A number of examples of invariant meshes is presented. In particular, and with numerous applications in numerics for continuous media, that most evolution PDEs need to be approximated on moving meshes.
Based on the developed method of finite-difference invariants, the practical sections of the book present dozens of examples of invariant schemes and meshes for physics and mechanics. In particular, there are new examples of invariant schemes for second-order ODEs, for the linear and nonlinear heat equation with a source, and for well-known equations including Burgers equation, the KdV equation, and the Schrödinger equation.
Preface
Introduction
Brief introduction to Lie group analysis of differential equations
Preliminaries: Heuristic approach in examples
Finite Differences and Transformation Groups in Space of Discrete Variables
The Taylor group and finite-difference derivatives
Difference analog of the Leibniz rule
Invariant difference meshes
Transformations preserving the geometric meaning of finite-difference derivatives
Newton’s group and Lagrange’s formula
Commutation properties and factorization of group operators on uniform difference meshes
Finite-difference integration and prolongation of the mesh space to nonlocal variables
Change of variables in the mesh space
Invariance of Finite-Difference Models
An invariance criterion for finite-difference equations on the difference mesh
Symmetry preservation in difference modeling: Method of finite-difference invariants
Examples of construction of difference models preserving the symmetry of the original continuous models
Invariant Difference Models of Ordinary Differential Equations
First-order invariant difference equations and lattices
Invariant second-order difference equations and lattices
Invariant Difference Models of Partial Differential Equations
Symmetry preserving difference schemes for the nonlinear heat equation with a source
Symmetry preserving difference schemes for the linear heat equation
Invariant difference models for the Burgers equation
Invariant difference model of the heat equation with heat flux relaxation
Invariant difference model of the Korteweg–de Vries equation
Invariant difference model of the nonlinear Shrödinger equation
Combined Mathematical Models and Some Generalizations
Second-order ordinary delay differential equations
Partial delay differential equations
Symmetry of differential-difference equations
Lagrangian Formalism for Difference Equations
Discrete representation of Euler’s operator
Criterion for the invariance of difference functionals
Invariance of difference Euler equations
Variation of difference functional and quasi-extremal equations
Invariance of global extremal equations and properties of quasiextremal equations
Conservation laws for difference equations
Noether-type identities and difference analog of Noether’s theorem
Necessary and sufficient conditions for global extremal equations to be invariant
Applications of Lagrangian formalism to second-order difference equations
Moving mesh schemes for the nonlinear Shrödinger equation
Hamiltonian Formalism for Difference Equations: Symmetries and First Integrals
Discrete Legendre transform
Variational statement of the difference Hamiltonian equations
Symplecticity of difference Hamiltonian equations
Invariance of the Hamiltonian action
Difference Hamiltonian identity and Noether-type theorem for difference Hamiltonian equations
Invariance of difference Hamiltonian equations
Examples
Discrete Representation of Ordinary Differential Equations with Symmetries
The discrete representation of ODE as a series
Three-point exact schemes for nonlinear ODE
Bibliography
Index
Biography
Vladimir Dorodnitsyn
The book provides a systematic application of Lie groups to difference equations, difference meshes, and difference functionals. Besides the well-explained theoretical background and motivations, there is also a large number of concrete examples discussed in reasonable details. Due to the fairly broad introductory part, the book is indeed self-contained. The main ideas and concepts appear understandable not only to experts.
—Vojtech Zadnik, Zentralblatt MATH 1236In recent years "difference geometry" and its applications to integrable systems and mathematical physics have attracted significant attention and this monograph will contribute to the ongoing developments in this general area. It is clearly written and largely self-contained …
—Peter J. Vassiliou, Mathematical Reviews, 2012e
