1st Edition

Applications of Lie Groups to Difference Equations

By Vladimir Dorodnitsyn Copyright 2010
    344 Pages 31 B/W Illustrations
    by Chapman & Hall

    344 Pages 31 B/W Illustrations
    by Chapman & Hall

    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.

    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
    Discrete Representation of Ordinary Differential Equations with Symmetries
    The discrete representation of ODE as a series
    Three-point exact schemes for nonlinear ODE


    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 1236

    In 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