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11 - Symmetry Preserving Discretization of Differential Equations and Lie Point Symmetries of Differential-Difference Equations

Published online by Cambridge University Press:  05 July 2011

By
Pavel Winternitz
Edited by
Decio Levi ,
Peter Olver ,
Zora Thomova  and
Pavel Winternitz
Pavel Winternitz
Affiliation:
Université de Montréal
Decio Levi
Affiliation:
Università degli Studi Roma Tre
Peter Olver
Affiliation:
University of Minnesota
Zora Thomova
Affiliation:
SUNY Institute of Technology
Pavel Winternitz
Affiliation:
Université de Montréal
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Summary

Abstract

In the first four sections of this chapter we consider an ordinary differential equation of any order invariant under some nontrivial group G of local point transformations. We show how such an ODE can be approximated by a difference scheme invariant under the same group G. Some advantages of such invariant schemes are pointed out. The schemes are exact for first-order equations. They can be solved analytically for some second-order equations. Used for numerical calculations the invariant schemes provide better qualitative descriptions of solutions than standard methods, specially close to singularities. The last two sections are devoted to methods of determining the Lie point symmetries of differential difference equations on fixed nontransforming lattices.

Introduction

Lie group theory started out as a theory of continuous transformations in the space of independent and dependent variables figuring in a system of differential equations. These point transformations were so constructed as to leave the space of solutions invariant, i.e., transform solutions into solutions. After Sophus Lie's seminal work in the end of the 19th and beginning of the 20th century. Lie theory developed in several directions, one being abstract group theory, another applications. In particular Lie group theory has evolved into a very general and powerful tool for obtaining exact (analytic) solutions of large classes of ordinary and partial differential equations. The symmetry theory of differential equations has been reviewed in modern books and review articles [5, 6, 25, 35, 36, 69, 82].

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Symmetries and Integrability of Difference Equations , pp. 292 - 341
Publisher: Cambridge University Press
Print publication year: 2011

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