Invariant difference schemes and their application to $SL(2,\mathbb{R})$ invariant ordinary differential equations

Abstract

We present an exposition of a method of discretizing ordinary differential equations while preserving their Lie point symmetries. This method is very general and can be applied to any ODE with a nontrivial symmetry group. The method is applied to obtain numerical slutions of second and third order ODEs invariant under two different realizations of $SL(2,\mathbb{R})$. The symmetry preserving method is shown to provide a better qualitative description of solutions than standard methods. In particular it provides solutions that are valid close to singularities and beyond them.

11 pages, 4 figures. Title, abstract, introduction and conclusion were rewritten for a better presentation and understanding of the article. Published: J. Phys. A: Math. Theor. 42 454016

References in corpus (5)

• Continuous Symmetries of Difference Equations
• Continuous symmetries of Lagrangians and exact solutions of discrete equations
• Difference schemes with point symmetries and their numerical tests
• Lie Symmetries and Exact Solutions of First Order Difference Schemes
• Symmetry preserving discretization of SL(2,R) invariant equations

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• Symmetry preserving discretization of ordinary differential equations. Large symmetry groups and higher order equations
• The Korteweg-de Vries equation and its symmetry-preserving discretization
• Conformally invariant elliptic Liouville equation and its symmetry preserving discretization