Lie Symmetries and Exact Solutions of First Order Difference Schemes

Abstract

We show that any first order ordinary differential equation with a known Lie point symmetry group can be discretized into a difference scheme with the same symmetry group. In general, the lattices are not regular ones, but must be adapted to the symmetries considered. The invariant difference schemes can be so chosen that their solutions coincide exactly with those of the original differential equation.

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References in corpus (4)

• Lie point symmetries of difference equations and lattices
• Continuous symmetries of Lagrangians and exact solutions of discrete equations
• Lie symmetries of multidimensional difference equations
• Lie Point Symmetries and Commuting Flows for Equations on Lattices

Cited by in corpus (10)

• Continuous Symmetries of Difference Equations
• Difference schemes with point symmetries and their numerical tests
• Discretization of partial differential equations preserving their physical symmetries
• Symmetry preserving discretization of SL(2,R) invariant equations
• Invariant difference schemes and their application to $SL(2,\mathbb{R})$ invariant ordinary differential equations
• Structure Preserving Discretizations of the Liouville Equation and their Numerical Tests
• The Korteweg-de Vries equation and its symmetry-preserving discretization
• Symmetry preserving discretization of ordinary differential equations. Large symmetry groups and higher order equations
• Invariant Discretization of Partial Differential Equations Admitting Infinite-Dimensional Symmetry Groups
• Conformally invariant elliptic Liouville equation and its symmetry preserving discretization