Discretization of partial differential equations preserving their physical symmetries

Abstract

A procedure for obtaining a "minimal" discretization of a partial differential equation, preserving all of its Lie point symmetries is presented. "Minimal" in this case means that the differential equation is replaced by a partial difference scheme involving N difference equations, where N is the number of independent and dependent variable. We restrict to one scalar function of two independent variables. As examples, invariant discretizations of the heat, Burgers and Korteweg-de Vries equations are presented. Some exact solutions of the discrete schemes are obtained.

29 pages, 3 figures

References in corpus (8)

• Lie point symmetries of difference equations and lattices
• Continuous symmetries of Lagrangians and exact solutions of discrete equations
• Lie symmetries of multidimensional difference equations
• Symmetry-preserving discrete schemes for some heat transfer equations
• Umbral Calculus, Difference Equations and the Discrete Schroedinger Equation
• A heat transfer with a source: the complete set of invariant difference schemes
• Lorentz and Galilei Invariance on Lattices
• Lie Symmetries and Exact Solutions of First Order Difference Schemes

Cited by in corpus (9)

• Continuous Symmetries of Difference Equations
• Invariant discretization schemes for the shallow-water equations
• Difference schemes with point symmetries and their numerical tests
• Invariant Discretization Schemes Using Evolution-Projection Techniques
• 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
• Convecting reference frames and invariant numerical models
• Multiscale expansions of difference equations in the small lattice spacing regime, and a vicinity and integrability test. I