Application of Lie Groups to Difference Equations: how to construct the invariant difference model

Symmetries are fundamental features of the differential equations of mathematical physics. They yield a number of useful properties such as symmetry reduction of PDEs, existence of various types of invariant solutions, integrability of ODEs, conservation laws for invariant variational problems, etc. Therefore, for preserving symmetries in discrete schemes, we retain qualitative properties of the underlying differential equations. We exhibit procedures that have been developed to preserve Lie group properties while constructing finite difference schemes and meshes. Examples for ODEs and PDEs will be considered.