Keldysh Institute of Applied Mathematics,

4 Miusskaya Pl.,

125047 Moscow,

RUSSIA

E-mail: dorod@spp.keldysh.ru

**Difference equations: symmetries, exact solutions,
conservation laws**

**Abstract:**

A review of several applications of the Lie groups of point transformations
to difference equations, meshes (lattices),
difference functionals is presented. Some examples of difference
models (i.e. difference equations and appropriate meshes) which
admit the same symmetry group as there continuous limits. In particularly
there were constructed the whole set of difference models for second order
ODEs, which posses symmetry group. For intagrable cases the discrete representations
of ODEs ("an exact finite-difference scheme") were developed.
The invariant variational problems for difference equations are considered.
The Lagrangian formalism and the Noether's type constructions for
difference fuctionals, meshes and difference equations are illustrated
by several examples.