Invariant difference schemes for second order ordinary differential equations possessing symmetries

A review of earlier publications referenced in an introduction of the paper is presented. There were developed and investigated invariant difference schemes for second order ordinary differential equations possessing symmetries in mentioned publications. Besides, a new example of such a scheme was added. It was also shown that for each case of invariant schemes there exists an exact scheme, which general solution coincides with a general solution of an appropriate differential equation in the lattice nodes with arbitrary density. Therefore a specific mathematical dualism is indicated: for the same physical process there exist two identical continuous and discrete mathematical models. The first model is described by continuous curves, whereas the second one is determined by lattice nodes on the same continuous curves.