Geometric integration of nonlinear dynamical systems

In modern literature, the geometric integration means numerical integration of differential equations that provides an accurate preservation of one or more “geometric” properties within rounding error. Among these properties we have to mention first conservation of energy, of momentum, of angular momentum, of volume of the phase space, of time-reversal symmetry, of symplectic structure (volume conservation) etc. In this article we consider the concept of geometrical integration using Lie transformations generated by dynamical system on the one hand, and matrix representation for corresponding evolution operators on the other hand. Examples of solutions for some test problems and of practical problems are given.