Differential equations play an important role in modeling natural phenomena in sciences and engineering. In this thesis, numerical integrations based on Magnus expansion are investigated for linear differential equations. Often, differential equations possess qualitative properties that we are interested in being preserved under discretization of a numerical method. This subject has been known under geometric numerical integration methods recently. In this thesis, we consider differential equations in a Lie algebra. The concepts of Lie algebras and Lie groups are important tools to solve differential equations with symmetric methods. A Lie group is a differentiable manifold which is also a group and such the group product and the inverse map, are differentiable. Familiar examples of Lie groups are matrix Lie groups.
