Magnus expansions and beyond

In this brief review we describe the coming of age of Magnus expansions as an asymptotic and numerical tool in the investigation of linear differential equations in a Lie-group and homogeneous-space setting. Special attention is afforded to the many connections between modern theory of geometric numerical integration and other parts of mathematics: from abstract algebra to differential geometry and combinatorics, all the way to classical numerical analysis. 1 Lie-group equations Numerical solution of evolutionary differential equations is as old as the theory of differential equations itself: although proper numerical analysis of differential equations commenced with Leonhard Euler, earlier ad hoc numerical ideas abound in the works of Sir Isaac Newton and of Gottfried von Leibnitz. (A brief, yet outstanding historical synopsis can be found in (Hairer, Nørsett & Wanner 1986).) In the last fifty years numerical analysis of differential equations has developed in leaps and bounds, in parallel with the evolution in computing power and speed. On the face of it, all is well in the numerical kingdom. However, a closer look reveals a worrying gap between the efforts of numerical analysts and of pure mathematicians. Thus, pure mathematicians expand a very great deal of effort to analyse qualitative properties of differential equations but they usually fall short of fleshing out numbers. At the same time, numerical analysts are extraordinarily successful in producing numbers and figures with appropriately small errors but these numbers and figures typically fail to respect qualitative properties of differential equations. This disparity between analysis and computation motivated in the last decade the emergence of a new paradigm of geometric numerical integration (GNI): to seek computational methods that render exactly important qualitative features of differential equations. Examples of qualitative features whose preservation under discretization is important include the symplectic structure of Hamiltonian and Lie–Poisson systems (Hairer, Lubich & Wanner 2003, Hairer, Lubich & Wanner 2006, Leimkuhler & Reich 2004, Marsden & West 2001), volume conservation of divergence-free differential systems (McLachlan &