Brief introduction to Lie-group methods

As mathematicians, we are excited by the profundity and {\ae}sthetic majesty of mathematical concepts. Yet, as numerical analysts, we cannot escape this elementary test: how much does a given construct, beautiful as it might be, help us to elucidate the nature of the computational phenomenon that we happen to investigate? Nothing can be taken for granted without justification and, by the same token, nothing that furthers our understanding can be discarded. Lie groups have played for more than a century a decisive role in our understanding of the geometry of differential equations. Their theory is extensive, deep and beautiful. It is the contention of this paper that the concept of Lie groups, within the wider terminology and machinery of differential geometry, is very helpful indeed in devising and analysing superior methods to discretise continuous dynamical systems. By sharing geometric structure and invariants with the original differential system, these methods are more accurate, more stable and occasionally cheaper for a significant range of differential problems.