Geometric integration and its applications

This review aims to give an introduction to the relatively new area of numerical analysis called geometric integration. This is an overall title applied to a series of numerical methods that aim to preserve the qualitative (and geometrical) features of a differential equation when it is discretised. As the importance of different qualitative features is a rather subjective judgement, a discussion of the qualitative theory of differential equations in general is given first. The article then develops both the underlying theory of geometrical integration methods and then illustrates their effectiveness on a wide ranging set of examples, looking at both ordinary and partial differential equations. Methods are developed for symplectic ODEs and PDEs, differential equations with symmetry, differential equations with conservation laws and problems which develop singularities and sharp interfaces. It is shown that there are clear links between geometrically based methods and adaptive methods (where adaptivity is applied both in space and in time). A case study is given at the end of this article on the application of geometrical integration methods to problems arising in meteorology. From this we can see the advantages (and the disadvantages) of the geometric approach.