Perspectives on geometric numerical integration

ABSTRACTGeometric numerical integration concerns the development, analysis, and use of algorithms for the numerical solution of differential equations that preserve a geometric or qualitative property of the exact solutions, such as an integral or symmetry, or preservation of a differential invariant such as symplecticity or phase space volume. The leapfrog method and the midpoint rule exemplify two major directions of development in the subject. It is an instance of the more general problem of structure-preserving model reduction: which features of reality should be preserved when building a model? We review some recent developments in geometric numerical integration, including the structure of Butcher series, symplectic integrators on non-Euclidean phase spaces, and symplectic integrators for boundary value problems.