A Concise Introduction to Geometric Numerical Integration

Book Preface

Dierential equations play an important role in applied mathematics andare omnipresent in the sciences and in technical applications. They appear in many di erent elds such as chemical reaction kinetics, molecular dynamics, electronic circuits, population dynamics, control theory and astrodynamical problems, to name just a few. However, since the early days of the subject, it has become evident that very often nding closed solutions is either simply impossible or extremely dicult. Therefore, computing or approximating solutions of dierential equations, partial as well as ordinary, linear or nonlinear, constitutes a crucial ingredient in all mathematical sciences.

Very often in applications, the dierential equation modeling the physical phenomenon one aims to study possesses qualitative (geometric) properties that are absolutely essential to preserve under discretization. Hamiltonian systems constitute a clear example. These appear in many dierent contexts (classical, statistical and quantum mechanics, molecular dynamics, celestial mechanics, etc.) and have a number of features that are not shared by generic dierential equations. These specic traits may be traced back to the fact that Hamiltonian  ows dene symplectic transformations in the underlying phase space.

The numerical integration of Hamiltonian systems by a conventional method results in discrete dynamics that are not symplectic, since there is a priori no reason whatsoever as to why numerical schemes should respect this property. If the time interval is short and the integration scheme provides a reasonable accuracy, the resulting violation of the symplectic character may be tolerable in practice. However, in many applications one needs to consider large time intervals so that the computed solution is useless due to its lack of symplecticity. One has then to construct special purpose integrators that when applied to a Hamiltonian problem do preserve the symplectic structure at the discrete level. These are known as symplectic integration algorithms, and they not only outperform standard methods from a qualitative point of view, but also the numerical error accumulates more slowly. This, of course, becomes very important in long-time computations.

Starting from the case of symplectic integration, the search for numerical integration methods that preserve the geometric structure of the problem was generalized to other types of dierential equations possessing a special structure worth being preserved under discretization. Examples include volumepreserving systems, dierential equations dened in Lie groups and homogeneous manifolds, systems possessing symmetries or reversing symmetries, etc. Although diverse, all these dierential equations have one important common feature, namely, that they all preserve some underlying geometric structure that in uences the qualitative nature of the phenomena they produce. The design and analysis of numerical integrators preserving this structure constitute the realm of Geometric Numerical Integration. In short, in geometric integration one is not only concerned with the classical accuracy and stability of the numerical algorithm, but the method must also incorporate into its very formulation the geometric properties of the system. This gives the integratornot only an improved qualitative behavior, but also allows for a  signicantly more accurate long-time integration than with general-purpose methods. In the analysis of the methods a number of techniques from dierent areas of mathematics, pure and applied, come into play, including Lie groups and Lie algebras, formal series of operators, dierential and symplectic geometry, etc.

In addition to the construction of new numerical algorithms, an important aspect of geometric integration is the explanation of the relationship between preservation of the geometric properties of a numerical method and the observed favorable error propagation in long-time integration.