GniCodes - Matlab programs for geometric numerical integration

Geometric numerical integration is synonymous with structure-preserving integration of ordinary differential equations. These notes, prepared for the Durham summer school 2002, are complementary to the monograph of Hairer, Lubich and Wanner "Geometric Numerical Integration". They give an introduction to the subject, and they discuss and explain the use of Matlab programs for experimenting with structure-preserving algorithms. We start with presenting some typical classes of problems having properties that are important to be conserved by the discretization (Section 1). The flow of Hamiltonian differential equations is symplectic and possesses conserved quantities. Conservative systems have a time-reversible flow. Differential equations with first integrals and problems on manifolds are also considered. We then introduce in Section 2 simple symplectic and symmetric integrators, (partitioned) Runge-Kutta methods, composition and splitting methods, linear multistep methods, and algorithms for Hamiltonian problems on manifolds. We briefly discuss their symplecticity and symmetry. The improved performance of such geometric integrators is best understood with the help of a backward error analysis (Section 3). We explain some implications for the long-time integration of Hamiltonian systems and of completely integrable problems. Section 4 is devoted to a presentation and explanation of Matlab codes for implicit Runge-Kutta, composition, and multistep methods. The final Section 5 gives a comparison of the different methods and illustrates the use of these programs at some typical interesting situations: the computation of Poincar\'e sections, and the simulation of the motion of two bodies on a sphere. The Matlab codes as well as their Fortran 77 counterparts can be downloaded at http://www.unige.ch/math/folks/hairer under the item ``software''.