GniCodes — Matlab Programs for Geometric Numerical Integration

Abstract

Geometric numerical integration is synonymous with structure-preser-ving integration of ordinary diierential equations. These notes, prepared for the Durham summer school 2002, are complementary to the monograph of Hairer, Lubich and Wanner They give a n i n troduction 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 t o b e c o n s e r v ed by the discretization (Section 1). The o f Hamiltonian diierential equations is symplectic and possesses conserved quanti-ties. Conservative systems have a time-reversible Diierential equations with integrals and problems on manifolds are also considered. We t h e n i n troduce 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 brieey discuss their sym-plecticity and symmetry. The improved performance of such geometric integrators is best understood with the help of a backward error analysis (Section 3). We e x -plain 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 nal Section 5 gives a comparison of the diierent 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