GniCodes
swMATH ID:  21086 
Software Authors:  Hairer, Ernst; Hairer, Martin 
Description:  GniCodes – Matlab programs for geometric numerical integration. Geometric numerical integration is synonymous with structurepreserving integration of ordinary differential equations. These notes, prepared for the Durham summer school 2002, are complementary to the monograph of E. Hairer, C. Lubich and G. Wanner [Geometric numerical integration. Structurepreserving algorithms for ordinary differential equations. (2002; Zbl 0994.65135)]. They give an introduction to the subject, and they discuss and explain the use of Matlab programs for experimenting with structurepreserving algorithms. We start by presenting some typical classes of problems having properties that are important and should be conserved by the discretization (Section 1). The flow of Hamiltonian differential equations is symplectic and possesses conserved quantities. Conservative systems have a timereversible 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) RungeKutta 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 longtime integration of Hamiltonian systems and of completely integrable problems. Section 4 is devoted to a presentation and explanation of Matlab codes for implicit RungeKutta, 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 Poincaré 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”. 
Homepage:  http://www.unige.ch/~hairer/software.html 
Dependencies:  Matlab 
Related Software:  Matlab; DASSL; DASPK 3.0; RODAS; RODES 
Cited in:  9 Publications 
Standard Articles
1 Publication describing the Software, including 1 Publication in zbMATH  Year 

GniCodes –Matlab programs for geometric numerical integration. Zbl 1028.65136 Hairer, Ernst; Hairer, Martin 
2003

all
top 5
Cited by 21 Authors
all
top 5
Cited in 6 Serials
3  Journal of Scientific Computing 
1  Journal of Computational Physics 
1  Nonlinearity 
1  Chaos, Solitons and Fractals 
1  Numerische Mathematik 
1  Universitext 
all
top 5