A technique to improve the error propagation when integrating relative equilibria
The aim of the paper is the study of a way to improve the propagation errors in the numerical integration of relative equilibria solution of Hamiltonian differential equations with symmetries. One knows that in this case and for stable equilibria, the error growth is typically quadratic for ``general'' schemes and linear for schemes that preserve the invariant quantities of the problem. The first part is an introduction concerning the existing studies and the analysed cases. The second part summarizes basic results about Hamiltonian relative equilibrium problems. The authors consider one parameter symmetry groups. They give also an example considering a Hamiltonian problem with four degrees of freedom. The third part presents the numerical problem and the study of the linearization around the relative equilibrium which is necessary to understand the error propagation. From this, the technique of modification is introduced, its implementation is discussed and the asymptotic behaviour of the new methods in the numerical integration of Hamiltonian relative equilibria is stated. The fourth part contains numerical examples, including small perturbations of the relative equilibrium, in order to illustrate the above technique and the gain in efficiency it implies.
scientific article; zbMATH DE number 2153699