We present a new class of integration methods for diierential equations on manifolds, in the framework of Lie group actions. Canonical coordinates of the second kind is used for representing the Lie group locally by means of its corresponding Lie algebra. The coordinate map itself can, in many cases, be computed inexpensively, but the approach also involves… (More)
RKMK methods and Crouch-Grossman methods are two classes of Lie group methods. The former is using flows and commutators of a Lie algebra of vector fields as a part of the method definition. The latter uses only compositions of flows of such vector fields, but the number of flows which needs to be computed is much higher than in the RKMK methods. We present… (More)
Continuous Explicit Runge-Kutta methods with the minimal number of stages are considered. These methods are continuously diierentiable if and only if one of the stages is the FSAL evaluation. A characterization of a subclass of these methods is developed for order 3,4 and 5. It is shown how the free parameters of these methods can be used either to minimize… (More)
There are several applications in which one needs to integrate a system of ODEs whose solution is an n × p matrix with orthonormal columns. In recent papers algorithms of arithmetic complexity order n× p 2 have been proposed. The class of Lie group integrators may seem like a worth while alternative for this class of problems, but it has not been clear how… (More)
We give a systematic method for discretizing Hamiltonian partial differential equations (PDEs) with constant symplectic structure, while preserving their energy exactly. The same method, applied to PDEs with constant dissipa-tive structure, also preserves the correct monotonic decrease of energy. The method is illustrated by many examples. In the… (More)
The Camassa–Holm equation is rich in geometric structures, it is completely integrable, bi-Hamiltonian, and it represents geodesics for a certain metric in the group of diffeomorphism. Here two new multi-symplectic formulations for the Camassa–Holm equation are presented, and the associated local conservation laws are shown to correspond to certain… (More)
In this article, we derive and study symmetric exponential integrators. Numerical experiments are performed for the cubic Schrödinger equation and comparisons with classical exponential integrators and other geometric methods are also given. Some of the proposed methods preserve the L 2-norm and/or the energy of the system.
We introduce a general format of numerical ODE-solvers which include many of the recently proposed exponential integrators. We derive a general order theory for these schemes in terms of B-series and bicoloured rooted trees. To ease the construction of specific schemes we generalise an idea of Zennaro and define Natural Continuous Extensions in the context… (More)