Sergio Blanes

PROCESSING SYMPLECTIC METHODS FOR NEAR-INTEGRABLE HAMILTONIAN SYSTEMS

Processing techniques are used to approximate the exact flow of near-integrable Hamiltonian systems depending on a small perturbation parameter. We study the reduction of the number of conditions for the kernel for this type of Hamiltonians and we build third, fourth and fifth order methods which are shown to be more efficient than previous algorithms for the same class of problems.

High-order Runge-Kutta-Nyström geometric methods with processing

Abstract We present new families of sixth- and eighth-order Runge–Kutta–Nystrom geometric integrators with processing for ordinary differential equations. Both the processor and the kernel are composed of explicitly computable flows associated with non trivial elements belonging to the Lie algebra involved in the problem. Their efficiency is found to be superior to other previously known algorithms of equivalent order, in some case up to four orders of magnitude.

Composition Methods for Differential Equations with Processing

We construct numerical integrators for differential equations up to order 12 obtained by composition of basic integrators. The following cases are considered: (i) composition for a system separable in two solvable parts, (ii) composition using as basic methods a first-order integrator and its adjoint, (iii) composition using second-order symmetric methods, and (iv) composition using fourth-order symmetric methods. Each scheme is implemented with a processor or corrector to improve their efficiency, and this can be done virtually cost-free.

Optimization of Lie group methods for differential equations

In this paper we present a technique for reducing to a minimum the number of commutators required in the practical implementation of Lie group methods for integrating numerically matrix differential equations. This technique is subsequently applied to the linear and nonlinear case for constructing new geometric integrators, optimal with respect to the number of commutators.