• Energy Research

This book deals with the numerical solution of differential problems within the framework of Geometric Integration, a branch of numerical analysis which aims to devise numerical methods able to reproduce, in the discrete solution, relevant geometric properties of the continuous vector field. Among them, a paramount role is played by the so called constants of motion, which are physical quantities that are conserved along the solution trajectories of a large set of differential systems, named Conservative Problems. In particular, the major emphasis will be on Hamiltonian systems, though more general problems will be also considered.

Hamiltonian Boundary Value Methods are one step schemes of high order where the internal stages are partly exploited to impose the order conditions (fundamental stages) and partly to confer the formula the property of conserving the Hamiltonian function when this is a polynomial with a given degree v. The term “silent stages” has been coined for these latter set of extra‐stages to mean that their presence does not cause an increase of the dimension of the associated nonlinear system to be solved at each step. By considering a specific method in this class, we give some details about how the solution of the nonlinear system may be conveniently carried out and how to compensate the effect of roundoff errors.

We introduce new methods for the numerical solution of general Hamiltonian boundary value problems. The main feature of the new formulae is to produce numerical solutions along which the energy is precisely conserved, as is the case with the analytical solution. We apply the methods to locate periodic orbits in the circular restricted three body problem by using their energy value rather than their pe- riod as input data. We also use the methods for solving optimal transfer problems in astrodynamics. 22 pages, 4 figures