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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 period as input data. We also use the methods for solving optimal transfer problems in astrodynamics. [ABSTRACT FROM AUTHOR]

One of the main features when dealing with Hamiltonian problems is the conservation of the energy. In this paper we review, at an elemental level, the main facts concerning the family of low-rank Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs) for the efficient numerical integration of these problems. Using these methods one can obtain, an at least "practical", conservation of the Hamiltonian. We also discuss the efficient implementation of HBVMs by means of two different procedures: the blended implementation of the methods and an iterative procedure based on a particular triangular splitting of the corresponding Butcher's matrix. We analyze the computational cost of these two procedures that result to be an excellent alternative to a classical fixed-point iteration when the problem at hand is a stiff one. A few numerical tests confirm all the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper we define an efficient implementation for the family of low-rank energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs), recently defined in the last years. The proposed implementation relies on the particular structure of the Butcher matrix defining such methods, for which we can derive an efficient splitting procedure. The very same procedure turns out to be automatically suited for the efficient implementation of Gauss-Legendre collocation methods, since these methods are a special instance of HBVMs. The linear convergence analysis of the splitting procedure exhibits excellent properties, which are confirmed by a few numerical tests. [ABSTRACT FROM AUTHOR]

Abstract: Recently, the class of Hamiltonian Boundary Value Methods (HBVMs) has been introduced with the aim of preserving the energy associated with polynomial Hamiltonian systems (and, more in general, with all suitably regular Hamiltonian systems). However, many interesting problems admit other invariants besides the Hamiltonian function. It would be therefore useful to have methods able to preserve any number of independent invariants. This goal is achieved by generalizing the line-integral approach which HBVMs rely on, thus obtaining a number of generalizations which we collectively name Line Integral Methods. In fact, it turns out that this approach is quite general, so that it can be applied to any numerical method whose discrete solution can be suitably associated with a polynomial, such as a collocation method, as well as to any conservative problem. In particular, a completely conservative variant of both HBVMs and Gauss collocation methods is presented. Numerical experiments confirm the effectiveness of the proposed methods. [Copyright &y& Elsevier]

Abstract: We discuss the efficient implementation of Hamiltonian BVMs (HBVMs), a recently introduced class of energy preserving methods for canonical Hamiltonian systems (see Brugnano et al.  and references therein), also sketching their blended formulation. We also discuss the case of separable problems, for which the structure of the problem can be exploited to gain efficiency. [Copyright &y& Elsevier]

The paper presents fourth order Runge–Kutta methods derived from symmetric Hermite–Obreshkov schemes by suitably approximating the involved higher derivatives. In particular, starting from the multi-derivative extension of the midpoint method we have obtained a new symmetric implicit Runge–Kutta method of order four, for the numerical solution of first-order differential equations. The new method is symplectic and is suitable for the solution of both initial and boundary value Hamiltonian problems. Moreover, starting from the conjugate class of multi-derivative trapezoidal schemes, we have derived a new method that is conjugate to the new symplectic method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a second order energy-conserving approximation procedure for Hamiltonian systems with holonomic constraints. The derivation of the procedure relies on the use of the so-called line integral framework . We provide numerical experiments to illustrate theoretical findings. [ABSTRACT FROM AUTHOR]