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In recent years, the class of Line Integral Methods has been devised for efficiently solving conservative problems, namely, problems with invariants of motion. Their main instance is given by Hamiltonian Boundary Value Methods (HBVMs), a class of energy-conserving Runge-Kutta methods for Hamiltonian problems. It turns out that the efficient derivation of such methods relies on a local Fourier expansion of the vector field along a suitable orthonormal basis. However, this procedure can be also regarded as a spectral expansion, which can be adapted to a number of differential problems. The foundation of the theory and relevant paths of investigation underlying this class of methods have been developed during the past editions of the ICNAAM Conference series. [ABSTRACT FROM AUTHOR]

In this study, an adaptive optimized one-step block Nyström method for solving second-order initial value problems (IVPs) of ODEs is proposed. The new strategy is constructed through a collocation approach with a new technique for choosing the collocation points. The theoretical analysis of the new method is addressed. An embedding-like procedure is considered to estimate the error in order to implement an accurate variable step-size strategy. The numerical tests show that the developed error estimation and the step-size control strategy have good performance compared with other schemes available in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we study recent results in the numerical solution of Hamiltonian partial differential equations (PDEs), by means of energy-conserving methods in the class of Line Integral Methods, in particular, the Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). We show that the use of energy-conserving methods, able to conserve a discrete counterpart of the Hamiltonian functional (which derives from a proper space semi-discretization), confers more robustness to the numerical solution of such problems. [ABSTRACT FROM AUTHOR]

The numerical solution of Hamiltonian PDEs has been the subject of many investigations in the last years, specially concerning the use of multi-symplectic methods. We shall here be concerned with the use of energy-conserving methods in the HBVMs class, when a spectral space discretization is considered. [ABSTRACT FROM AUTHOR]

In this paper we define a class of modified line integral methods, which are a suitable modification of energy conserving methods in the HBVMs class, able to cope with conservative problems possessing multiple invariants. The analysis of the methods is sketched, along with some numerical tests. [ABSTRACT FROM AUTHOR]

In this paper we show that energy conserving methods, in particular those in the class of Hamiltonian Boundary Value Methods, can be conveniently used for the numerical solution of Hamiltonian Partial Differential Equations, after a suitable space semi-discretization. [ABSTRACT FROM AUTHOR]

We consider the issue of energy conservation in the numerical solution of Hamiltonian systems coupled with boundary conditions and discuss a few examples arising from astrodynamics. [ABSTRACT FROM AUTHOR]

The numerical solution of conservative problems, i.e., problems characterized by the presence of constants of motion, is of great interest in the computational practice. Such problems, indeed, occur in many real-life applications, ranging from the nano-scale of molecular dynamics to the macro-scale of celestial mechanics. Often, they are formulated as Hamiltonian problems. Concerning such problems, recently the energy conserving methods named Hamiltonian Boundary Value Methods (HBVMs) have been introduced. In this paper we review the main facts about HBVMs, as well as the existing connections with other approaches to the problem. A few new directions of investigation will be also outlined. In particular, we will place emphasis on the last contributions to the field of Prof. Donato Trigiante, passed away last year. [ABSTRACT FROM AUTHOR]

We introduce a new formulation of Gauss collocation methods for the numerical solution of ordinary differential equations. These formulae may be thought of as Runge-Kutta methods with rank-deficient array and may be specified in order to allow an easy parallel implementation. We show some preliminary results on Gauss methods of order 4, 6 and 8. [ABSTRACT FROM AUTHOR]

We show that many Runge-Kutta methods derived in the framework of Geometric Integration can be elegantly formalized by using special matrices defined by a suitable polynomial basis. [ABSTRACT FROM AUTHOR]

A new class of geometric integrators, able to preserve any number of independent invariants of a general conservative problem, is here sketched, by suitably generalizing the line-integral approach which has recently led to the definition of Hamiltonian BVMs (HBVMs), a class of energy-preserving methods for canonical Hamiltonian systems. Because of this reason, the new methods are collectively named Line Integral Methods. We here sketch the main results contained in [1]. [ABSTRACT FROM AUTHOR]

The article describes some numerical methods for Hamiltonian problems. It presents a few numerical tests comparing geometric integrators in various formulas and analyzes an error in the Hamiltonian. Using variable stepsize with a standard stepsize selection strategy, it demonstrates the numerical results.

Recently, a whole class of evergy-preserving integrators has been derived for the numerical solution of Hamiltonian problems [3, 2, 4]. In the mainstream of this research [6], we have defined a new family of symplectic integrators depending on a real parameter α [7]. For α = 0, the corresponding method in the family becomes the classical Gauss collocation formula of order 2s, where s denotes the number of the internal stages. For any given non-null α, the corresponding method remains symplectic and has order 2s-2: hence it may be interpreted as a O(h2s-2) (symplectic) perturbation of the Gauss method. Under suitable assumptions, it can be shown that the parameter α may be properly tuned, at each step of the integration procedure, so as to guarantee energy conservation in the numerical solution. The resulting method shares the same order 2s as the generating Gauss formula, and is able to preserve both energy and quadratic invariants. [ABSTRACT FROM AUTHOR]