Donato Trigiante

A simple framework for the derivation and analysis of effective one-step methods for ODEs☆

In this paper, we provide a simple framework to derive and analyse a class of one-step methods that may be conceived as a generalization of the class of Gauss methods. The framework consists in coupling two simple tools: firstly a local Fourier expansion of the continuous problem is truncated after a finite number of terms and secondly the coefficients of the expansion are computed by a suitable quadrature formula. Different choices of the basis lead to different classes of methods, even though we shall here consider only the case of an orthonormal polynomial basis, from which a large subclass of Runge–Kutta methods can be derived. The obtained results are then applied to prove, in a simplified way, the order and stability properties of Hamiltonian BVMs (HBVMs), a recently introduced class of energy preserving methods for canonical Hamiltonian systems (see [2] and references therein). A few numerical tests are also included, in order to confirm the effectiveness of the methods resulting from our analysis.

Convergence and stability of boundary value methods for ordinary differential equations

A usual way to approximate the solution of initial value problems for ordinary differential equations is the use of a linear multistep formula. If the formula has k steps, k values are needed to obtain the discrete solution. The continuous problem provides only the initial value. It is customary to impose the additional k - 1 conditions at the successive k - 1 initial points. However, the class of methods obtained in this way suffers from heavy limitations summarized by the two Dahlquist barriers. It is also possible to impose the additional conditions at different grid-points. For example, some conditions can be imposed at the initial points and the remaining ones at the final points. The obtained methods, called boundary value methods (BVMs), do not have barriers whatsoever. In this paper the question of convergence of BVMs is discussed, along with the linear stability theory. Some numerical examples on stiff test problems are also presented.

Analysis of Hamiltonian Boundary Value Methods (HBVMs): a class of energy-preserving Runge-Kutta methods for the numerical solution of polynomial Hamiltonian systems

Abstract One main issue, when numerically integrating autonomous Hamiltonian systems, is the long-term conservation of some of its invariants; among them the Hamiltonian function itself. For example, it is well known that classical symplectic methods can only exactly preserve, at most, quadratic Hamiltonians. In this paper, we report the theoretical foundations which have led to the definition of the new family of methods, called Hamiltonian Boundary Value Methods (HBVMs). HBVMs are able to exactly preserve, in the discrete solution, Hamiltonian functions of polynomial type of arbitrarily high degree. These methods turn out to be symmetric and can have arbitrarily high order. A few numerical tests confirm the theoretical results.

Energy- and Quadratic Invariants--Preserving Integrators Based upon Gauss Collocation Formulae

We introduce a new family of symplectic integrators depending on a real parameter α. 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(h) (symplectic) perturbation of the Gauss method. Under suitable assumptions, we show 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.We introduce a new family of symplectic integrators for canonical Hamiltonian systems. Each method in the family depends on a real parameter

A Hybrid Mesh Selection Strategy Based on Conditioning for Boundary Value ODE Problems

\alpha

The lack of continuity and the role of infinite and infinitesimal in numerical methods for ODEs: The case of symplecticity

. When

Hamiltonian BVMs (HBVMs): A Family of "Drift Free" Methods for Integrating polynomial Hamiltonian problems'

\alpha=0

Stability of some boundary value methods for the solution of initial value problems

we obtain the classical Gauss collocation formula of order

B-Spline Linear Multistep Methods and their Continuous Extensions

2s

Boundary value methods: The third way between linear multistep and Runge-Kutta methods

, where