A unifying framework for the derivation and analysis of effective classes of one-step methods for ODEs

TL;DR

This paper introduces a unified framework based on Fourier expansions for deriving and analyzing various effective one-step methods for solving ordinary differential equations, including energy-preserving Hamiltonian methods.

Contribution

It presents a simple, Fourier-based framework to derive and analyze classes of one-step methods, including a new simplified proof of properties for Hamiltonian BVMs.

Findings

Framework successfully derives various classes of methods

Proves order and stability properties of HBVMs

Numerical tests confirm effectiveness of the methods

Abstract

In this paper, we provide a simple framework to derive and analyse several classes of effective one-step methods. The framework consists in the discretization of a local Fourier expansion of the continuous problem. 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 is 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. A few numerical tests with such methods are also included, in order to confirm the effectiveness of the methods.