Analysis of Spectral Hamiltonian Boundary Value Methods (SHBVMs) for the numerical solution of ODE problems
Pierluigi Amodio, Luigi Brugnano, Felice Iavernaro
TL;DR
This work analyzes SHBVMs, i.e., Hamiltonian Boundary Value Methods used as spectral methods in time, for solving ODEs arising from Hamiltonian systems. It develops a spectral time discretization by projecting the vector field onto a Legendre subspace and establishes decay bounds for the Fourier coefficients $\gamma_j$ under analyticity, supporting spectral convergence. SHBVMs are then constructed by computing $\gamma_j(\sigma)$ with Gauss–Legendre quadrature (order $2k$, with $k=\max\{20,s+2\}$) and solving a reduced $s$-dimensional discrete system via a blended Newton iteration that leverages a small, precomputed matrix $\Sigma$, enabling high-order, spectrally accurate time stepping. Numerical tests on the Kepler problem, a Lotka–Volterra Poisson system, and a stiff ODE validate the theory, showing invariant conservation and favorable efficiency relative to standard ODE solvers, thus supporting SHBVMs as robust general-purpose ODE solvers for challenging Hamiltonian problems.
Abstract
Recently, the numerical solution of stiffly/highly-oscillatory Hamiltonian problems has been attacked by using Hamiltonian Boundary Value Methods (HBVMs) as spectral methods in time. While a theoretical analysis of this spectral approach has been only partially addressed, there is enough numerical evidence that it turns out to be very effective even when applied to a wider range of problems. Here we fill this gap by providing a thorough convergence analysis of the methods and confirm the theoretical results with the aid of a few numerical tests.