Analysis of spectral Hamiltonian boundary value methods (SHBVMs) for the numerical solution of ODE 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.

Line Integral Solution of Hamiltonian PDEs

“…Summing all up, overall SHBVMs will result to be extremely effective and competitive, as is testified by the numerical tests reported in Section 6 (see also [2,21,38]).…”

“…HBVMs are energy-conserving methods derived within the framework of (discrete) line integral methods, initially proposed in [54][55][56][57][58], and later refined in [22][23][24][29][30][31]. The approach has also been extended along several directions [10,14,18,24,25,27,28,39], including Hamiltonian BVPs [1], constrained Hamiltonian problems [15], highly-oscillatory problems [2,21,38], and Hamiltonian PDEs [3,13,16,17,21,40]. We also refer to the review paper [20] and to the monograph [19].…”

“…To conclude this quick introduction to HBVMs, we mention their use as spectral methods in time, which has been the subject of recent investigations [2,21,38]. We mention that the use of Runge-Kutta methods as spectral methods in time has been considered previously in [4,5,7,69] (see also [30]).…”

“…with tol ≈ ε. Moreover, the analysis in [2] shows that one could even use tol ≈ √ ε in (38), still obtaining full machine accuracy at t = h. At last (see (9)), by choosing k large enough, one may obtain full machine accuracy in the approximation of γ j (σ) by means ofγ j , j = 0, . .…”

“…so that (48) continues formally to hold, even though now u is no more the solution of (39). 2 Nevertheless, in the spirit of Fourier-Galerkin methods [8], by requiring the residual obtained by plugging u into (39) be orthogonal to the functional subspace 3…”