Spectrally accurate space-time solution of Hamiltonian PDEs

Abstract

Recently, the numerical solution of multi-frequency, highly-oscillatory Hamiltonian problems has been attacked by using Hamiltonian Boundary Value Methods (HBVMs) as spectral methods in time. When the problem derives from the space semi-discretization of (possibly Hamiltonian) partial differential equations (PDEs), the resulting problem may be stiffly-oscillatory, rather than highly-oscillatory. In such a case, a different implementation of the methods is needed, in order to gain the maximum efficiency.Keywords Multi-frequency highly-oscillatory problems · Stiffly-oscillatory problems · Hamiltonian problems · Energy-conserving methods · Spectral methods · Legendre polynomials · Hamiltonian Boundary Value Methods Mathematics Subject Classification (2010) 65P10 · 65L05 · 65N35 IntroductionMulti-frequency highly-oscillatory problems have been recently attacked by using Hamiltonian Boundary Value Methods (HBVMs) as spectral methods in time [8]. The proposed approach has proven to be very efficient when solving a number of severe highly-oscillatory problems, allowing to effectively and accurately "resolve" all high-frequency components in the solution. Sometimes,

Spectrally accurate space–time solution of Manakov systems

“…As one may see, this is the most efficient one, allowing the use of very large time-step (h = 1, in the present case), and a very small execution time. 8 This further confirms what observed in [18].…”

“…This, in turn, allows considering relatively large time-steps. The use of HBVMs in this fashion has been considered in [29] for highly oscillatory problems, and in [18] for deriving a spectrally accurate space-time numerical solution of some Hamiltonian PDEs. A thorough convergence analysis of the methods, when used as spectral methods in time, has been made in [3].…”

“…Moreover, when using a double precision IEEE arithmetic, the value of k can be conveniently chosen by means of the following heuristics [18,29] k = max{20, s + 2}.…”

Line Integral Solution of Hamiltonian PDEs

“…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].…”

Spectrally accurate energy‐preserving methods for the numerical solution of the “good” Boussinesq equation

“…The previous blended implementation of HBVMs is particularly interesting, since it allows the use of relatively large values of s. This, in turn, allows to use HBVMs as spectral methods in time [44,45], so that one obtains, for the used finite precision arithmetic, the maximum possible accuracy compatible with the considered time-step. We here sketch the use of HBVMs as spectral methods (which we shall refer to as spectral HBVMs or, in short, SHBVMs): further details can be found in the previous references [44,45].…”