A note on the efficient implementation of Hamiltonian BVMs

Abstract

We discuss the efficient implementation of Hamiltonian BVMs (HBVMs), a recently introduced class of energy preserving methods for canonical Hamiltonian systems, via their blended formulation. We also discuss the case of separable problems, for which the structure of the problem can be exploited to gain efficiency.Comment: 10 pages, 4 figure

Energy-conserving methods for Hamiltonian boundary value problems and applications in astrodynamics

“…In the actual implementation, their distribution is chosen according to what explained in[7] (see also[8]), i.e., the corresponding s abscissae are approximately uniformly spaced in [0, 1].…”

Line Integral Solution of Differential Problems

“…We here sketch the main facts concerning the so called blended implementation of HBVMs, a Newton-like iteration alternative to (36), which only requires to factor a matrix having the same size as that of the continuous problem, thus resulting into a much more efficient implementation of the methods [1,16]. This technique derives from the definition of blended implicit methods, which have been at first considered in [41,42], and then developed in [43][44][45].…”

“…• in the special case of separable Hamiltonian problems, the blended implementation of the methods can be made even more efficient, since the discrete problem can be cast in terms of the generalized coordinates only (see [16] [51]). As matter of fact, it has been actually implemented in the Matlab code hbvm, which is freely available on the internet at the url [52].…”

“…Line integral methods were at first studied to derive energy-conserving methods for Hamiltonian problems: a coarse idea of the methods can be found in [5,6]; the first instances of such methods were then studied in [7][8][9]; later on, the approach has been refined in [10][11][12][13] and developed in [14][15][16][17][18]. Further generalizations, along several directions, have been considered in [19][20][21][22][23][24][25][26][27][28]: in particular, in [19] Hamiltonian boundary value problems have been considered, which are not covered in this review.…”

Recent Advances on Preserving Methods for General Conservative Systems

“…More recently, energy preserving Runge-Kutta methods of order two have been derived in [12], based on the concept of discrete line integral. This idea, further developed, has led to fourth order examples of conservative Runge-Kutta methods [13,14] and, finally, to Hamiltonian Boundary Value Methods (HBVMs) [3,2,4,5,6,7,8], a class of energy-preserving Runge-Kutta methods of any high order. Even though energy-conservation is an important feature for the discrete dynamical system induced by the methods, many Hamiltonian problems (and, in general, conservative problems) are characterized by the presence of multiple invariants.…”