Line Integral Solution of Differential Problems

Abstract

Abstract:In recent years, the numerical solution of differential problems, possessing constants of motion, has been attacked by imposing the vanishing of a corresponding line integral. The resulting methods have been, therefore, collectively named (discrete) line integral methods, where it is taken into account that a suitable numerical quadrature is used. The methods, at first devised for the numerical solution of Hamiltonian problems, have been later generalized along several directions and, actually, the research is still very active. In this paper we collect the main facts about line integral methods, also sketching various research trends, and provide a comprehensive set of references.

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

“…providing, in the case of Hamiltonian problems, relevant conservation properties, as is specified by the following theorem [30,31,34]. Remark 1 From the last two points in Theorem 7, one has that, by choosing k large enough, either an exact energy-conservation can be gained, in the polynomial case, or a "practical" energy-conservation can be obtained in the general case.…”

“…In particular, we shall see that the system (4) has an Hamiltonian structure, which can be preserved by a suitable space semi-discretization. The time integration will be then performed by using energy-conserving methods in the HBVMs class [30][31][32][33][34][35][36][37], and this will allow us to retain many geometric properties of the solution, as later specified: as matter of fact, this paper follows a systematic study of the application of HBVMs for efficiently solving Hamiltonian partial differential equations (PDEs) [30,31,[38][39][40].…”

Line Integral Solution of Hamiltonian PDEs

“…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]. The basic idea line integral methods rely on is that the conservation of an invariant can be recast as the vanishing of a corresponding line-integral.…”

A note on the continuous-stage Runge–Kutta(–Nyström) formulation of Hamiltonian Boundary Value Methods (HBVMs)

“…which, according to [18, Definition 1] (see also [14]), is the Master Functional Equation defining a HBVM(∞, s) method.…”