Analysis of Hamiltonian Boundary Value Methods (HBVMs): A class of energy-preserving Runge–Kutta methods for the numerical solution of polynomial Hamiltonian systems

Abstract

One main issue, when numerically integrating autonomous Hamiltonian systems, is the long-term conservation of some of its invariants, among which the Hamiltonian function itself. For example, it is well known that classical symplectic methods can only exactly preserve, at most, quadratic Hamiltonians. In this paper, a new family of methods, called Hamiltonian Boundary Value Methods (HBVMs), is introduced and analyzed. HBVMs are able to exactly preserve, in the discrete solution, Hamiltonian functions of polynomial type of arbitrarily high degree. These methods turn out to be symmetric, precisely A-stable, and can have arbitrarily high order. A few numerical tests confirm the theoretical results.

High‐order energy‐preserving schemes for the improved Boussinesq equation

“…These proposed schemes have several attractive properties in contrast to the known studied methods. For instance, these schemes may have arbitrarily high order of accuracy in both the spatial and temporal variable, and the schemes are precisely A‐stable according to . Moreover, these schemes can conserve the discrete mass and energy to within machine precision.…”

Computational models and optimal control strategies for emotion contagion in the human population in emergencies

“…The optimality system is solved using an iterative method with a Runge-Kutta fourth order scheme [44]. The state system with an initial guess is solved forward in time, and then the adjoint system is solved backward in time.…”

Preface to the special issue NUMTA 2013

“…In this group of papers, the reader will find stochastic and deterministic methods developed for solving both global and local optimization problems as well as algorithms developed for dealing with multiobjective optimization. The third group of papers (see [2][3][4]) is dedicated to the numerical solution of differential problems. In particular, energy-conserving methods for Hamiltonian problems, multigrid methods for reaction-diffusion problems, and finite-difference methods for Sturm-Liouville problems.…”

“…It also considers numerical techniques and problems arising when natural phenomena are modelled on computers. The papers included in this special issue (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]) have been carefully selected by the guest editors among the submissions reflecting the talks presented at the international conference ''Numerical computations: Theory and Algorithms (NUMTA)'' held in June 17-23, 2013, Falerna (CZ), Italy. The NUMTA 2013 has been organized by the University of Calabria, Rende (CS), Italy and the N.I.…”