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

摘要

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