Hamiltonian boundary value methods (HBVMs) and their efficient implementation

One of the main features when dealing with Hamiltonian problems is the conservation of the energy. In this paper we review, at an elemental level, the main facts concerning the family of low-rank Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs) for the efficient numerical integration of these problems. Using these methods one can obtain, an at least “practical”, conservation of the Hamiltonian. We also discuss the efficient implementation of HBVMs by means of two different procedures: the blended implementation of the methods and an iterative procedure based on a particular triangular splitting of the corresponding Butcher’s matrix. We analyze the computational cost of these two procedures that result to be an excellent alternative to a classical fixed-point iteration when the problem at hand is a stiff one. A few numerical tests confirm all the theoretical findings.