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

One main issue, when numerically integrating autonomous Hamiltonian systems, is the long-term conservation of some of its invariants; among them 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, we report the theoretical foundations which have led to the definition of the new family of methods, called Hamiltonian Boundary Value Methods (HBVMs). 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 and can have arbitrarily high order. A few numerical tests confirm the theoretical results.