Analisys of Hamiltonian Boundary Value Methods (HBVMs): a class of energy-preserving Runge-Kutta methods for the numerical solution of polynomial Hamiltonian systems
For researchers in numerical integration of Hamiltonian systems, this provides a novel method that overcomes the limitation of symplectic methods to quadratic Hamiltonians, enabling exact preservation of higher-degree polynomial Hamiltonians.
This paper introduces and analyzes Hamiltonian Boundary Value Methods (HBVMs), a new family of energy-preserving Runge-Kutta methods that exactly preserve polynomial Hamiltonians of arbitrarily high degree, while being symmetric, A-stable, and of arbitrarily high order.
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.