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

TL;DR

This paper introduces Hamiltonian Boundary Value Methods (HBVMs), a new class of energy-preserving Runge-Kutta methods that can exactly conserve polynomial Hamiltonians of any degree, ensuring long-term stability in numerical solutions.

Contribution

The paper presents HBVMs, a novel family of methods capable of exactly preserving high-degree polynomial Hamiltonians, advancing numerical integration of Hamiltonian systems.

Findings

HBVMs are symmetric and A-stable.

They can achieve arbitrarily high order accuracy.

Numerical tests confirm energy preservation and stability.

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.