Efficient implementation of Gauss collocation and Hamiltonian Boundary Value Methods

TL;DR

This paper presents an efficient implementation technique for Hamiltonian Boundary Value Methods (HBVMs) and Gauss-Legendre collocation methods, leveraging their structural properties to improve computational efficiency and convergence.

Contribution

The paper introduces a splitting procedure exploiting the structure of HBVMs, enabling efficient implementation of both HBVMs and Gauss-Legendre collocation methods.

Findings

Splitting procedure significantly improves computational efficiency.

Linear convergence of the splitting method is confirmed by numerical tests.

Gauss-Legendre collocation methods are effectively implemented as a special case of HBVMs.

Abstract

In this paper we define an efficient implementation for the family of low-rank energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs), recently defined in the last years. The proposed implementation relies on the particular structure of the Butcher matrix defining such methods, for which we can derive an efficient splitting procedure. The very same procedure turns out to be automatically suited for the efficient implementation of Gauss-Legendre collocation methods, since these methods are a special instance of HBVMs. The linear convergence analysis of the splitting procedure exhibits excellent properties, which are confirmed by a few numerical tests.