Continuous-Stage Runge-Kutta approximation to Differential Problems

TL;DR

This paper explores continuous-stage Runge-Kutta methods, particularly Hamiltonian Boundary Value Methods, for efficiently solving differential problems while conserving energy, offering a new perspective on numerical integration techniques.

Contribution

It revisits and generalizes the interpretation of energy-conserving HBVMs as continuous-stage Runge-Kutta methods for broader differential problems.

Findings

Provides a unified framework for energy-conserving methods

Reveals connections between HBVMs and continuous-stage Runge-Kutta methods

Enhances understanding of numerical solutions for Hamiltonian and general differential problems

Abstract

In recent years, the efficient numerical solution of Hamiltonian problems has led to the definition of a class of energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Such methods admit an interesting interpretation in terms of continuous-stage Runge-Kutta methods, which is here recalled and revisited for general differential problems.