Construction of symplectic (partitioned) Runge-Kutta methods with continuous stage

TL;DR

This paper introduces a new, simplified approach to constructing symplectic (partitioned) Runge-Kutta methods with continuous stages, leveraging orthogonal polynomial expansions to preserve geometric structure in Hamiltonian systems.

Contribution

It presents a novel method for constructing symplectic (partitioned) Runge-Kutta methods using continuous stages and orthogonal polynomial expansions, simplifying previous approaches.

Findings

Provides a new construction framework for symplectic methods

Utilizes orthogonal polynomial expansions for method development

Simplifies the process of designing structure-preserving integrators

Abstract

Hamiltonian systems are one of the most important class of dynamical systems with a geometric structure called symplecticity and the numerical algorithms which can preserve such geometric structure are of interest. In this article we study the construction of symplectic (partitioned) Runge-Kutta methods with continuous stage, which provides a new and simple way to construct symplectic (partitioned) Runge-Kutta methods in classical sense. This line of construction of symplectic methods relies heavily on the expansion of orthogonal polynomials and the simplifying assumptions for (partitioned) Runge-Kutta type methods.