An extended framework of continuous-stage Runge-Kutta methods

TL;DR

This paper introduces an extended framework for continuous-stage Runge-Kutta methods that incorporates weighted orthogonal polynomials, enabling the construction of new integrators with specific geometric properties such as symplecticity.

Contribution

The paper develops a generalized framework allowing the use of various weighted orthogonal polynomials in continuous-stage Runge-Kutta methods, facilitating the design of integrators with desired geometric features.

Findings

Constructed new symplectic integrators using Legendre, Laguerre, and Hermite polynomials.

Extended the applicability of continuous-stage Runge-Kutta methods to more complex cases.

Demonstrated the potential for tailored geometric properties in numerical integrators.

Abstract

We propose an extended framework for continuous-stage Runge-Kutta methods which enables us to treat more complicated cases especially for the case weighting on infinite intervals. By doing this, various types of weighted orthogonal polynomials (e.g., Jacobi polynomials, Laguerre polynomials, Hermite polynomials etc.) can be used in the construction of Runge-Kutta-type methods. Particularly, families of Runge-Kutta-type methods with geometric properties can be constructed in this new framework. As examples, some new symplectic integrators by using Legendre polynomials, Laguerre polynomials and Hermite polynomials are constructed.