Symplectic P-stable Additive Runge--Kutta Methods

TL;DR

This paper introduces a new family of symplectic additive Runge--Kutta methods that utilize different quadrature formulas for various terms in the Lagrangian, enhancing stability for oscillatory problems.

Contribution

It develops a novel class of symplectic methods combining Lobatto and Gauss--Legendre quadratures, which are P-stable and suitable for oscillatory systems.

Findings

Methods are symplectic and P-stable.

Compatible with highly oscillatory problems.

Maintain the implicitness of Lobatto IIIA-B methods.

Abstract

Symplectic partitioned Runge--Kutta methods can be obtained from a variational formulation where all the terms in the discrete Lagrangian are treated with the same quadrature formula. We construct a family of symplectic methods allowing the use of different quadrature formulas (primary and secondary) for different terms of the Lagrangian. In particular, we study a family of methods using Lobatto quadrature (with corresponding Lobatto IIIA-B symplectic pair) as a primary method and Gauss--Legendre quadrature as a secondary method. The methods have the same favourable implicitness as the underlying Lobatto IIIA-B pair, and, in addition, they are \emph{P-stable}, therefore suitable for application to highly oscillatory problems.