Linear Stability of Partitioned Runge-Kutta Methods

We study the linear stability of partitioned Runge–Kutta (PRK) methods applied to linear separable Hamiltonian ODEs and to the semidiscretization of certain Hamiltonian PDEs. We extend the] by presenting simplified expressions of the trace of the stability matrix, tr Ms, for the Lobatto IIIA–IIIB family of symplectic PRK methods. By making the connection to Padé approximants and continued fractions, we study the asymptotic behavior of tr Ms(ω) as a function of the frequency ω and stage number s. 1. Introduction. Partitioned Runge–Kutta (PRK) methods have a checkered history. They were first introduced in the 1970s for the integration of certain stiff differential equations. This area did not develop, partly because of a lack of naturally partitioned stiff systems. There was renewed interest in the 1990s with the advent of symplectic integration of Hamiltonian systems, with their natural partitioning into position (q) and momentum (p) variables. In 1993, Sanz-Serna and Abia [27] and Sun [28] found conditions on the parameters for the s-stage PRK method