Symplectic P-stable additive Runge—Kutta methods

<p style='text-indent:20px;'>Classical symplectic partitioned Runge–Kutta methods can be obtained from a variational formulation where all the terms in discrete Lagrangian are treated with same quadrature formula. We construct family of allowing use different formulas (primary and secondary) for Lagrangian. In particular, we study using Lobatto (with corresponding IIIA-B pair) as primary method Gauss–Legendre secondary method. The have implicitness underlying pair, and, addition, they <i>P-stable</i>, therefore suitable application to highly oscillatory problems.</p>