Starting algorithms for Gauss Runge-Kutta methods for Hamiltonian systems

Among the symplectic integrators for the numerical solution of general Hamiltonian systems, implicit Runge-Kutta methods of Gauss type (RKG) play an important role. To improve the efficiency of the algorithms to be used in the solution of the nonlinear equations of stages, accurate starting values for the iterative process are required. In this paper, a class of starting algorithms, which are based on numerical information computed in two previous steps, is studied. For two- and three-stages RKG methods, explicit starting algorithms for the stage equations with orders three and four are derived. Finally, some numerical experiments comparing the behaviour of the new starting algorithms with the standard first iterant based on Lagrange interpolation of stages in the previous step are presented.