Initializers for RK-Gauss methods based on pseudo-symplecticity

Symplectic Runge–Kutta (RK) methods for general Hamiltonian systems are implicit and an iterative scheme must be used to obtain the solution at each step. In this paper the classical order and the pseudo-symplecticity order [Pseudo-symplectic Runge–Kutta methods, BIT 38 (1998) 439–461] of the one step method that results after σ fixed point iterations for solving the implicit equations of stages in an implicit RK method are studied. In the numerical experiments with some RK-Gauss methods, σ is chosen so that the pseudo-symplecticity order is twice the classical order. Thus, the pseudo-symplectic method retains some important properties of the original symplectic one. Further, new starting algorithms are constructed taking into account their pseudo-symplecticity properties and are compared with other initializers existing in the literature.