Efficient and robust numerical methods for solving the periodic Riccati differential equation (PRDE) are addressed. Such methods are essential, for example, when deriving feedback controllers for orbital stabilization of underactuated mechanical systems. Two recently proposed methods for solving the PRDE are presented and evaluated on artificial systems and on two stabilization problems originating from mechanical systems with unstable dynamics. The first method is of the type multiple shooting and relies on computing the stable invariant subspace of an associated Hamiltonian system. The stable subspace is determined using algorithms for computing a reordered periodic real Schur form of a cyclic matrix sequence, and a recently proposed method which implicitly constructs a stable subspace from an associated lifted pencil. The second method reformulates the PRDE as a maximization problem where the stabilizing solution is approximated with finite dimensional trigonometric base functions. By doing this reformulation the problem turns into a semidefinite programming problem with linear matrix inequality constraints.