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Efficient and accurate structure exploiting numerical methods for solvingthe periodic Riccati differential equation (PRDE) are addressed. Such methods areessential, for example, to design periodic feedback controllers for periodic controlsystems. Three recently proposed methods for solving the PRDE are presented andevaluated on challenging periodic linear artificial systems with known solutions and applied to the stabilization of periodic motions of mechanical systems. The first twomethods are of the type multiple shooting and rely on computing the stable invariantsubspace of an associated Hamiltonian system. The stable subspace is determinedusing either algorithms for computing an ordered periodic real Schur form of a cyclicmatrix sequence, or a recently proposed method which implicitly constructs a stabledeflating subspace from an associated lifted pencil. The third method reformulatesthe PRDE as a convex optimization problem where the stabilizing solution is approximatedby its truncated Fourier series. As known, this reformulation leads to a semidefiniteprogramming problem with linear matrix inequality constraints admitting aneffective numerical realization. The numerical evaluation of the PRDE methods, withfocus on the number of states (n) and the length of the period (T ) of the periodicsystems considered, includes both quantitative and qualitative results.