In this paper, two high order numerical methods, the CCI method and the decomposition method, are propose to simulate wave propagation in locally perturbed periodic closed waveguides. As is well known the problem is not always uniquely solvable due to the existence of guided modes, the limiting absorption principle is a standard way to get the unique physical solution. Both methods are based on the Floquet-Bloch transform which transforms the original problem to an equivalent family of cell problems. The CCI method is based on a modification of integral contours of the inverse transform, and the decomposition method comes from an explicit definition of the radiation condition. Due to the local perturbation, the family of cell problems are coupled thus the whole system is actually defined in 3D. Based on different types of singularities, high order methods are developed for faster convergence rates. Finally we show the convergence results by both theoretical explanations and numerical examples.