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In order to efficiently study the complex nonlinear beam-wave interaction in high-power klystrons, we have developed a 2.5-D code, named, KLY2D, based on a theoretical model combining particle-in-cell (PIC) method and finite-difference time-domain (FDTD) algorithm together. In the model, the particle charge and the beam current are properly assigned onto the grids based on the PIC method; the FDTD algorithm is introduced to solve Maxwell's equations so that the space charge of the electron beam can be accurately calculated, and the port-approximation method is employed to simulate high-frequency cavity fields; finally, the Lorentz motion equation is solved to further advance particle motion. This 2.5-D model is more accurate than the simple 1-D nonlinear code. For the cylindrical structure of a normal klystron, the port-approximation cavity model is accurate enough to model field-to-beam effect and saves much more computer resource than the full 3-D PIC model. This code is benchmarked on an S-band 50-MW high-peak-power klystron. The good consistency between the theoretical results and the experimental data indicates the reliability of the theoretical model and the simulation code, which is of importance for further promoting the design and the development of high-power klystrons.