The waveguide is decomposed into several substructures in the longitudinal direction

and PML absorbing boundary is treated as one of them.The transverse section of the substructures is discretized with the finite element method

a precise integration method based on the Riccati equations is used for the longitudinal integration to generate the export stiffness matrices

and then the waveguide discontinuity problems are expected to be solved through the combination of the substructures.Numerical examples demonstrate that

the waveguide discontinuity problems can be effectively solved with this method

the disadvantage that the artificial boundary must be far away from the non-uniform region can be overcome

and the field near the non-uniform region can be worked out.